Research ArticleConditional Stability for an Inverse Problem ofDetermining a Space-Dependent Source Coefficient in theAdvection-Dispersion Equation with RobinrsquosBoundary Condition

Shunqin Wang1 Chunlong Sun2 and Gongsheng Li2

1 College of Mathematics and Statistics Nanyang Normal University Nanyang Henan 473061 China2 Institute of Applied Mathematics Shandong University of Technology Zibo Shandong 255049 China

Correspondence should be addressed to Gongsheng Li ligssduteducn

Received 3 December 2013 Revised 27 February 2014 Accepted 27 February 2014 Published 24 April 2014

Academic Editor Sergei V Pereverzyev

Copyright copy 2014 Shunqin Wang et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper deals with an inverse problem of determining the space-dependent source coefficient in one-dimensional advection-dispersion equationwithRobinrsquos boundary conditionData compatibility for the inverse problem is analyzed bywhich an admissibleset for the unknown is set forth Furthermore with the help of an integral identity a conditional Lipschitz stability is establishedby suitably controlling the solution of an adjoint problem

1 Introduction

The process of solute transport and transformation in thesoils and groundwater is always involving some complicatedphysical andor chemical reactions By themass conservationlaw the process can often be described by mathematicalmodels of advection-dispersion andor reaction diffusionequations with sourcesink terms In many cases for thesolute transport model the dispersiondiffusion coefficientthe sourcesink term and other physical quantities are oftenunknown and cannot be measured easily So the methodof inverse problem and parameter identification has hadwide applications in the research of soil and groundwaterproblems

Denoting Ω119879

= (119909 119905) 0 lt 119909 lt 119897 0 lt 119905 lt 119879 the one-dimensional advection-dispersion equation usually utilized isgiven as

119877120597119888

120597119905minus

120597

120597119909(119863

120597119888

120597119909) + V

120597119888

120597119909+ 119891 (119909 119905 119888) = 0 (119909 119905) isin Ω

119879

(1)

where 119877 ge 1 is the retardation factor 119888 = 119888(119909 119905) isthe volume-averaged concentration at point 119909 and time

119905 119863 = 119863(119909) is the dispersion coefficient V gt 0 is theaverage flow velocity and 119891 = 119891(119909 119905 119888) represents a generalsourcesink term in the solute transportation

Based on (1) a well-posed forward problem can beobtained together with suitable initial boundary value condi-tions As for inverse problems of identifying the source termin the advection-dispersion equation there are quite a fewresearches on the uniqueness and existence but conditionalstability seems to be paid less attention to our knowledgeespecially for construction of Lipschitz stability The reasonmaybe comes from the methodology suitable for the stabilityanalysisThe variational integral identitymethod also knownas monotonicity method or the adjoint method see forexample [1ndash5] has been applied to parameter identificationproblems in the parabolic equations by which uniquenessresults can be obtained in a general way Nevertheless theintegral identity method can also be utilized to construct sta-bility for the inverse problems based on (1) In [6] conditionalstability has ever been discussed for the inverse problem ofdetermining the source coefficient 119902(119909) in (1) in the case of119891 = 119902(119909)119888 with Dirichlet boundary conditions Howeverthere is a conceptional fault in the proof of the stability in [6]when defining the norm of the unknown The norm could

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 598135 6 pageshttpdxdoiorg1011552014598135

2 Abstract and Applied Analysis

have no meanings when using an unsuitable bilinear formto give its definition Another difference of this paper withthe work of [6] lies in using different boundary conditionsWe will consider the corresponding inverse problem for(1) with Robinrsquos boundary condition at the left-hand sideof the domain and the homogeneous Neumann boundarycondition at the right-hand side of the domain

Therefore we have reason to consider the inverse problemof determining the source coefficient 119902 = 119902(119909) in (1) alsoin the case of 119891 = 119902(119909)119888 but with the boundary conditionof Robinrsquos type By data compatibility analysis and using theintegral identitymethod the solution of the forward problemcan be positive and monotone on the time variable and anadmissible set for the unknown source coefficient is givenMost important of all a new bilinear form is put forwardwith the aid of the solution of the adjoint problem and the 119871

2

norm for the unknown is well-defined by which a conditionalLipschitz stability for the inverse problem is established Therest of this paper is organized as follows

In Section 2 the forward problem and the inverse prob-lem of determining the space-dependent source coefficientwith final observations are introduced In Section 3 datacompatibility is analyzed based on an integral identity bywhich the solution of the forward problem can be monotoneand positive and an admissible set for the unknown is setforth In Section 4 an integral identity combining variationsof the known functions with the changes of the unknownis established with the aid of an adjoint problem and asuitable bilinear form and the corresponding norm for theunknown are defined by which Lipschitz stability for theinverse problem is constructed

2 The Forward Problem and the InverseProblem

Let us begin with the forward problem before discussingthe inverse problem In this paper for (1) suppose that theretardation factor 119877 = 1 and the dispersal coefficient 119863 is apositive constant and the source term has the form of 119891 =

119902(119909)119888 For the constant dispersion coefficient119863 it always hasthe representation of 119863 = 119886

119871V where 119886

119871gt 0 is the longitude

dispersion coefficient and V is also the average flow velocityThen the model we are to deal with is given as [7]

120597119888

120597119905minus 119886119871V1205972119888

1205971199092+ V

120597119888

120597119909+ 119902 (119909) 119888 = 0 (119909 119905) isin Ω

119879 (2)

The initial value condition is given as

119888 (119909 0) = 1198880 (119909) 0 le 119909 le 119897 (3)

The left boundary at 119909 = 0 is given with Robinrsquos condition

V119888 (0 119905) minus 119863119888119909 (0 119905) = 119892 (119905) 0 le 119905 le 119879 (4)

where the function 119892 = 119892(119905) represents the solute fluxthrough the left boundary because of the difference ofconcentration at the two sides of the boundary At the rightboundary 119909 = 119897 we assume it is an outflow boundary or theboundary is imperceivable that is we have the condition

119888119909 (119897 119905) = 0 0 le 119905 le 119879 (5)

By (2) together with the initial boundary value condi-tions (3)ndash(5) we get the so-called forward problem Further-more under suitable conditions for the initial boundary valuefunctions for example 119888

0(119909) isin 119862([0 119897]) 119892(119905) isin 119862([0 119879]) and

119902(119909) is boundedmeasurable we know from [8] that the aboveforward problem has unique solution 119888(119909 119905) isin 119862

21(Ω119879) On

the other hand if the source coefficient function 119902 = 119902(119909) in(2) is unknown we have to determine it by some additionalinformation of the solution The additional data we have arethe final observations at one final time 119905 = 119879 given as [9ndash12]

119888 (119909 119879) = 120579 (119909) 0 le 119909 le 119897 (6)

As a result an inverse problem of determining the sourcecoefficient 119902(119909) is formulated by (2)ndash(5) together with theadditional information (6) To the best of our knowledgeconditional stability for the above inverse source problem for(2) with Robinrsquos boundary condition has not been studied inthe known literature In what follows we will firstly give datacompatibility analysis for the inverse problem and then putforward an integral identity with which a conditional stabilityis constructed by suitably controlling an adjoint problem

3 Data Compatibility Analysis

Consider the inverse problem (2)ndash(6) Noting the realityof real problems the initial value function 119888

0(119909) the flux

function 119892(119905) the additional function 120579(119909) and the hydraulicparameters 119886

119871and V are all claimed to be nonnegative on

(119909 119905) isin Ω119879throughout the paper if there is no specification

In addition by sdot 2we denote the 119871

2-norm in the corre-sponding space

Suppose that 1198880(119909) 119892(119905) and 120579(119909) satisfy condition (A)

(A) 1198880(119909) isin 119862

2[0 119897] 119892(119905) isin 119862

1[0 119879] and 120579(119909) is piece-

wise continuous differential on 119909 isin [0 119897] and thereare positive constants 119872

0and 120576 such that 119888

02le 1198720

and 1205792ge 120576

Moreover the data should satisfy consistency condition(B)

(B) V1198880(0) minus 119863119888

1015840

0(0) = 119892(0) 119888

1015840

0(119897) = 120579

1015840(119897) = 0

Denote 119888 = 119888(119909 119905 119902) as the unique solution of the forwardproblem (2)ndash(5) for any prescribed source coefficient 119902 =

119902(119909) belonging to suitable spaces With a similar method asused in [4 6 13 14] we can prove the following theorem toreveal the monotonicity and positivity of the solution to theforward problem

Theorem 1 Suppose that the conditions (A) and (B) aresatisfied and 119888 = 119888(119909 119905 119902) is a priori bounded then thefollowing statements hold

(1) If the solute flux 119892(119905) is monotone decreasing on 119905 isin

[0 119879] and 119902 = 119902(119909) has the property (P1)

(P1) 119902(119909)1198880(119909) minus V1198881015840

0(119909) + 119863119888

10158401015840

0(119909) ge 0 0 le 119909 le 119897

Abstract and Applied Analysis 3

then for each given 119909 isin [0 119897] it follows that0 le 120579 (119909) = 119888 (119909 119879) le 119888 (119909 119905)

le 119888 (119909 0) = 1198880 (119909) 0 lt 119905 lt 119879

(7)

(2) If the solute flux 119892(119905) is monotone increasing on 119905 isin

[0 119879] and 119902 = 119902(119909) has the property (P2)

(P2) 119902(119909)1198880(119909) minus V1198881015840

0(119909) + 119863119888

10158401015840

0(119909) le 0 0 le 119909 le 119897

then for each given 119909 isin [0 119897] it follows that0 le 1198880 (119909) = 119888 (119909 0) le 119888 (119909 119905)

le 119888 (119909 119879) = 120579 (119909) 0 lt 119905 lt 119879

(8)

Proof We only prove the assertion (1) and (2) can be provedsimilarly For 119888 = 119888(119909 119905 119902) and any smooth test function120593(119909 119905) we have

intΩ119879

(119888119905minus 119863119888119909119909

+ V119888119909+ 119902 (119909) 119888) 120593119905119889119909 119889119905 = 0 (9)

Integration by parts leads to

intΩ119879

119888119905(120593119905+ 119863120593119909119909

+ V120593119909minus 119902 (119909) 120593) 119889119909 119889119905

= int

119879

0

[119863 (119888119909120593119905minus 119888120593119909119905) minus V119888120593

119905]119909=119897

119909=0119889119905

+ int

119897

0

[119863119888120593119909119909

+ V119888120593119909minus 119902 (119909) 119888120593]

119905=119879

119905=0119889119909

(10)

Denote ℎ = V119863 = 1119886119871 If there is

120593119909 (119897 119905) + ℎ120593 (119897 119905) = 0 (11)

then120593119909119905 (119897 119905) + ℎ120593

119905 (119897 119905) = 0 (12)and we get

119863120593119909119905 (119897 119905) + V120593

119905 (119897 119905) = 0 (13)Let 119866 = 119866(119909 119905) be any arbitrary nonnegative function

on (119909 119905) isin Ω119879and 120593 = 120593(119909 119905) solve the following adjoint

problem120593119905+ 119863120593119909119909

+ V120593119909minus 119902 (119909) 120593 = 119866 (119909 119905) (119909 119905) isin Ω

119879

120593119909 (0 119905) = 0 120593

119909 (119897 119905) + ℎ120593 (119897 119905) = 0 0 le 119905 le 119879

120593 (119909 119879) = 0 0 le 119909 le 119897

(14)

Then together with the initial boundary value conditionsgiven by (3) (4) and (5) the equality (10) is reduced to

intΩ119879

119888119905119866 (119909 119905) 119889119909 119889119905

= int

119897

0

119888 (119909 0) [minus119863120593119909119909 (119909 0) minus V120593

119909 (119909 0)

+119902 (119909) 120593 (119909 0)] 119889119909

+ int

119879

0

119892 (119905) 120593119905 (0 119905) 119889119905 = 1198681+ 1198682

(15)

where

1198681= int

119897

0

1198880 (119909)

times [119902 (119909) 120593 (119909 0) minus V120593119909 (119909 0) minus 119863120593

119909119909 (119909 0)] 119889119909

1198682= int

119879

0

119892 (119905) 120593119905 (0 119905) 119889119905

(16)

Firstly by using integration by parts for 1198681 we have

1198681= minus 119863[119888

0 (119909) 120593119909 (119909 0) minus 1198881015840

0(119909) 120593 (119909 0)]

119897

0

minus V[1198880(119909)120593(119909 0)]

119897

0

+ int

119897

0

[119902 (119909) 1198880 (119909) + V11988810158400(119909) minus 119863119888

10158401015840

0(119909)]

times 120593 (119909 0) 119889119909

(17)

Thanks to the consistency condition (B) and the boundaryvalue conditions of the adjoint problem (14) we can get

1198681= 119892 (0) 120593 (0 0)

+ int

119897

0

[119902 (119909) 1198880 (119909) + V11988810158400(119909) minus 119863119888

10158401015840

0(119909)] 120593 (119909 0) 119889119909

(18)

Also by integration by parts for 1198682and noting 120593(0 119879) = 0 we

have

1198682= minus119892 (0) 120593 (0 0) minus int

119879

0

1198921015840(119905) 120593 (0 119905) 119889119905 (19)

So we get

intΩ119879

119888119905119866 (119909 119905) 119889119909 119889119905

= int

119897

0

[119902 (119909) 1198880 (119909) + V11988810158400(119909) minus 119863119888

10158401015840

0(119909)] 120593 (119909 0) 119889119909

+ int

119879

0

[minus1198921015840(119905)] 120593 (0 119905) 119889119905

(20)

By applying the maximum principle of parabolic typeof PDE see for example [15] to the adjoint problem (14)we conclude that if 119866(119909 119905) is nonnegative but is otherwisearbitrary then 120593(119909 119905) is negative on Ω

119879 and so does for

120593(0 119905) and 120593(119909 0) Then together with the property (P1) weknow that the sign of the first term of the right-hand side of(20) is nonpositive In addition by virtue of the monotonedecreasing property of the boundary flux 119892(119905) the sign of thesecond term of the right-hand side of (20) is also nonpositiveHence the sign of expression (20) or (15) is nonpositive thatis we have

intΩ119879

119888119905119866 (119909 119905) 119889119909 119889119905 le 0 (21)

Since 119866 = 119866(119909 119905) is nonnegative but otherwise arbitraryit follows that if there is any positive measure subset of Ω

119879

where 119888119905(119909 119905) is positive then a contradiction of (21) can be

achieved by choosing the support 119866 = 119866(119909 119905) in this positivemeasure set This proves that for each 119909 isin [0 119897] there is119888119905(119909 119905) le 0 for 0 le 119905 le 119879 and the assertion (1) is valid The

proof is completed

4 Abstract and Applied Analysis

Remark 2 By setting V = 0 and 119891(119909 119905 119888) = 0 in (1) we getthe model ever discussed in [4] where Neumannrsquos boundaryconditions are employed and the initial value is zero Thanksto 1198880(119909) = 0 we have by equality (15)

intΩ119879

119888119905119866 (119909 119905) 119889119909 119889119905 = int

119879

0

119892 (119905) 120593119905 (0 119905) 119889119905 (22)

Since 120593(0 119905) le 120593(0 119879) for 0 lt 119905 lt 119879 there is 120593119905(0 119905) ge 0

Hence we deduce that assertion (1) is also valid if 119892(119905) lt 0

and assertion (2) is valid for 119892(119905) gt 0 However if consideringnonzero initial condition the property (P1)(P2) is neededand the assumption of monotonicity for 119892(119905) can be replacedby the properties of keeping its positivenegative sign for 119905 isin

(0 119879) and 119892(0) = 0 such that the assertions of this theoremcan be proved too

According to Theorem 1 we can get two sufficientconditions under which the inverse problem (2)ndash(6) is of datacompatibility That is the unknown source coefficient 119902(119909)

should have one of the following conditions with the knowndata functions 119888

0(119909) and 119892(119905)

(C1) 1198921015840(119905) le 0 for 119905 isin [0 119879] and 119902(119909) has property (P1)

(C2) 1198921015840(119905) ge 0 119905 isin [0 119879] and 119902(119909) has property (P2)

In summary if the conditions (A) (B) and (C1) aresatisfied the solution 119888 = 119888(119909 119905 119902) is monotone decreasingon 119905 isin [0 119879] for each 119909 isin [0 119897] If the conditions (A) (B) and(C2) are satisfied then the solution 119888 = 119888(119909 119905 119902) is monotoneincreasing on 119905 isin [0 119879] for each 119909 isin [0 119897]

In the following discussionswe assume that the condition(C1) is satisfied and the unknown source coefficient 119902 =

119902(119909) is continuous and has property (P1) Moreover let thefunction 119902 = 119902(119909) take nonnegative values for 119909 isin [0 119897]Thus an admissible set for the unknown source coefficientsis defined as follows

119878ad = 119902 119902 is continuous 119902 ge 0 for 119909 isin [0 119897]

and has property (P1) (23)

In the next section wewill establish a conditional stabilityfor the inverse problem of determining 119902 isin 119878ad with thehelp of an integral identity by suitably controlling an adjointproblem

4 Conditional Stability of the Inverse Problem

It is more difficult to prove stability than to prove existenceand uniqueness for an inverse problem and it always needsadditional conditions and suitable topology to construct astability which is called conditional stability for the inverseproblem

41 Construction of Integral Identity In this section a vari-ational integral identity with the aid of an adjoint problemis established which reflects a corresponding relation of theunknown source coefficientwith those of the initial boundaryvalues and the additional data

Theorem 3 Let 119888119894

= 119888(119909 119905 119902119894) (119894 = 1 2) be two solutions

corresponding to the initial boundary value functions 1198881198940(119909) and

119892119894(119905) (119894 = 1 2) and let 120579

119894(119909) (119894 = 1 2) be the corresponding

additional observations Then it follows that

intΩ119879

1198882(1199021minus 1199022) 120593 (119908) (119909 119905) 119889119909 119889119905

= int

119897

0

(1205792minus 1205791) 119908 (119909) 119889119909 + int

119897

0

(1198881

0minus 1198882

0) 120593 (119909 0) 119889119909

+ int

119879

0

(1198921minus 1198922) 120593 (0 119905) 119889119905

(24)

where 120593 = 120593(119908)(119909 119905) is the solution of a suitable adjointproblem with input data 119908 = 119908(119909) given in the proof of thistheorem

Proof Denoting 119862 = 1198881minus 1198882and noting that 119888

1and 1198882both

satisfy (2) and (3)ndash(6) we have

119862119905minus 119863119862

119909119909+ V119862119909+ 1199021119862 = (119902

2minus 1199021) 1198882 (25)

119862 (119909 0) = 1198881

0minus 1198882

0 (26)

V119862 (0 119905) minus 119863119862119909 (0 119905) = 119892

1minus 1198922 119862119909 (119897 119905) = 0 (27)

119862 (119909 119879) = 1205791minus 1205792 (28)

By smooth test function 120593 = 120593(119909 119905) and by multiplying twosides of (25) and integrating on Ω

119879 respectively we have

intΩ119879

[119862119905minus 119863119862

119909119909+ V119862119909+ 1199021119862] 120593 (119909 119905) 119889119909 119889119905

= intΩ119879

1198882[1199022minus 1199021] 120593 (119909 119905) 119889119909 119889119905

(29)

Integration by parts for the left-hand side of equality (29)and with a similar method as used in the proof of expression(15) we get the identity (24) as long as choosing 120593(119909 119905) as thesolution of the adjoint problem

120593119905+ 119863120593119909119909

+ V120593119909minus 1199021 (119909) 120593 = 0 (119909 119905) isin Ω

119879

120593119909 (0 119905) = 0 120593

119909 (119897 119905) + ℎ120593 (119897 119905) = 0 119905 isin [0 119879]

120593 (119909 119879) = 119908 (119909) 119909 isin [0 119897]

(30)

where ℎ = V119863 = 1119886119871and 119908 = 119908(119909) is called a controllable

input Since the adjoint problem is completely controlled bythe unique input 119908 = 119908(119909) we denote the solution of theadjoint problem (30) by120593 = 120593(119908)(119909 119905)The proof is over

Obviously by setting 120591 = 119879 minus 119905 and 119909 = 119909 for the adjointproblem (30) and also denoting 120591 as 119905 it is transformed to anormal initial boundary value problem given as follows

120593119905minus 119863120593119909119909

minus V120593119909+ 1199021 (119909) 120593 = 0 (119909 119905) isin Ω

119879

120593119909 (0 119905) = 0 120593

119909 (119897 119905) + ℎ120593 (119897 119905) = 0 119905 isin [0 119879]

120593 (119909 0) = 119908 (119909) 119909 isin [0 119897]

(31)

Abstract and Applied Analysis 5

In what follows we cope with problem (31) instead ofproblem (30) For this adjoint problem we can see belowthat its solution is really determined by the unique input119908 = 119908(119909)

42 Lipschitz Stability Firstly by applying variable separa-tion method and Sturm-Liouville eigenvalue theory [16] wecan get an explicit expression of the solution for the adjointproblem (31)

Lemma 4 Suppose that the source coefficient 119902(119909) isin 119878119886119889

andthe hydraulic parameters 119863 and V are positive then there is

120593 (119908) (119909 119905) =

infin

sum

119899=1

119886119899exp (minus120582

119899119863119905)119883

119899 (119909) (32)

where120582119899119883119899(119909) (119899 = 1 2 ) are eigenvalues and correspond-

ing eigenfunctions of the following Sturm-Liouville problemrespectively

11988310158401015840

+ ℎ1198831015840minus

1199021 (119909)

119863119883 + 120582119883 = 0

1198831015840(0) = 0 119883

1015840(119897) + ℎ119883 (119897) = 0

(33)

119886119899=

(119908119883119899)

(119883119899 119883119899)

=

int119897

0120588 (119909)119908 (119909)119883119899 (119909) 119889119909

10038171003817100381710038171198831198991003817100381710038171003817

2

2

(34)

where 120588(119909) = exp(ℎ119909) is the weighted function of theeigenvalue problem (33) and the eigenfunctions series 119883

119899(119909)

are orthogonal and complete on 1198712(0 119897) with the weighted

function 120588(119909)

Remark 5 By multiplying 120588(119909) = exp(ℎ119909) at the two sides ofthe differential equation in (33) we have

minus119889

119889119909(120588 (119909)119883

1015840) + 119876 (119909)119883 = 120582120588 (119909)119883 (35)

where 119876(119909) = 120588(119909)1199021(119909)119863 ge 0 for 119909 isin [0 119897] Combing

with the homogeneous boundary conditions 1198831015840(0) = 0

and 1198831015840(119897) + ℎ119883(119897) = 0 we deduce that the operator

minus(119889119889119909)[120588(119909)(119889119889119909)] + 119876(119909) is self-adjoint and the problem(33) is a regular Sturm-Liouville eigenvalue problem and sothe eigenfunctions119883

119899(119909) 119899 = 1 2 are orthogonal with the

weighted function 120588(119909) for 119909 isin [0 119897]

With the aid of Lemma 4 and identity (24) we define abilinear formB(119902 119908) 119871

2(0 1) times 119871

2(0 1) rarr 119877 by

B (119902 119908) = intΩ119879

1198882119902 (119909) 120593 (119908) (119909 119905) 119889119909 119889119905 (36)

where 120593(119908) is the solution of the adjoint problem (31) for119908 isin 119871

2(0 119897) and 119888

2= 119888(119909 119905 119902

2) is a definite solution of

the forward problem (2)ndash(5) for any given 1199022

isin 119878ad and 1198882

is nonnegative and bounded for (119909 119905) isin Ω119879which can be

regarded as a weighted function of the bilinear form It isnoticeable that B is defined with 119902(119909) and 119908(119909) which isdifferent from those definitions given in [6 17]

By using maximum-minimum principle in the adjointproblem (31) we know that the solution 120593 = 120593(119908)(119909 119905) takesnonnegative values as long as 119908 = 119908(119909) isin 119871

2(0 119897) taking

nonnegative values on 119909 isin [0 119897] In addition by generaltheory of parabolic equation [8] there exists positive constant1198721such that

100381710038171003817100381712059310038171003817100381710038171198712(Ω119879)

le 11987211199081198712(0119897) (37)

Then we get the following lemma

Lemma 6 The functional B defined by (36) is a boundedbilinear form

Proof In fact B is linear on 119902 from (36) and together withthe expressions (32) and (34) it follows that 120593(119908) is linearon 119908 and then B is linear on 119908 too Next using Holderinequality for the right-hand side of (36) we have

1003816100381610038161003816B (119902 119908)1003816100381610038161003816 le 119872

1(intΩ119879

[119902 (119909)]2119889119909 119889119905)

12

times (intΩ119879

[120593 (119908)]2119889119909 119889119905)

12

= 1198722radic119879

100381710038171003817100381711990210038171003817100381710038171198712(0119897)

100381710038171003817100381712059310038171003817100381710038171198712(Ω119879)

le 1198723

100381710038171003817100381711990210038171003817100381710038171198712(0119897)

1199081198712(0119897)

(38)

where 1198723

= 11987211198722radic119879 and 119872

2here is a positive constant

depending on the domain and the prescribed solution 1198882 then

B is bounded on 119902 and 119908 The proof is over

By definition (36) and Lemmas 4 and 6 we define anadmissible space for the inputs as 119882 = 119908 isin 119871

2(0 119897) 119908 ge

0 1199082le 119864 and a norm for the source coefficient via

100381710038171003817100381711990210038171003817100381710038172

= sup119908isin119882 119908 = 0

1003816100381610038161003816B (119902 119908)1003816100381610038161003816

1003817100381710038171003817119888210038171003817100381710038172

1199082

(39)

where 119864 gt 0 is a constant and 11988822

= (intΩ119879

[119888(119909 119905 1199022)]2

119889119909 119889119905)12

It is not difficult to testify that the above definition is well-defined by which we can construct a conditional stability forthe inversion of the source coefficient based on the integralidentity (24)

Theorem 7 Suppose that the assumptions (A) (B) and (C1)are satisfied and (119888

119894 119902119894) (119894 = 1 2) are two pairs of solutions to

the inverse problem (2)ndash(6) corresponding to the data 119888119894

0and

119892119894(119894 = 1 2) and 120579

119894(119894 = 1 2) are the additional observations

Then for 119902 isin 119878119886119889

and 119908 isin 119882 there exists constant 119872 =

119872(119863 V Ω119879 1205792) such that

10038171003817100381710038171199021 minus 1199022

10038171003817100381710038172

le 119872(10038171003817100381710038171205791 minus 120579

2

10038171003817100381710038172+

100381710038171003817100381710038171198881

0minus 1198882

0

100381710038171003817100381710038172+

10038171003817100381710038171198921 minus 1198922

10038171003817100381710038172)

(40)

6 Abstract and Applied Analysis

Proof Firstly by assertion (1) of Theorem 1 there is

[120579 (119909)]2le [119888 (119909 119905)]

2le [1198880 (119909)]2

0 lt 119909 lt 119897 0 lt 119905 lt 119879

(41)

for any solution 119888(119909 119905) of the forward problem (2)ndash(5) withthe final function 120579(119909) = 119888(119909 119879) and the initial function1198880(119909) = 119888(119909 0) Then noting condition (A) we have for the

solution 1198882= 119888(119909 119905 119902

2)

1

1198720radic119879

le1

1003817100381710038171003817119888010038171003817100381710038172

radic119879

le1

1003817100381710038171003817119888210038171003817100381710038172

le1

1205792radic119879

le1

120576radic119879

(42)

Next by definition (39) and utilizing the integral identity (24)and Cauchy-Schwartz inequality we get

10038171003817100381710038171199021 minus 1199022

10038171003817100381710038172le

1

1205792radic119879

sdot 10038171003817100381710038171205791 minus 120579

2

10038171003817100381710038172sup119908isin119882 119908 = 0

1199082

1199082

+100381710038171003817100381710038171198881

0minus 1198882

0

100381710038171003817100381710038172sup119908isin119882 119908 = 0

1003817100381710038171003817120593(119909 0)10038171003817100381710038172

1199082

+10038171003817100381710038171198921 minus 119892

2

10038171003817100381710038172sup119908isin119882 119908 = 0

1003817100381710038171003817120593 (0 119905)10038171003817100381710038172

1199082

(43)

Also by the general theory of 1D parabolic equation we knowthat there exists a constant119872 gt 0 depending on119863 V and thedomain Ω

119879such that

1003817100381710038171003817120593(119909 0)10038171003817100381710038172

1003817100381710038171003817120593(0 119905)

10038171003817100381710038172le 1198721199082 (44)

Therefore by setting

119872 =1

1205792radic119879

max 1119872 (45)

and noting (43) it follows that (40) is valid The proof is over

Remark 8 This theorem shows a conditional Lipschitz sta-bility for the inverse problem (2)ndash(6) Furthermore by thistheorem we can see that the uniqueness in the meaning of 1198712for the inverse problem can be easily deduced by the stabilityestimate (40)

Conflict of Interests

The authors declare that there is no conflict of interests withany commercial identities regarding the publication of thispaper

Acknowledgments

The authors thank the anonymous referees for valuablecommentsThiswork is supported by the Basic andAdvancedResearch Program of Henan Science Committee (Research

on Some Problems in the Category of Quantic Lattices)and the Research Award for Teachers in Nanyang NormalUniversity (no nynu200749) and is partially supported by theNational Natural Science Foundation of China (nos 11071148and 11371231)

References

[1] P DuChateau ldquoMonotonicity and invertibility of coefficient-to-datamappings for parabolic inverse problemsrdquo SIAM Journal onMathematical Analysis vol 26 no 6 pp 1473ndash1487 1995

[2] P DuChateau ldquoAn inverse problem for the hydraulic propertiesof porous mediardquo SIAM Journal on Mathematical Analysis vol28 no 3 pp 611ndash632 1997

[3] P DuChateau ldquoAn adjoint method for proving identifiability ofcoefficients in parabolic equationsrdquo Journal of Inverse and Ill-Posed Problems vol 21 no 5 pp 639ndash663 2013

[4] A Hasanov P DuChateau and B Pektas ldquoAn adjoint problemapproach and coarse-fine mesh method for identification of thediffusion coefficient in a linear parabolic equationrdquo Journal ofInverse and Ill-Posed Problems vol 14 no 5 pp 435ndash463 2006

[5] Y H Ou A Hasanov and Z H Liu ldquoInverse coefficientproblems for nonlinear parabolic differential equationsrdquo ActaMathematica Sinica vol 24 no 10 pp 1617ndash1624 2008

[6] G Li D Yao and F Yang ldquoAn inverse problem of identifyingsource coefficient in solute transportationrdquo Journal of Inverseand Ill-Posed Problems vol 16 no 1 pp 51ndash63 2008

[7] N-Z Sun Mathematical Modeling of Groundwater PollutionSpringer New York NY USA 1996

[8] J R Cannon The One-Dimensional Heat Equation vol 23Addison-Wesley London UK 1984

[9] I Bushuyev ldquoGlobal uniqueness for inverse parabolic problemswith final observationrdquo Inverse Problems vol 11 no 4 pp L11ndashL16 1995

[10] M Choulli and M Yamamoto ldquoConditional stability in deter-mining a heat sourcerdquo Journal of Inverse and Ill-Posed Problemsvol 12 no 3 pp 233ndash243 2004

[11] V Isakov ldquoSome inverse problems for the diffusion equationrdquoInverse Problems vol 15 no 1 pp 3ndash10 1999

[12] E G Savateev and R Riganti ldquoInverse problem for thenonlinear heat equation with the final overdeterminationrdquoMathematical and Computer Modelling vol 22 no 1 pp 29ndash431995

[13] J R Cannon and P DuChateau ldquoStructural identification of anunknown source term in a heat equationrdquo Inverse Problems vol14 no 3 pp 535ndash551 1998

[14] G Li ldquoData compatibility and conditional stability for aninverse source problem in the heat equationrdquo Applied Mathe-matics and Computation vol 173 no 1 pp 566ndash581 2006

[15] R P SperbMaximumPrinciples andTheir Applications vol 157Academic Press New York NY USA 1981

[16] R Courant and D Hilbert Methods of Mathematical Physics IInterscience New York NY USA 1953

[17] L Gongsheng and M Yamamoto ldquoStability analysis for deter-mining a source term in a 1-D advection-dispersion equationrdquoJournal of Inverse and Ill-Posed Problems vol 14 no 2 pp 147ndash155 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

have no meanings when using an unsuitable bilinear formto give its definition Another difference of this paper withthe work of [6] lies in using different boundary conditionsWe will consider the corresponding inverse problem for(1) with Robinrsquos boundary condition at the left-hand sideof the domain and the homogeneous Neumann boundarycondition at the right-hand side of the domain

Therefore we have reason to consider the inverse problemof determining the source coefficient 119902 = 119902(119909) in (1) alsoin the case of 119891 = 119902(119909)119888 but with the boundary conditionof Robinrsquos type By data compatibility analysis and using theintegral identitymethod the solution of the forward problemcan be positive and monotone on the time variable and anadmissible set for the unknown source coefficient is givenMost important of all a new bilinear form is put forwardwith the aid of the solution of the adjoint problem and the 119871

2

norm for the unknown is well-defined by which a conditionalLipschitz stability for the inverse problem is established Therest of this paper is organized as follows

In Section 2 the forward problem and the inverse prob-lem of determining the space-dependent source coefficientwith final observations are introduced In Section 3 datacompatibility is analyzed based on an integral identity bywhich the solution of the forward problem can be monotoneand positive and an admissible set for the unknown is setforth In Section 4 an integral identity combining variationsof the known functions with the changes of the unknownis established with the aid of an adjoint problem and asuitable bilinear form and the corresponding norm for theunknown are defined by which Lipschitz stability for theinverse problem is constructed

2 The Forward Problem and the InverseProblem

Let us begin with the forward problem before discussingthe inverse problem In this paper for (1) suppose that theretardation factor 119877 = 1 and the dispersal coefficient 119863 is apositive constant and the source term has the form of 119891 =

119902(119909)119888 For the constant dispersion coefficient119863 it always hasthe representation of 119863 = 119886

119871V where 119886

119871gt 0 is the longitude

dispersion coefficient and V is also the average flow velocityThen the model we are to deal with is given as [7]

120597119888

120597119905minus 119886119871V1205972119888

1205971199092+ V

120597119888

120597119909+ 119902 (119909) 119888 = 0 (119909 119905) isin Ω

119879 (2)

The initial value condition is given as

119888 (119909 0) = 1198880 (119909) 0 le 119909 le 119897 (3)

The left boundary at 119909 = 0 is given with Robinrsquos condition

V119888 (0 119905) minus 119863119888119909 (0 119905) = 119892 (119905) 0 le 119905 le 119879 (4)

where the function 119892 = 119892(119905) represents the solute fluxthrough the left boundary because of the difference ofconcentration at the two sides of the boundary At the rightboundary 119909 = 119897 we assume it is an outflow boundary or theboundary is imperceivable that is we have the condition

119888119909 (119897 119905) = 0 0 le 119905 le 119879 (5)

By (2) together with the initial boundary value condi-tions (3)ndash(5) we get the so-called forward problem Further-more under suitable conditions for the initial boundary valuefunctions for example 119888

0(119909) isin 119862([0 119897]) 119892(119905) isin 119862([0 119879]) and

119902(119909) is boundedmeasurable we know from [8] that the aboveforward problem has unique solution 119888(119909 119905) isin 119862

21(Ω119879) On

the other hand if the source coefficient function 119902 = 119902(119909) in(2) is unknown we have to determine it by some additionalinformation of the solution The additional data we have arethe final observations at one final time 119905 = 119879 given as [9ndash12]

119888 (119909 119879) = 120579 (119909) 0 le 119909 le 119897 (6)

As a result an inverse problem of determining the sourcecoefficient 119902(119909) is formulated by (2)ndash(5) together with theadditional information (6) To the best of our knowledgeconditional stability for the above inverse source problem for(2) with Robinrsquos boundary condition has not been studied inthe known literature In what follows we will firstly give datacompatibility analysis for the inverse problem and then putforward an integral identity with which a conditional stabilityis constructed by suitably controlling an adjoint problem

3 Data Compatibility Analysis

Consider the inverse problem (2)ndash(6) Noting the realityof real problems the initial value function 119888

0(119909) the flux

function 119892(119905) the additional function 120579(119909) and the hydraulicparameters 119886

119871and V are all claimed to be nonnegative on

(119909 119905) isin Ω119879throughout the paper if there is no specification

In addition by sdot 2we denote the 119871

2-norm in the corre-sponding space

Suppose that 1198880(119909) 119892(119905) and 120579(119909) satisfy condition (A)

(A) 1198880(119909) isin 119862

2[0 119897] 119892(119905) isin 119862

1[0 119879] and 120579(119909) is piece-

wise continuous differential on 119909 isin [0 119897] and thereare positive constants 119872

0and 120576 such that 119888

02le 1198720

and 1205792ge 120576

Moreover the data should satisfy consistency condition(B)

(B) V1198880(0) minus 119863119888

1015840

0(0) = 119892(0) 119888

1015840

0(119897) = 120579

1015840(119897) = 0

Denote 119888 = 119888(119909 119905 119902) as the unique solution of the forwardproblem (2)ndash(5) for any prescribed source coefficient 119902 =

119902(119909) belonging to suitable spaces With a similar method asused in [4 6 13 14] we can prove the following theorem toreveal the monotonicity and positivity of the solution to theforward problem

Theorem 1 Suppose that the conditions (A) and (B) aresatisfied and 119888 = 119888(119909 119905 119902) is a priori bounded then thefollowing statements hold

(1) If the solute flux 119892(119905) is monotone decreasing on 119905 isin

[0 119879] and 119902 = 119902(119909) has the property (P1)

(P1) 119902(119909)1198880(119909) minus V1198881015840

0(119909) + 119863119888

10158401015840

0(119909) ge 0 0 le 119909 le 119897

Abstract and Applied Analysis 3

then for each given 119909 isin [0 119897] it follows that0 le 120579 (119909) = 119888 (119909 119879) le 119888 (119909 119905)

le 119888 (119909 0) = 1198880 (119909) 0 lt 119905 lt 119879

(7)

(2) If the solute flux 119892(119905) is monotone increasing on 119905 isin

[0 119879] and 119902 = 119902(119909) has the property (P2)

(P2) 119902(119909)1198880(119909) minus V1198881015840

0(119909) + 119863119888

10158401015840

0(119909) le 0 0 le 119909 le 119897

then for each given 119909 isin [0 119897] it follows that0 le 1198880 (119909) = 119888 (119909 0) le 119888 (119909 119905)

le 119888 (119909 119879) = 120579 (119909) 0 lt 119905 lt 119879

(8)

Proof We only prove the assertion (1) and (2) can be provedsimilarly For 119888 = 119888(119909 119905 119902) and any smooth test function120593(119909 119905) we have

intΩ119879

(119888119905minus 119863119888119909119909

+ V119888119909+ 119902 (119909) 119888) 120593119905119889119909 119889119905 = 0 (9)

Integration by parts leads to

intΩ119879

119888119905(120593119905+ 119863120593119909119909

+ V120593119909minus 119902 (119909) 120593) 119889119909 119889119905

= int

119879

0

[119863 (119888119909120593119905minus 119888120593119909119905) minus V119888120593

119905]119909=119897

119909=0119889119905

+ int

119897

0

[119863119888120593119909119909

+ V119888120593119909minus 119902 (119909) 119888120593]

119905=119879

119905=0119889119909

(10)

Denote ℎ = V119863 = 1119886119871 If there is

120593119909 (119897 119905) + ℎ120593 (119897 119905) = 0 (11)

then120593119909119905 (119897 119905) + ℎ120593

119905 (119897 119905) = 0 (12)and we get

119863120593119909119905 (119897 119905) + V120593

119905 (119897 119905) = 0 (13)Let 119866 = 119866(119909 119905) be any arbitrary nonnegative function

on (119909 119905) isin Ω119879and 120593 = 120593(119909 119905) solve the following adjoint

problem120593119905+ 119863120593119909119909

+ V120593119909minus 119902 (119909) 120593 = 119866 (119909 119905) (119909 119905) isin Ω

119879

120593119909 (0 119905) = 0 120593

119909 (119897 119905) + ℎ120593 (119897 119905) = 0 0 le 119905 le 119879

120593 (119909 119879) = 0 0 le 119909 le 119897

(14)

Then together with the initial boundary value conditionsgiven by (3) (4) and (5) the equality (10) is reduced to

intΩ119879

119888119905119866 (119909 119905) 119889119909 119889119905

= int

119897

0

119888 (119909 0) [minus119863120593119909119909 (119909 0) minus V120593

119909 (119909 0)

+119902 (119909) 120593 (119909 0)] 119889119909

+ int

119879

0

119892 (119905) 120593119905 (0 119905) 119889119905 = 1198681+ 1198682

(15)

where

1198681= int

119897

0

1198880 (119909)

times [119902 (119909) 120593 (119909 0) minus V120593119909 (119909 0) minus 119863120593

119909119909 (119909 0)] 119889119909

1198682= int

119879

0

119892 (119905) 120593119905 (0 119905) 119889119905

(16)

Firstly by using integration by parts for 1198681 we have

1198681= minus 119863[119888

0 (119909) 120593119909 (119909 0) minus 1198881015840

0(119909) 120593 (119909 0)]

119897

0

minus V[1198880(119909)120593(119909 0)]

119897

0

+ int

119897

0

[119902 (119909) 1198880 (119909) + V11988810158400(119909) minus 119863119888

10158401015840

0(119909)]

times 120593 (119909 0) 119889119909

(17)

Thanks to the consistency condition (B) and the boundaryvalue conditions of the adjoint problem (14) we can get

1198681= 119892 (0) 120593 (0 0)

+ int

119897

0

[119902 (119909) 1198880 (119909) + V11988810158400(119909) minus 119863119888

10158401015840

0(119909)] 120593 (119909 0) 119889119909

(18)

Also by integration by parts for 1198682and noting 120593(0 119879) = 0 we

have

1198682= minus119892 (0) 120593 (0 0) minus int

119879

0

1198921015840(119905) 120593 (0 119905) 119889119905 (19)

So we get

intΩ119879

119888119905119866 (119909 119905) 119889119909 119889119905

= int

119897

0

[119902 (119909) 1198880 (119909) + V11988810158400(119909) minus 119863119888

10158401015840

0(119909)] 120593 (119909 0) 119889119909

+ int

119879

0

[minus1198921015840(119905)] 120593 (0 119905) 119889119905

(20)

By applying the maximum principle of parabolic typeof PDE see for example [15] to the adjoint problem (14)we conclude that if 119866(119909 119905) is nonnegative but is otherwisearbitrary then 120593(119909 119905) is negative on Ω

119879 and so does for

120593(0 119905) and 120593(119909 0) Then together with the property (P1) weknow that the sign of the first term of the right-hand side of(20) is nonpositive In addition by virtue of the monotonedecreasing property of the boundary flux 119892(119905) the sign of thesecond term of the right-hand side of (20) is also nonpositiveHence the sign of expression (20) or (15) is nonpositive thatis we have

intΩ119879

119888119905119866 (119909 119905) 119889119909 119889119905 le 0 (21)

Since 119866 = 119866(119909 119905) is nonnegative but otherwise arbitraryit follows that if there is any positive measure subset of Ω

119879

where 119888119905(119909 119905) is positive then a contradiction of (21) can be

achieved by choosing the support 119866 = 119866(119909 119905) in this positivemeasure set This proves that for each 119909 isin [0 119897] there is119888119905(119909 119905) le 0 for 0 le 119905 le 119879 and the assertion (1) is valid The

proof is completed

4 Abstract and Applied Analysis

Remark 2 By setting V = 0 and 119891(119909 119905 119888) = 0 in (1) we getthe model ever discussed in [4] where Neumannrsquos boundaryconditions are employed and the initial value is zero Thanksto 1198880(119909) = 0 we have by equality (15)

intΩ119879

119888119905119866 (119909 119905) 119889119909 119889119905 = int

119879

0

119892 (119905) 120593119905 (0 119905) 119889119905 (22)

Since 120593(0 119905) le 120593(0 119879) for 0 lt 119905 lt 119879 there is 120593119905(0 119905) ge 0

Hence we deduce that assertion (1) is also valid if 119892(119905) lt 0

and assertion (2) is valid for 119892(119905) gt 0 However if consideringnonzero initial condition the property (P1)(P2) is neededand the assumption of monotonicity for 119892(119905) can be replacedby the properties of keeping its positivenegative sign for 119905 isin

(0 119879) and 119892(0) = 0 such that the assertions of this theoremcan be proved too

According to Theorem 1 we can get two sufficientconditions under which the inverse problem (2)ndash(6) is of datacompatibility That is the unknown source coefficient 119902(119909)

should have one of the following conditions with the knowndata functions 119888

0(119909) and 119892(119905)

(C1) 1198921015840(119905) le 0 for 119905 isin [0 119879] and 119902(119909) has property (P1)

(C2) 1198921015840(119905) ge 0 119905 isin [0 119879] and 119902(119909) has property (P2)

In summary if the conditions (A) (B) and (C1) aresatisfied the solution 119888 = 119888(119909 119905 119902) is monotone decreasingon 119905 isin [0 119879] for each 119909 isin [0 119897] If the conditions (A) (B) and(C2) are satisfied then the solution 119888 = 119888(119909 119905 119902) is monotoneincreasing on 119905 isin [0 119879] for each 119909 isin [0 119897]

In the following discussionswe assume that the condition(C1) is satisfied and the unknown source coefficient 119902 =

119902(119909) is continuous and has property (P1) Moreover let thefunction 119902 = 119902(119909) take nonnegative values for 119909 isin [0 119897]Thus an admissible set for the unknown source coefficientsis defined as follows

119878ad = 119902 119902 is continuous 119902 ge 0 for 119909 isin [0 119897]

and has property (P1) (23)

In the next section wewill establish a conditional stabilityfor the inverse problem of determining 119902 isin 119878ad with thehelp of an integral identity by suitably controlling an adjointproblem

4 Conditional Stability of the Inverse Problem

It is more difficult to prove stability than to prove existenceand uniqueness for an inverse problem and it always needsadditional conditions and suitable topology to construct astability which is called conditional stability for the inverseproblem

41 Construction of Integral Identity In this section a vari-ational integral identity with the aid of an adjoint problemis established which reflects a corresponding relation of theunknown source coefficientwith those of the initial boundaryvalues and the additional data

Theorem 3 Let 119888119894

= 119888(119909 119905 119902119894) (119894 = 1 2) be two solutions

corresponding to the initial boundary value functions 1198881198940(119909) and

119892119894(119905) (119894 = 1 2) and let 120579

119894(119909) (119894 = 1 2) be the corresponding

additional observations Then it follows that

intΩ119879

1198882(1199021minus 1199022) 120593 (119908) (119909 119905) 119889119909 119889119905

= int

119897

0

(1205792minus 1205791) 119908 (119909) 119889119909 + int

119897

0

(1198881

0minus 1198882

0) 120593 (119909 0) 119889119909

+ int

119879

0

(1198921minus 1198922) 120593 (0 119905) 119889119905

(24)

where 120593 = 120593(119908)(119909 119905) is the solution of a suitable adjointproblem with input data 119908 = 119908(119909) given in the proof of thistheorem

Proof Denoting 119862 = 1198881minus 1198882and noting that 119888

1and 1198882both

satisfy (2) and (3)ndash(6) we have

119862119905minus 119863119862

119909119909+ V119862119909+ 1199021119862 = (119902

2minus 1199021) 1198882 (25)

119862 (119909 0) = 1198881

0minus 1198882

0 (26)

V119862 (0 119905) minus 119863119862119909 (0 119905) = 119892

1minus 1198922 119862119909 (119897 119905) = 0 (27)

119862 (119909 119879) = 1205791minus 1205792 (28)

By smooth test function 120593 = 120593(119909 119905) and by multiplying twosides of (25) and integrating on Ω

119879 respectively we have

intΩ119879

[119862119905minus 119863119862

119909119909+ V119862119909+ 1199021119862] 120593 (119909 119905) 119889119909 119889119905

= intΩ119879

1198882[1199022minus 1199021] 120593 (119909 119905) 119889119909 119889119905

(29)

Integration by parts for the left-hand side of equality (29)and with a similar method as used in the proof of expression(15) we get the identity (24) as long as choosing 120593(119909 119905) as thesolution of the adjoint problem

120593119905+ 119863120593119909119909

+ V120593119909minus 1199021 (119909) 120593 = 0 (119909 119905) isin Ω

119879

120593119909 (0 119905) = 0 120593

119909 (119897 119905) + ℎ120593 (119897 119905) = 0 119905 isin [0 119879]

120593 (119909 119879) = 119908 (119909) 119909 isin [0 119897]

(30)

where ℎ = V119863 = 1119886119871and 119908 = 119908(119909) is called a controllable

input Since the adjoint problem is completely controlled bythe unique input 119908 = 119908(119909) we denote the solution of theadjoint problem (30) by120593 = 120593(119908)(119909 119905)The proof is over

Obviously by setting 120591 = 119879 minus 119905 and 119909 = 119909 for the adjointproblem (30) and also denoting 120591 as 119905 it is transformed to anormal initial boundary value problem given as follows

120593119905minus 119863120593119909119909

minus V120593119909+ 1199021 (119909) 120593 = 0 (119909 119905) isin Ω

119879

120593119909 (0 119905) = 0 120593

119909 (119897 119905) + ℎ120593 (119897 119905) = 0 119905 isin [0 119879]

120593 (119909 0) = 119908 (119909) 119909 isin [0 119897]

(31)

Abstract and Applied Analysis 5

In what follows we cope with problem (31) instead ofproblem (30) For this adjoint problem we can see belowthat its solution is really determined by the unique input119908 = 119908(119909)

42 Lipschitz Stability Firstly by applying variable separa-tion method and Sturm-Liouville eigenvalue theory [16] wecan get an explicit expression of the solution for the adjointproblem (31)

Lemma 4 Suppose that the source coefficient 119902(119909) isin 119878119886119889

andthe hydraulic parameters 119863 and V are positive then there is

120593 (119908) (119909 119905) =

infin

sum

119899=1

119886119899exp (minus120582

119899119863119905)119883

119899 (119909) (32)

where120582119899119883119899(119909) (119899 = 1 2 ) are eigenvalues and correspond-

ing eigenfunctions of the following Sturm-Liouville problemrespectively

11988310158401015840

+ ℎ1198831015840minus

1199021 (119909)

119863119883 + 120582119883 = 0

1198831015840(0) = 0 119883

1015840(119897) + ℎ119883 (119897) = 0

(33)

119886119899=

(119908119883119899)

(119883119899 119883119899)

=

int119897

0120588 (119909)119908 (119909)119883119899 (119909) 119889119909

10038171003817100381710038171198831198991003817100381710038171003817

2

2

(34)

where 120588(119909) = exp(ℎ119909) is the weighted function of theeigenvalue problem (33) and the eigenfunctions series 119883

119899(119909)

are orthogonal and complete on 1198712(0 119897) with the weighted

function 120588(119909)

Remark 5 By multiplying 120588(119909) = exp(ℎ119909) at the two sides ofthe differential equation in (33) we have

minus119889

119889119909(120588 (119909)119883

1015840) + 119876 (119909)119883 = 120582120588 (119909)119883 (35)

where 119876(119909) = 120588(119909)1199021(119909)119863 ge 0 for 119909 isin [0 119897] Combing

with the homogeneous boundary conditions 1198831015840(0) = 0

and 1198831015840(119897) + ℎ119883(119897) = 0 we deduce that the operator

minus(119889119889119909)[120588(119909)(119889119889119909)] + 119876(119909) is self-adjoint and the problem(33) is a regular Sturm-Liouville eigenvalue problem and sothe eigenfunctions119883

119899(119909) 119899 = 1 2 are orthogonal with the

weighted function 120588(119909) for 119909 isin [0 119897]

With the aid of Lemma 4 and identity (24) we define abilinear formB(119902 119908) 119871

2(0 1) times 119871

2(0 1) rarr 119877 by

B (119902 119908) = intΩ119879

1198882119902 (119909) 120593 (119908) (119909 119905) 119889119909 119889119905 (36)

where 120593(119908) is the solution of the adjoint problem (31) for119908 isin 119871

2(0 119897) and 119888

2= 119888(119909 119905 119902

2) is a definite solution of

the forward problem (2)ndash(5) for any given 1199022

isin 119878ad and 1198882

is nonnegative and bounded for (119909 119905) isin Ω119879which can be

regarded as a weighted function of the bilinear form It isnoticeable that B is defined with 119902(119909) and 119908(119909) which isdifferent from those definitions given in [6 17]

By using maximum-minimum principle in the adjointproblem (31) we know that the solution 120593 = 120593(119908)(119909 119905) takesnonnegative values as long as 119908 = 119908(119909) isin 119871

2(0 119897) taking

nonnegative values on 119909 isin [0 119897] In addition by generaltheory of parabolic equation [8] there exists positive constant1198721such that

100381710038171003817100381712059310038171003817100381710038171198712(Ω119879)

le 11987211199081198712(0119897) (37)

Then we get the following lemma

Lemma 6 The functional B defined by (36) is a boundedbilinear form

Proof In fact B is linear on 119902 from (36) and together withthe expressions (32) and (34) it follows that 120593(119908) is linearon 119908 and then B is linear on 119908 too Next using Holderinequality for the right-hand side of (36) we have

1003816100381610038161003816B (119902 119908)1003816100381610038161003816 le 119872

1(intΩ119879

[119902 (119909)]2119889119909 119889119905)

12

times (intΩ119879

[120593 (119908)]2119889119909 119889119905)

12

= 1198722radic119879

100381710038171003817100381711990210038171003817100381710038171198712(0119897)

100381710038171003817100381712059310038171003817100381710038171198712(Ω119879)

le 1198723

100381710038171003817100381711990210038171003817100381710038171198712(0119897)

1199081198712(0119897)

(38)

where 1198723

= 11987211198722radic119879 and 119872

2here is a positive constant

depending on the domain and the prescribed solution 1198882 then

B is bounded on 119902 and 119908 The proof is over

By definition (36) and Lemmas 4 and 6 we define anadmissible space for the inputs as 119882 = 119908 isin 119871

2(0 119897) 119908 ge

0 1199082le 119864 and a norm for the source coefficient via

100381710038171003817100381711990210038171003817100381710038172

= sup119908isin119882 119908 = 0

1003816100381610038161003816B (119902 119908)1003816100381610038161003816

1003817100381710038171003817119888210038171003817100381710038172

1199082

(39)

where 119864 gt 0 is a constant and 11988822

= (intΩ119879

[119888(119909 119905 1199022)]2

119889119909 119889119905)12

It is not difficult to testify that the above definition is well-defined by which we can construct a conditional stability forthe inversion of the source coefficient based on the integralidentity (24)

Theorem 7 Suppose that the assumptions (A) (B) and (C1)are satisfied and (119888

119894 119902119894) (119894 = 1 2) are two pairs of solutions to

the inverse problem (2)ndash(6) corresponding to the data 119888119894

0and

119892119894(119894 = 1 2) and 120579

119894(119894 = 1 2) are the additional observations

Then for 119902 isin 119878119886119889

and 119908 isin 119882 there exists constant 119872 =

119872(119863 V Ω119879 1205792) such that

10038171003817100381710038171199021 minus 1199022

10038171003817100381710038172

le 119872(10038171003817100381710038171205791 minus 120579

2

10038171003817100381710038172+

100381710038171003817100381710038171198881

0minus 1198882

0

100381710038171003817100381710038172+

10038171003817100381710038171198921 minus 1198922

10038171003817100381710038172)

(40)

6 Abstract and Applied Analysis

Proof Firstly by assertion (1) of Theorem 1 there is

[120579 (119909)]2le [119888 (119909 119905)]

2le [1198880 (119909)]2

0 lt 119909 lt 119897 0 lt 119905 lt 119879

(41)

for any solution 119888(119909 119905) of the forward problem (2)ndash(5) withthe final function 120579(119909) = 119888(119909 119879) and the initial function1198880(119909) = 119888(119909 0) Then noting condition (A) we have for the

solution 1198882= 119888(119909 119905 119902

2)

1

1198720radic119879

le1

1003817100381710038171003817119888010038171003817100381710038172

radic119879

le1

1003817100381710038171003817119888210038171003817100381710038172

le1

1205792radic119879

le1

120576radic119879

(42)

Next by definition (39) and utilizing the integral identity (24)and Cauchy-Schwartz inequality we get

10038171003817100381710038171199021 minus 1199022

10038171003817100381710038172le

1

1205792radic119879

sdot 10038171003817100381710038171205791 minus 120579

2

10038171003817100381710038172sup119908isin119882 119908 = 0

1199082

1199082

+100381710038171003817100381710038171198881

0minus 1198882

0

100381710038171003817100381710038172sup119908isin119882 119908 = 0

1003817100381710038171003817120593(119909 0)10038171003817100381710038172

1199082

+10038171003817100381710038171198921 minus 119892

2

10038171003817100381710038172sup119908isin119882 119908 = 0

1003817100381710038171003817120593 (0 119905)10038171003817100381710038172

1199082

(43)

Also by the general theory of 1D parabolic equation we knowthat there exists a constant119872 gt 0 depending on119863 V and thedomain Ω

119879such that

1003817100381710038171003817120593(119909 0)10038171003817100381710038172

1003817100381710038171003817120593(0 119905)

10038171003817100381710038172le 1198721199082 (44)

Therefore by setting

119872 =1

1205792radic119879

max 1119872 (45)

and noting (43) it follows that (40) is valid The proof is over

Remark 8 This theorem shows a conditional Lipschitz sta-bility for the inverse problem (2)ndash(6) Furthermore by thistheorem we can see that the uniqueness in the meaning of 1198712for the inverse problem can be easily deduced by the stabilityestimate (40)

Conflict of Interests

The authors declare that there is no conflict of interests withany commercial identities regarding the publication of thispaper

Acknowledgments

The authors thank the anonymous referees for valuablecommentsThiswork is supported by the Basic andAdvancedResearch Program of Henan Science Committee (Research

on Some Problems in the Category of Quantic Lattices)and the Research Award for Teachers in Nanyang NormalUniversity (no nynu200749) and is partially supported by theNational Natural Science Foundation of China (nos 11071148and 11371231)

References

[1] P DuChateau ldquoMonotonicity and invertibility of coefficient-to-datamappings for parabolic inverse problemsrdquo SIAM Journal onMathematical Analysis vol 26 no 6 pp 1473ndash1487 1995

[2] P DuChateau ldquoAn inverse problem for the hydraulic propertiesof porous mediardquo SIAM Journal on Mathematical Analysis vol28 no 3 pp 611ndash632 1997

[3] P DuChateau ldquoAn adjoint method for proving identifiability ofcoefficients in parabolic equationsrdquo Journal of Inverse and Ill-Posed Problems vol 21 no 5 pp 639ndash663 2013

[4] A Hasanov P DuChateau and B Pektas ldquoAn adjoint problemapproach and coarse-fine mesh method for identification of thediffusion coefficient in a linear parabolic equationrdquo Journal ofInverse and Ill-Posed Problems vol 14 no 5 pp 435ndash463 2006

[5] Y H Ou A Hasanov and Z H Liu ldquoInverse coefficientproblems for nonlinear parabolic differential equationsrdquo ActaMathematica Sinica vol 24 no 10 pp 1617ndash1624 2008

[6] G Li D Yao and F Yang ldquoAn inverse problem of identifyingsource coefficient in solute transportationrdquo Journal of Inverseand Ill-Posed Problems vol 16 no 1 pp 51ndash63 2008

[7] N-Z Sun Mathematical Modeling of Groundwater PollutionSpringer New York NY USA 1996

[8] J R Cannon The One-Dimensional Heat Equation vol 23Addison-Wesley London UK 1984

[9] I Bushuyev ldquoGlobal uniqueness for inverse parabolic problemswith final observationrdquo Inverse Problems vol 11 no 4 pp L11ndashL16 1995

[10] M Choulli and M Yamamoto ldquoConditional stability in deter-mining a heat sourcerdquo Journal of Inverse and Ill-Posed Problemsvol 12 no 3 pp 233ndash243 2004

[11] V Isakov ldquoSome inverse problems for the diffusion equationrdquoInverse Problems vol 15 no 1 pp 3ndash10 1999

[12] E G Savateev and R Riganti ldquoInverse problem for thenonlinear heat equation with the final overdeterminationrdquoMathematical and Computer Modelling vol 22 no 1 pp 29ndash431995

[13] J R Cannon and P DuChateau ldquoStructural identification of anunknown source term in a heat equationrdquo Inverse Problems vol14 no 3 pp 535ndash551 1998

[14] G Li ldquoData compatibility and conditional stability for aninverse source problem in the heat equationrdquo Applied Mathe-matics and Computation vol 173 no 1 pp 566ndash581 2006

[15] R P SperbMaximumPrinciples andTheir Applications vol 157Academic Press New York NY USA 1981

[16] R Courant and D Hilbert Methods of Mathematical Physics IInterscience New York NY USA 1953

[17] L Gongsheng and M Yamamoto ldquoStability analysis for deter-mining a source term in a 1-D advection-dispersion equationrdquoJournal of Inverse and Ill-Posed Problems vol 14 no 2 pp 147ndash155 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

then for each given 119909 isin [0 119897] it follows that0 le 120579 (119909) = 119888 (119909 119879) le 119888 (119909 119905)

le 119888 (119909 0) = 1198880 (119909) 0 lt 119905 lt 119879

(7)

(2) If the solute flux 119892(119905) is monotone increasing on 119905 isin

[0 119879] and 119902 = 119902(119909) has the property (P2)

(P2) 119902(119909)1198880(119909) minus V1198881015840

0(119909) + 119863119888

10158401015840

0(119909) le 0 0 le 119909 le 119897

then for each given 119909 isin [0 119897] it follows that0 le 1198880 (119909) = 119888 (119909 0) le 119888 (119909 119905)

le 119888 (119909 119879) = 120579 (119909) 0 lt 119905 lt 119879

(8)

Proof We only prove the assertion (1) and (2) can be provedsimilarly For 119888 = 119888(119909 119905 119902) and any smooth test function120593(119909 119905) we have

intΩ119879

(119888119905minus 119863119888119909119909

+ V119888119909+ 119902 (119909) 119888) 120593119905119889119909 119889119905 = 0 (9)

Integration by parts leads to

intΩ119879

119888119905(120593119905+ 119863120593119909119909

+ V120593119909minus 119902 (119909) 120593) 119889119909 119889119905

= int

119879

0

[119863 (119888119909120593119905minus 119888120593119909119905) minus V119888120593

119905]119909=119897

119909=0119889119905

+ int

119897

0

[119863119888120593119909119909

+ V119888120593119909minus 119902 (119909) 119888120593]

119905=119879

119905=0119889119909

(10)

Denote ℎ = V119863 = 1119886119871 If there is

120593119909 (119897 119905) + ℎ120593 (119897 119905) = 0 (11)

then120593119909119905 (119897 119905) + ℎ120593

119905 (119897 119905) = 0 (12)and we get

119863120593119909119905 (119897 119905) + V120593

119905 (119897 119905) = 0 (13)Let 119866 = 119866(119909 119905) be any arbitrary nonnegative function

on (119909 119905) isin Ω119879and 120593 = 120593(119909 119905) solve the following adjoint

problem120593119905+ 119863120593119909119909

+ V120593119909minus 119902 (119909) 120593 = 119866 (119909 119905) (119909 119905) isin Ω

119879

120593119909 (0 119905) = 0 120593

119909 (119897 119905) + ℎ120593 (119897 119905) = 0 0 le 119905 le 119879

120593 (119909 119879) = 0 0 le 119909 le 119897

(14)

Then together with the initial boundary value conditionsgiven by (3) (4) and (5) the equality (10) is reduced to

intΩ119879

119888119905119866 (119909 119905) 119889119909 119889119905

= int

119897

0

119888 (119909 0) [minus119863120593119909119909 (119909 0) minus V120593

119909 (119909 0)

+119902 (119909) 120593 (119909 0)] 119889119909

+ int

119879

0

119892 (119905) 120593119905 (0 119905) 119889119905 = 1198681+ 1198682

(15)

where

1198681= int

119897

0

1198880 (119909)

times [119902 (119909) 120593 (119909 0) minus V120593119909 (119909 0) minus 119863120593

119909119909 (119909 0)] 119889119909

1198682= int

119879

0

119892 (119905) 120593119905 (0 119905) 119889119905

(16)

Firstly by using integration by parts for 1198681 we have

1198681= minus 119863[119888

0 (119909) 120593119909 (119909 0) minus 1198881015840

0(119909) 120593 (119909 0)]

119897

0

minus V[1198880(119909)120593(119909 0)]

119897

0

+ int

119897

0

[119902 (119909) 1198880 (119909) + V11988810158400(119909) minus 119863119888

10158401015840

0(119909)]

times 120593 (119909 0) 119889119909

(17)

Thanks to the consistency condition (B) and the boundaryvalue conditions of the adjoint problem (14) we can get

1198681= 119892 (0) 120593 (0 0)

+ int

119897

0

[119902 (119909) 1198880 (119909) + V11988810158400(119909) minus 119863119888

10158401015840

0(119909)] 120593 (119909 0) 119889119909

(18)

Also by integration by parts for 1198682and noting 120593(0 119879) = 0 we

have

1198682= minus119892 (0) 120593 (0 0) minus int

119879

0

1198921015840(119905) 120593 (0 119905) 119889119905 (19)

So we get

intΩ119879

119888119905119866 (119909 119905) 119889119909 119889119905

= int

119897

0

[119902 (119909) 1198880 (119909) + V11988810158400(119909) minus 119863119888

10158401015840

0(119909)] 120593 (119909 0) 119889119909

+ int

119879

0

[minus1198921015840(119905)] 120593 (0 119905) 119889119905

(20)

By applying the maximum principle of parabolic typeof PDE see for example [15] to the adjoint problem (14)we conclude that if 119866(119909 119905) is nonnegative but is otherwisearbitrary then 120593(119909 119905) is negative on Ω

119879 and so does for

120593(0 119905) and 120593(119909 0) Then together with the property (P1) weknow that the sign of the first term of the right-hand side of(20) is nonpositive In addition by virtue of the monotonedecreasing property of the boundary flux 119892(119905) the sign of thesecond term of the right-hand side of (20) is also nonpositiveHence the sign of expression (20) or (15) is nonpositive thatis we have

intΩ119879

119888119905119866 (119909 119905) 119889119909 119889119905 le 0 (21)

Since 119866 = 119866(119909 119905) is nonnegative but otherwise arbitraryit follows that if there is any positive measure subset of Ω

119879

where 119888119905(119909 119905) is positive then a contradiction of (21) can be

achieved by choosing the support 119866 = 119866(119909 119905) in this positivemeasure set This proves that for each 119909 isin [0 119897] there is119888119905(119909 119905) le 0 for 0 le 119905 le 119879 and the assertion (1) is valid The

proof is completed

4 Abstract and Applied Analysis

Remark 2 By setting V = 0 and 119891(119909 119905 119888) = 0 in (1) we getthe model ever discussed in [4] where Neumannrsquos boundaryconditions are employed and the initial value is zero Thanksto 1198880(119909) = 0 we have by equality (15)

intΩ119879

119888119905119866 (119909 119905) 119889119909 119889119905 = int

119879

0

119892 (119905) 120593119905 (0 119905) 119889119905 (22)

Since 120593(0 119905) le 120593(0 119879) for 0 lt 119905 lt 119879 there is 120593119905(0 119905) ge 0

Hence we deduce that assertion (1) is also valid if 119892(119905) lt 0

and assertion (2) is valid for 119892(119905) gt 0 However if consideringnonzero initial condition the property (P1)(P2) is neededand the assumption of monotonicity for 119892(119905) can be replacedby the properties of keeping its positivenegative sign for 119905 isin

(0 119879) and 119892(0) = 0 such that the assertions of this theoremcan be proved too

According to Theorem 1 we can get two sufficientconditions under which the inverse problem (2)ndash(6) is of datacompatibility That is the unknown source coefficient 119902(119909)

should have one of the following conditions with the knowndata functions 119888

0(119909) and 119892(119905)

(C1) 1198921015840(119905) le 0 for 119905 isin [0 119879] and 119902(119909) has property (P1)

(C2) 1198921015840(119905) ge 0 119905 isin [0 119879] and 119902(119909) has property (P2)

In summary if the conditions (A) (B) and (C1) aresatisfied the solution 119888 = 119888(119909 119905 119902) is monotone decreasingon 119905 isin [0 119879] for each 119909 isin [0 119897] If the conditions (A) (B) and(C2) are satisfied then the solution 119888 = 119888(119909 119905 119902) is monotoneincreasing on 119905 isin [0 119879] for each 119909 isin [0 119897]

In the following discussionswe assume that the condition(C1) is satisfied and the unknown source coefficient 119902 =

119902(119909) is continuous and has property (P1) Moreover let thefunction 119902 = 119902(119909) take nonnegative values for 119909 isin [0 119897]Thus an admissible set for the unknown source coefficientsis defined as follows

119878ad = 119902 119902 is continuous 119902 ge 0 for 119909 isin [0 119897]

and has property (P1) (23)

In the next section wewill establish a conditional stabilityfor the inverse problem of determining 119902 isin 119878ad with thehelp of an integral identity by suitably controlling an adjointproblem

4 Conditional Stability of the Inverse Problem

It is more difficult to prove stability than to prove existenceand uniqueness for an inverse problem and it always needsadditional conditions and suitable topology to construct astability which is called conditional stability for the inverseproblem

41 Construction of Integral Identity In this section a vari-ational integral identity with the aid of an adjoint problemis established which reflects a corresponding relation of theunknown source coefficientwith those of the initial boundaryvalues and the additional data

Theorem 3 Let 119888119894

= 119888(119909 119905 119902119894) (119894 = 1 2) be two solutions

corresponding to the initial boundary value functions 1198881198940(119909) and

119892119894(119905) (119894 = 1 2) and let 120579

119894(119909) (119894 = 1 2) be the corresponding

additional observations Then it follows that

intΩ119879

1198882(1199021minus 1199022) 120593 (119908) (119909 119905) 119889119909 119889119905

= int

119897

0

(1205792minus 1205791) 119908 (119909) 119889119909 + int

119897

0

(1198881

0minus 1198882

0) 120593 (119909 0) 119889119909

+ int

119879

0

(1198921minus 1198922) 120593 (0 119905) 119889119905

(24)

where 120593 = 120593(119908)(119909 119905) is the solution of a suitable adjointproblem with input data 119908 = 119908(119909) given in the proof of thistheorem

Proof Denoting 119862 = 1198881minus 1198882and noting that 119888

1and 1198882both

satisfy (2) and (3)ndash(6) we have

119862119905minus 119863119862

119909119909+ V119862119909+ 1199021119862 = (119902

2minus 1199021) 1198882 (25)

119862 (119909 0) = 1198881

0minus 1198882

0 (26)

V119862 (0 119905) minus 119863119862119909 (0 119905) = 119892

1minus 1198922 119862119909 (119897 119905) = 0 (27)

119862 (119909 119879) = 1205791minus 1205792 (28)

By smooth test function 120593 = 120593(119909 119905) and by multiplying twosides of (25) and integrating on Ω

119879 respectively we have

intΩ119879

[119862119905minus 119863119862

119909119909+ V119862119909+ 1199021119862] 120593 (119909 119905) 119889119909 119889119905

= intΩ119879

1198882[1199022minus 1199021] 120593 (119909 119905) 119889119909 119889119905

(29)

Integration by parts for the left-hand side of equality (29)and with a similar method as used in the proof of expression(15) we get the identity (24) as long as choosing 120593(119909 119905) as thesolution of the adjoint problem

120593119905+ 119863120593119909119909

+ V120593119909minus 1199021 (119909) 120593 = 0 (119909 119905) isin Ω

119879

120593119909 (0 119905) = 0 120593

119909 (119897 119905) + ℎ120593 (119897 119905) = 0 119905 isin [0 119879]

120593 (119909 119879) = 119908 (119909) 119909 isin [0 119897]

(30)

where ℎ = V119863 = 1119886119871and 119908 = 119908(119909) is called a controllable

input Since the adjoint problem is completely controlled bythe unique input 119908 = 119908(119909) we denote the solution of theadjoint problem (30) by120593 = 120593(119908)(119909 119905)The proof is over

Obviously by setting 120591 = 119879 minus 119905 and 119909 = 119909 for the adjointproblem (30) and also denoting 120591 as 119905 it is transformed to anormal initial boundary value problem given as follows

120593119905minus 119863120593119909119909

minus V120593119909+ 1199021 (119909) 120593 = 0 (119909 119905) isin Ω

119879

120593119909 (0 119905) = 0 120593

119909 (119897 119905) + ℎ120593 (119897 119905) = 0 119905 isin [0 119879]

120593 (119909 0) = 119908 (119909) 119909 isin [0 119897]

(31)

Abstract and Applied Analysis 5

In what follows we cope with problem (31) instead ofproblem (30) For this adjoint problem we can see belowthat its solution is really determined by the unique input119908 = 119908(119909)

42 Lipschitz Stability Firstly by applying variable separa-tion method and Sturm-Liouville eigenvalue theory [16] wecan get an explicit expression of the solution for the adjointproblem (31)

Lemma 4 Suppose that the source coefficient 119902(119909) isin 119878119886119889

andthe hydraulic parameters 119863 and V are positive then there is

120593 (119908) (119909 119905) =

infin

sum

119899=1

119886119899exp (minus120582

119899119863119905)119883

119899 (119909) (32)

where120582119899119883119899(119909) (119899 = 1 2 ) are eigenvalues and correspond-

ing eigenfunctions of the following Sturm-Liouville problemrespectively

11988310158401015840

+ ℎ1198831015840minus

1199021 (119909)

119863119883 + 120582119883 = 0

1198831015840(0) = 0 119883

1015840(119897) + ℎ119883 (119897) = 0

(33)

119886119899=

(119908119883119899)

(119883119899 119883119899)

=

int119897

0120588 (119909)119908 (119909)119883119899 (119909) 119889119909

10038171003817100381710038171198831198991003817100381710038171003817

2

2

(34)

where 120588(119909) = exp(ℎ119909) is the weighted function of theeigenvalue problem (33) and the eigenfunctions series 119883

119899(119909)

are orthogonal and complete on 1198712(0 119897) with the weighted

function 120588(119909)

Remark 5 By multiplying 120588(119909) = exp(ℎ119909) at the two sides ofthe differential equation in (33) we have

minus119889

119889119909(120588 (119909)119883

1015840) + 119876 (119909)119883 = 120582120588 (119909)119883 (35)

where 119876(119909) = 120588(119909)1199021(119909)119863 ge 0 for 119909 isin [0 119897] Combing

with the homogeneous boundary conditions 1198831015840(0) = 0

and 1198831015840(119897) + ℎ119883(119897) = 0 we deduce that the operator

minus(119889119889119909)[120588(119909)(119889119889119909)] + 119876(119909) is self-adjoint and the problem(33) is a regular Sturm-Liouville eigenvalue problem and sothe eigenfunctions119883

119899(119909) 119899 = 1 2 are orthogonal with the

weighted function 120588(119909) for 119909 isin [0 119897]

With the aid of Lemma 4 and identity (24) we define abilinear formB(119902 119908) 119871

2(0 1) times 119871

2(0 1) rarr 119877 by

B (119902 119908) = intΩ119879

1198882119902 (119909) 120593 (119908) (119909 119905) 119889119909 119889119905 (36)

where 120593(119908) is the solution of the adjoint problem (31) for119908 isin 119871

2(0 119897) and 119888

2= 119888(119909 119905 119902

2) is a definite solution of

the forward problem (2)ndash(5) for any given 1199022

isin 119878ad and 1198882

is nonnegative and bounded for (119909 119905) isin Ω119879which can be

regarded as a weighted function of the bilinear form It isnoticeable that B is defined with 119902(119909) and 119908(119909) which isdifferent from those definitions given in [6 17]

By using maximum-minimum principle in the adjointproblem (31) we know that the solution 120593 = 120593(119908)(119909 119905) takesnonnegative values as long as 119908 = 119908(119909) isin 119871

2(0 119897) taking

nonnegative values on 119909 isin [0 119897] In addition by generaltheory of parabolic equation [8] there exists positive constant1198721such that

100381710038171003817100381712059310038171003817100381710038171198712(Ω119879)

le 11987211199081198712(0119897) (37)

Then we get the following lemma

Lemma 6 The functional B defined by (36) is a boundedbilinear form

Proof In fact B is linear on 119902 from (36) and together withthe expressions (32) and (34) it follows that 120593(119908) is linearon 119908 and then B is linear on 119908 too Next using Holderinequality for the right-hand side of (36) we have

1003816100381610038161003816B (119902 119908)1003816100381610038161003816 le 119872

1(intΩ119879

[119902 (119909)]2119889119909 119889119905)

12

times (intΩ119879

[120593 (119908)]2119889119909 119889119905)

12

= 1198722radic119879

100381710038171003817100381711990210038171003817100381710038171198712(0119897)

100381710038171003817100381712059310038171003817100381710038171198712(Ω119879)

le 1198723

100381710038171003817100381711990210038171003817100381710038171198712(0119897)

1199081198712(0119897)

(38)

where 1198723

= 11987211198722radic119879 and 119872

2here is a positive constant

depending on the domain and the prescribed solution 1198882 then

B is bounded on 119902 and 119908 The proof is over

By definition (36) and Lemmas 4 and 6 we define anadmissible space for the inputs as 119882 = 119908 isin 119871

2(0 119897) 119908 ge

0 1199082le 119864 and a norm for the source coefficient via

100381710038171003817100381711990210038171003817100381710038172

= sup119908isin119882 119908 = 0

1003816100381610038161003816B (119902 119908)1003816100381610038161003816

1003817100381710038171003817119888210038171003817100381710038172

1199082

(39)

where 119864 gt 0 is a constant and 11988822

= (intΩ119879

[119888(119909 119905 1199022)]2

119889119909 119889119905)12

It is not difficult to testify that the above definition is well-defined by which we can construct a conditional stability forthe inversion of the source coefficient based on the integralidentity (24)

Theorem 7 Suppose that the assumptions (A) (B) and (C1)are satisfied and (119888

119894 119902119894) (119894 = 1 2) are two pairs of solutions to

the inverse problem (2)ndash(6) corresponding to the data 119888119894

0and

119892119894(119894 = 1 2) and 120579

119894(119894 = 1 2) are the additional observations

Then for 119902 isin 119878119886119889

and 119908 isin 119882 there exists constant 119872 =

119872(119863 V Ω119879 1205792) such that

10038171003817100381710038171199021 minus 1199022

10038171003817100381710038172

le 119872(10038171003817100381710038171205791 minus 120579

2

10038171003817100381710038172+

100381710038171003817100381710038171198881

0minus 1198882

0

100381710038171003817100381710038172+

10038171003817100381710038171198921 minus 1198922

10038171003817100381710038172)

(40)

6 Abstract and Applied Analysis

Proof Firstly by assertion (1) of Theorem 1 there is

[120579 (119909)]2le [119888 (119909 119905)]

2le [1198880 (119909)]2

0 lt 119909 lt 119897 0 lt 119905 lt 119879

(41)

for any solution 119888(119909 119905) of the forward problem (2)ndash(5) withthe final function 120579(119909) = 119888(119909 119879) and the initial function1198880(119909) = 119888(119909 0) Then noting condition (A) we have for the

solution 1198882= 119888(119909 119905 119902

2)

1

1198720radic119879

le1

1003817100381710038171003817119888010038171003817100381710038172

radic119879

le1

1003817100381710038171003817119888210038171003817100381710038172

le1

1205792radic119879

le1

120576radic119879

(42)

Next by definition (39) and utilizing the integral identity (24)and Cauchy-Schwartz inequality we get

10038171003817100381710038171199021 minus 1199022

10038171003817100381710038172le

1

1205792radic119879

sdot 10038171003817100381710038171205791 minus 120579

2

10038171003817100381710038172sup119908isin119882 119908 = 0

1199082

1199082

+100381710038171003817100381710038171198881

0minus 1198882

0

100381710038171003817100381710038172sup119908isin119882 119908 = 0

1003817100381710038171003817120593(119909 0)10038171003817100381710038172

1199082

+10038171003817100381710038171198921 minus 119892

2

10038171003817100381710038172sup119908isin119882 119908 = 0

1003817100381710038171003817120593 (0 119905)10038171003817100381710038172

1199082

(43)

Also by the general theory of 1D parabolic equation we knowthat there exists a constant119872 gt 0 depending on119863 V and thedomain Ω

119879such that

1003817100381710038171003817120593(119909 0)10038171003817100381710038172

1003817100381710038171003817120593(0 119905)

10038171003817100381710038172le 1198721199082 (44)

Therefore by setting

119872 =1

1205792radic119879

max 1119872 (45)

and noting (43) it follows that (40) is valid The proof is over

Remark 8 This theorem shows a conditional Lipschitz sta-bility for the inverse problem (2)ndash(6) Furthermore by thistheorem we can see that the uniqueness in the meaning of 1198712for the inverse problem can be easily deduced by the stabilityestimate (40)

Conflict of Interests

The authors declare that there is no conflict of interests withany commercial identities regarding the publication of thispaper

Acknowledgments

The authors thank the anonymous referees for valuablecommentsThiswork is supported by the Basic andAdvancedResearch Program of Henan Science Committee (Research

on Some Problems in the Category of Quantic Lattices)and the Research Award for Teachers in Nanyang NormalUniversity (no nynu200749) and is partially supported by theNational Natural Science Foundation of China (nos 11071148and 11371231)

References

[1] P DuChateau ldquoMonotonicity and invertibility of coefficient-to-datamappings for parabolic inverse problemsrdquo SIAM Journal onMathematical Analysis vol 26 no 6 pp 1473ndash1487 1995

[2] P DuChateau ldquoAn inverse problem for the hydraulic propertiesof porous mediardquo SIAM Journal on Mathematical Analysis vol28 no 3 pp 611ndash632 1997

[3] P DuChateau ldquoAn adjoint method for proving identifiability ofcoefficients in parabolic equationsrdquo Journal of Inverse and Ill-Posed Problems vol 21 no 5 pp 639ndash663 2013

[4] A Hasanov P DuChateau and B Pektas ldquoAn adjoint problemapproach and coarse-fine mesh method for identification of thediffusion coefficient in a linear parabolic equationrdquo Journal ofInverse and Ill-Posed Problems vol 14 no 5 pp 435ndash463 2006

[5] Y H Ou A Hasanov and Z H Liu ldquoInverse coefficientproblems for nonlinear parabolic differential equationsrdquo ActaMathematica Sinica vol 24 no 10 pp 1617ndash1624 2008

[6] G Li D Yao and F Yang ldquoAn inverse problem of identifyingsource coefficient in solute transportationrdquo Journal of Inverseand Ill-Posed Problems vol 16 no 1 pp 51ndash63 2008

[7] N-Z Sun Mathematical Modeling of Groundwater PollutionSpringer New York NY USA 1996

[8] J R Cannon The One-Dimensional Heat Equation vol 23Addison-Wesley London UK 1984

[9] I Bushuyev ldquoGlobal uniqueness for inverse parabolic problemswith final observationrdquo Inverse Problems vol 11 no 4 pp L11ndashL16 1995

[10] M Choulli and M Yamamoto ldquoConditional stability in deter-mining a heat sourcerdquo Journal of Inverse and Ill-Posed Problemsvol 12 no 3 pp 233ndash243 2004

[11] V Isakov ldquoSome inverse problems for the diffusion equationrdquoInverse Problems vol 15 no 1 pp 3ndash10 1999

[12] E G Savateev and R Riganti ldquoInverse problem for thenonlinear heat equation with the final overdeterminationrdquoMathematical and Computer Modelling vol 22 no 1 pp 29ndash431995

[13] J R Cannon and P DuChateau ldquoStructural identification of anunknown source term in a heat equationrdquo Inverse Problems vol14 no 3 pp 535ndash551 1998

[14] G Li ldquoData compatibility and conditional stability for aninverse source problem in the heat equationrdquo Applied Mathe-matics and Computation vol 173 no 1 pp 566ndash581 2006

[15] R P SperbMaximumPrinciples andTheir Applications vol 157Academic Press New York NY USA 1981

[16] R Courant and D Hilbert Methods of Mathematical Physics IInterscience New York NY USA 1953

[17] L Gongsheng and M Yamamoto ldquoStability analysis for deter-mining a source term in a 1-D advection-dispersion equationrdquoJournal of Inverse and Ill-Posed Problems vol 14 no 2 pp 147ndash155 2006

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

Remark 2 By setting V = 0 and 119891(119909 119905 119888) = 0 in (1) we getthe model ever discussed in [4] where Neumannrsquos boundaryconditions are employed and the initial value is zero Thanksto 1198880(119909) = 0 we have by equality (15)

intΩ119879

119888119905119866 (119909 119905) 119889119909 119889119905 = int

119879

0

119892 (119905) 120593119905 (0 119905) 119889119905 (22)

Since 120593(0 119905) le 120593(0 119879) for 0 lt 119905 lt 119879 there is 120593119905(0 119905) ge 0

Hence we deduce that assertion (1) is also valid if 119892(119905) lt 0

and assertion (2) is valid for 119892(119905) gt 0 However if consideringnonzero initial condition the property (P1)(P2) is neededand the assumption of monotonicity for 119892(119905) can be replacedby the properties of keeping its positivenegative sign for 119905 isin

(0 119879) and 119892(0) = 0 such that the assertions of this theoremcan be proved too

According to Theorem 1 we can get two sufficientconditions under which the inverse problem (2)ndash(6) is of datacompatibility That is the unknown source coefficient 119902(119909)

should have one of the following conditions with the knowndata functions 119888

0(119909) and 119892(119905)

(C1) 1198921015840(119905) le 0 for 119905 isin [0 119879] and 119902(119909) has property (P1)

(C2) 1198921015840(119905) ge 0 119905 isin [0 119879] and 119902(119909) has property (P2)

In summary if the conditions (A) (B) and (C1) aresatisfied the solution 119888 = 119888(119909 119905 119902) is monotone decreasingon 119905 isin [0 119879] for each 119909 isin [0 119897] If the conditions (A) (B) and(C2) are satisfied then the solution 119888 = 119888(119909 119905 119902) is monotoneincreasing on 119905 isin [0 119879] for each 119909 isin [0 119897]

In the following discussionswe assume that the condition(C1) is satisfied and the unknown source coefficient 119902 =

119902(119909) is continuous and has property (P1) Moreover let thefunction 119902 = 119902(119909) take nonnegative values for 119909 isin [0 119897]Thus an admissible set for the unknown source coefficientsis defined as follows

119878ad = 119902 119902 is continuous 119902 ge 0 for 119909 isin [0 119897]

and has property (P1) (23)

In the next section wewill establish a conditional stabilityfor the inverse problem of determining 119902 isin 119878ad with thehelp of an integral identity by suitably controlling an adjointproblem

4 Conditional Stability of the Inverse Problem

It is more difficult to prove stability than to prove existenceand uniqueness for an inverse problem and it always needsadditional conditions and suitable topology to construct astability which is called conditional stability for the inverseproblem

41 Construction of Integral Identity In this section a vari-ational integral identity with the aid of an adjoint problemis established which reflects a corresponding relation of theunknown source coefficientwith those of the initial boundaryvalues and the additional data

Theorem 3 Let 119888119894

= 119888(119909 119905 119902119894) (119894 = 1 2) be two solutions

corresponding to the initial boundary value functions 1198881198940(119909) and

119892119894(119905) (119894 = 1 2) and let 120579

119894(119909) (119894 = 1 2) be the corresponding

additional observations Then it follows that

intΩ119879

1198882(1199021minus 1199022) 120593 (119908) (119909 119905) 119889119909 119889119905

= int

119897

0

(1205792minus 1205791) 119908 (119909) 119889119909 + int

119897

0

(1198881

0minus 1198882

0) 120593 (119909 0) 119889119909

+ int

119879

0

(1198921minus 1198922) 120593 (0 119905) 119889119905

(24)

where 120593 = 120593(119908)(119909 119905) is the solution of a suitable adjointproblem with input data 119908 = 119908(119909) given in the proof of thistheorem

Proof Denoting 119862 = 1198881minus 1198882and noting that 119888

1and 1198882both

satisfy (2) and (3)ndash(6) we have

119862119905minus 119863119862

119909119909+ V119862119909+ 1199021119862 = (119902

2minus 1199021) 1198882 (25)

119862 (119909 0) = 1198881

0minus 1198882

0 (26)

V119862 (0 119905) minus 119863119862119909 (0 119905) = 119892

1minus 1198922 119862119909 (119897 119905) = 0 (27)

119862 (119909 119879) = 1205791minus 1205792 (28)

By smooth test function 120593 = 120593(119909 119905) and by multiplying twosides of (25) and integrating on Ω

119879 respectively we have

intΩ119879

[119862119905minus 119863119862

119909119909+ V119862119909+ 1199021119862] 120593 (119909 119905) 119889119909 119889119905

= intΩ119879

1198882[1199022minus 1199021] 120593 (119909 119905) 119889119909 119889119905

(29)

Integration by parts for the left-hand side of equality (29)and with a similar method as used in the proof of expression(15) we get the identity (24) as long as choosing 120593(119909 119905) as thesolution of the adjoint problem

120593119905+ 119863120593119909119909

+ V120593119909minus 1199021 (119909) 120593 = 0 (119909 119905) isin Ω

119879

120593119909 (0 119905) = 0 120593

119909 (119897 119905) + ℎ120593 (119897 119905) = 0 119905 isin [0 119879]

120593 (119909 119879) = 119908 (119909) 119909 isin [0 119897]

(30)

where ℎ = V119863 = 1119886119871and 119908 = 119908(119909) is called a controllable

input Since the adjoint problem is completely controlled bythe unique input 119908 = 119908(119909) we denote the solution of theadjoint problem (30) by120593 = 120593(119908)(119909 119905)The proof is over

Obviously by setting 120591 = 119879 minus 119905 and 119909 = 119909 for the adjointproblem (30) and also denoting 120591 as 119905 it is transformed to anormal initial boundary value problem given as follows

120593119905minus 119863120593119909119909

minus V120593119909+ 1199021 (119909) 120593 = 0 (119909 119905) isin Ω

119879

120593119909 (0 119905) = 0 120593

119909 (119897 119905) + ℎ120593 (119897 119905) = 0 119905 isin [0 119879]

120593 (119909 0) = 119908 (119909) 119909 isin [0 119897]

(31)

Abstract and Applied Analysis 5

In what follows we cope with problem (31) instead ofproblem (30) For this adjoint problem we can see belowthat its solution is really determined by the unique input119908 = 119908(119909)

42 Lipschitz Stability Firstly by applying variable separa-tion method and Sturm-Liouville eigenvalue theory [16] wecan get an explicit expression of the solution for the adjointproblem (31)

Lemma 4 Suppose that the source coefficient 119902(119909) isin 119878119886119889

andthe hydraulic parameters 119863 and V are positive then there is

120593 (119908) (119909 119905) =

infin

sum

119899=1

119886119899exp (minus120582

119899119863119905)119883

119899 (119909) (32)

where120582119899119883119899(119909) (119899 = 1 2 ) are eigenvalues and correspond-

ing eigenfunctions of the following Sturm-Liouville problemrespectively

11988310158401015840

+ ℎ1198831015840minus

1199021 (119909)

119863119883 + 120582119883 = 0

1198831015840(0) = 0 119883

1015840(119897) + ℎ119883 (119897) = 0

(33)

119886119899=

(119908119883119899)

(119883119899 119883119899)

=

int119897

0120588 (119909)119908 (119909)119883119899 (119909) 119889119909

10038171003817100381710038171198831198991003817100381710038171003817

2

2

(34)

where 120588(119909) = exp(ℎ119909) is the weighted function of theeigenvalue problem (33) and the eigenfunctions series 119883

119899(119909)

are orthogonal and complete on 1198712(0 119897) with the weighted

function 120588(119909)

Remark 5 By multiplying 120588(119909) = exp(ℎ119909) at the two sides ofthe differential equation in (33) we have

minus119889

119889119909(120588 (119909)119883

1015840) + 119876 (119909)119883 = 120582120588 (119909)119883 (35)

where 119876(119909) = 120588(119909)1199021(119909)119863 ge 0 for 119909 isin [0 119897] Combing

with the homogeneous boundary conditions 1198831015840(0) = 0

and 1198831015840(119897) + ℎ119883(119897) = 0 we deduce that the operator

minus(119889119889119909)[120588(119909)(119889119889119909)] + 119876(119909) is self-adjoint and the problem(33) is a regular Sturm-Liouville eigenvalue problem and sothe eigenfunctions119883

119899(119909) 119899 = 1 2 are orthogonal with the

weighted function 120588(119909) for 119909 isin [0 119897]

With the aid of Lemma 4 and identity (24) we define abilinear formB(119902 119908) 119871

2(0 1) times 119871

2(0 1) rarr 119877 by

B (119902 119908) = intΩ119879

1198882119902 (119909) 120593 (119908) (119909 119905) 119889119909 119889119905 (36)

where 120593(119908) is the solution of the adjoint problem (31) for119908 isin 119871

2(0 119897) and 119888

2= 119888(119909 119905 119902

2) is a definite solution of

the forward problem (2)ndash(5) for any given 1199022

isin 119878ad and 1198882

is nonnegative and bounded for (119909 119905) isin Ω119879which can be

regarded as a weighted function of the bilinear form It isnoticeable that B is defined with 119902(119909) and 119908(119909) which isdifferent from those definitions given in [6 17]

By using maximum-minimum principle in the adjointproblem (31) we know that the solution 120593 = 120593(119908)(119909 119905) takesnonnegative values as long as 119908 = 119908(119909) isin 119871

2(0 119897) taking

nonnegative values on 119909 isin [0 119897] In addition by generaltheory of parabolic equation [8] there exists positive constant1198721such that

100381710038171003817100381712059310038171003817100381710038171198712(Ω119879)

le 11987211199081198712(0119897) (37)

Then we get the following lemma

Lemma 6 The functional B defined by (36) is a boundedbilinear form

Proof In fact B is linear on 119902 from (36) and together withthe expressions (32) and (34) it follows that 120593(119908) is linearon 119908 and then B is linear on 119908 too Next using Holderinequality for the right-hand side of (36) we have

1003816100381610038161003816B (119902 119908)1003816100381610038161003816 le 119872

1(intΩ119879

[119902 (119909)]2119889119909 119889119905)

12

times (intΩ119879

[120593 (119908)]2119889119909 119889119905)

12

= 1198722radic119879

100381710038171003817100381711990210038171003817100381710038171198712(0119897)

100381710038171003817100381712059310038171003817100381710038171198712(Ω119879)

le 1198723

100381710038171003817100381711990210038171003817100381710038171198712(0119897)

1199081198712(0119897)

(38)

where 1198723

= 11987211198722radic119879 and 119872

2here is a positive constant

depending on the domain and the prescribed solution 1198882 then

B is bounded on 119902 and 119908 The proof is over

By definition (36) and Lemmas 4 and 6 we define anadmissible space for the inputs as 119882 = 119908 isin 119871

2(0 119897) 119908 ge

0 1199082le 119864 and a norm for the source coefficient via

100381710038171003817100381711990210038171003817100381710038172

= sup119908isin119882 119908 = 0

1003816100381610038161003816B (119902 119908)1003816100381610038161003816

1003817100381710038171003817119888210038171003817100381710038172

1199082

(39)

where 119864 gt 0 is a constant and 11988822

= (intΩ119879

[119888(119909 119905 1199022)]2

119889119909 119889119905)12

It is not difficult to testify that the above definition is well-defined by which we can construct a conditional stability forthe inversion of the source coefficient based on the integralidentity (24)

Theorem 7 Suppose that the assumptions (A) (B) and (C1)are satisfied and (119888

119894 119902119894) (119894 = 1 2) are two pairs of solutions to

the inverse problem (2)ndash(6) corresponding to the data 119888119894

0and

119892119894(119894 = 1 2) and 120579

119894(119894 = 1 2) are the additional observations

Then for 119902 isin 119878119886119889

and 119908 isin 119882 there exists constant 119872 =

119872(119863 V Ω119879 1205792) such that

10038171003817100381710038171199021 minus 1199022

10038171003817100381710038172

le 119872(10038171003817100381710038171205791 minus 120579

2

10038171003817100381710038172+

100381710038171003817100381710038171198881

0minus 1198882

0

100381710038171003817100381710038172+

10038171003817100381710038171198921 minus 1198922

10038171003817100381710038172)

(40)

6 Abstract and Applied Analysis

Proof Firstly by assertion (1) of Theorem 1 there is

[120579 (119909)]2le [119888 (119909 119905)]

2le [1198880 (119909)]2

0 lt 119909 lt 119897 0 lt 119905 lt 119879

(41)

for any solution 119888(119909 119905) of the forward problem (2)ndash(5) withthe final function 120579(119909) = 119888(119909 119879) and the initial function1198880(119909) = 119888(119909 0) Then noting condition (A) we have for the

solution 1198882= 119888(119909 119905 119902

2)

1

1198720radic119879

le1

1003817100381710038171003817119888010038171003817100381710038172

radic119879

le1

1003817100381710038171003817119888210038171003817100381710038172

le1

1205792radic119879

le1

120576radic119879

(42)

Next by definition (39) and utilizing the integral identity (24)and Cauchy-Schwartz inequality we get

10038171003817100381710038171199021 minus 1199022

10038171003817100381710038172le

1

1205792radic119879

sdot 10038171003817100381710038171205791 minus 120579

2

10038171003817100381710038172sup119908isin119882 119908 = 0

1199082

1199082

+100381710038171003817100381710038171198881

0minus 1198882

0

100381710038171003817100381710038172sup119908isin119882 119908 = 0

1003817100381710038171003817120593(119909 0)10038171003817100381710038172

1199082

+10038171003817100381710038171198921 minus 119892

2

10038171003817100381710038172sup119908isin119882 119908 = 0

1003817100381710038171003817120593 (0 119905)10038171003817100381710038172

1199082

(43)

Also by the general theory of 1D parabolic equation we knowthat there exists a constant119872 gt 0 depending on119863 V and thedomain Ω

119879such that

1003817100381710038171003817120593(119909 0)10038171003817100381710038172

1003817100381710038171003817120593(0 119905)

10038171003817100381710038172le 1198721199082 (44)

Therefore by setting

119872 =1

1205792radic119879

max 1119872 (45)

and noting (43) it follows that (40) is valid The proof is over

Remark 8 This theorem shows a conditional Lipschitz sta-bility for the inverse problem (2)ndash(6) Furthermore by thistheorem we can see that the uniqueness in the meaning of 1198712for the inverse problem can be easily deduced by the stabilityestimate (40)

Conflict of Interests

The authors declare that there is no conflict of interests withany commercial identities regarding the publication of thispaper

Acknowledgments

The authors thank the anonymous referees for valuablecommentsThiswork is supported by the Basic andAdvancedResearch Program of Henan Science Committee (Research

on Some Problems in the Category of Quantic Lattices)and the Research Award for Teachers in Nanyang NormalUniversity (no nynu200749) and is partially supported by theNational Natural Science Foundation of China (nos 11071148and 11371231)

References

[1] P DuChateau ldquoMonotonicity and invertibility of coefficient-to-datamappings for parabolic inverse problemsrdquo SIAM Journal onMathematical Analysis vol 26 no 6 pp 1473ndash1487 1995

[2] P DuChateau ldquoAn inverse problem for the hydraulic propertiesof porous mediardquo SIAM Journal on Mathematical Analysis vol28 no 3 pp 611ndash632 1997

[3] P DuChateau ldquoAn adjoint method for proving identifiability ofcoefficients in parabolic equationsrdquo Journal of Inverse and Ill-Posed Problems vol 21 no 5 pp 639ndash663 2013

[4] A Hasanov P DuChateau and B Pektas ldquoAn adjoint problemapproach and coarse-fine mesh method for identification of thediffusion coefficient in a linear parabolic equationrdquo Journal ofInverse and Ill-Posed Problems vol 14 no 5 pp 435ndash463 2006

[5] Y H Ou A Hasanov and Z H Liu ldquoInverse coefficientproblems for nonlinear parabolic differential equationsrdquo ActaMathematica Sinica vol 24 no 10 pp 1617ndash1624 2008

[6] G Li D Yao and F Yang ldquoAn inverse problem of identifyingsource coefficient in solute transportationrdquo Journal of Inverseand Ill-Posed Problems vol 16 no 1 pp 51ndash63 2008

[7] N-Z Sun Mathematical Modeling of Groundwater PollutionSpringer New York NY USA 1996

[8] J R Cannon The One-Dimensional Heat Equation vol 23Addison-Wesley London UK 1984

[9] I Bushuyev ldquoGlobal uniqueness for inverse parabolic problemswith final observationrdquo Inverse Problems vol 11 no 4 pp L11ndashL16 1995

[10] M Choulli and M Yamamoto ldquoConditional stability in deter-mining a heat sourcerdquo Journal of Inverse and Ill-Posed Problemsvol 12 no 3 pp 233ndash243 2004

[11] V Isakov ldquoSome inverse problems for the diffusion equationrdquoInverse Problems vol 15 no 1 pp 3ndash10 1999

[12] E G Savateev and R Riganti ldquoInverse problem for thenonlinear heat equation with the final overdeterminationrdquoMathematical and Computer Modelling vol 22 no 1 pp 29ndash431995

[13] J R Cannon and P DuChateau ldquoStructural identification of anunknown source term in a heat equationrdquo Inverse Problems vol14 no 3 pp 535ndash551 1998

[14] G Li ldquoData compatibility and conditional stability for aninverse source problem in the heat equationrdquo Applied Mathe-matics and Computation vol 173 no 1 pp 566ndash581 2006

[15] R P SperbMaximumPrinciples andTheir Applications vol 157Academic Press New York NY USA 1981

[16] R Courant and D Hilbert Methods of Mathematical Physics IInterscience New York NY USA 1953

[17] L Gongsheng and M Yamamoto ldquoStability analysis for deter-mining a source term in a 1-D advection-dispersion equationrdquoJournal of Inverse and Ill-Posed Problems vol 14 no 2 pp 147ndash155 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

In what follows we cope with problem (31) instead ofproblem (30) For this adjoint problem we can see belowthat its solution is really determined by the unique input119908 = 119908(119909)

42 Lipschitz Stability Firstly by applying variable separa-tion method and Sturm-Liouville eigenvalue theory [16] wecan get an explicit expression of the solution for the adjointproblem (31)

Lemma 4 Suppose that the source coefficient 119902(119909) isin 119878119886119889

andthe hydraulic parameters 119863 and V are positive then there is

120593 (119908) (119909 119905) =

infin

sum

119899=1

119886119899exp (minus120582

119899119863119905)119883

119899 (119909) (32)

where120582119899119883119899(119909) (119899 = 1 2 ) are eigenvalues and correspond-

ing eigenfunctions of the following Sturm-Liouville problemrespectively

11988310158401015840

+ ℎ1198831015840minus

1199021 (119909)

119863119883 + 120582119883 = 0

1198831015840(0) = 0 119883

1015840(119897) + ℎ119883 (119897) = 0

(33)

119886119899=

(119908119883119899)

(119883119899 119883119899)

=

int119897

0120588 (119909)119908 (119909)119883119899 (119909) 119889119909

10038171003817100381710038171198831198991003817100381710038171003817

2

2

(34)

where 120588(119909) = exp(ℎ119909) is the weighted function of theeigenvalue problem (33) and the eigenfunctions series 119883

119899(119909)

are orthogonal and complete on 1198712(0 119897) with the weighted

function 120588(119909)

Remark 5 By multiplying 120588(119909) = exp(ℎ119909) at the two sides ofthe differential equation in (33) we have

minus119889

119889119909(120588 (119909)119883

1015840) + 119876 (119909)119883 = 120582120588 (119909)119883 (35)

where 119876(119909) = 120588(119909)1199021(119909)119863 ge 0 for 119909 isin [0 119897] Combing

with the homogeneous boundary conditions 1198831015840(0) = 0

and 1198831015840(119897) + ℎ119883(119897) = 0 we deduce that the operator

minus(119889119889119909)[120588(119909)(119889119889119909)] + 119876(119909) is self-adjoint and the problem(33) is a regular Sturm-Liouville eigenvalue problem and sothe eigenfunctions119883

119899(119909) 119899 = 1 2 are orthogonal with the

weighted function 120588(119909) for 119909 isin [0 119897]

With the aid of Lemma 4 and identity (24) we define abilinear formB(119902 119908) 119871

2(0 1) times 119871

2(0 1) rarr 119877 by

B (119902 119908) = intΩ119879

1198882119902 (119909) 120593 (119908) (119909 119905) 119889119909 119889119905 (36)

where 120593(119908) is the solution of the adjoint problem (31) for119908 isin 119871

2(0 119897) and 119888

2= 119888(119909 119905 119902

2) is a definite solution of

the forward problem (2)ndash(5) for any given 1199022

isin 119878ad and 1198882

is nonnegative and bounded for (119909 119905) isin Ω119879which can be

regarded as a weighted function of the bilinear form It isnoticeable that B is defined with 119902(119909) and 119908(119909) which isdifferent from those definitions given in [6 17]

By using maximum-minimum principle in the adjointproblem (31) we know that the solution 120593 = 120593(119908)(119909 119905) takesnonnegative values as long as 119908 = 119908(119909) isin 119871

2(0 119897) taking

nonnegative values on 119909 isin [0 119897] In addition by generaltheory of parabolic equation [8] there exists positive constant1198721such that

100381710038171003817100381712059310038171003817100381710038171198712(Ω119879)

le 11987211199081198712(0119897) (37)

Then we get the following lemma

Lemma 6 The functional B defined by (36) is a boundedbilinear form

Proof In fact B is linear on 119902 from (36) and together withthe expressions (32) and (34) it follows that 120593(119908) is linearon 119908 and then B is linear on 119908 too Next using Holderinequality for the right-hand side of (36) we have

1003816100381610038161003816B (119902 119908)1003816100381610038161003816 le 119872

1(intΩ119879

[119902 (119909)]2119889119909 119889119905)

12

times (intΩ119879

[120593 (119908)]2119889119909 119889119905)

12

= 1198722radic119879

100381710038171003817100381711990210038171003817100381710038171198712(0119897)

100381710038171003817100381712059310038171003817100381710038171198712(Ω119879)

le 1198723

100381710038171003817100381711990210038171003817100381710038171198712(0119897)

1199081198712(0119897)

(38)

where 1198723

= 11987211198722radic119879 and 119872

2here is a positive constant

depending on the domain and the prescribed solution 1198882 then

B is bounded on 119902 and 119908 The proof is over

By definition (36) and Lemmas 4 and 6 we define anadmissible space for the inputs as 119882 = 119908 isin 119871

2(0 119897) 119908 ge

0 1199082le 119864 and a norm for the source coefficient via

100381710038171003817100381711990210038171003817100381710038172

= sup119908isin119882 119908 = 0

1003816100381610038161003816B (119902 119908)1003816100381610038161003816

1003817100381710038171003817119888210038171003817100381710038172

1199082

(39)

where 119864 gt 0 is a constant and 11988822

= (intΩ119879

[119888(119909 119905 1199022)]2

119889119909 119889119905)12

It is not difficult to testify that the above definition is well-defined by which we can construct a conditional stability forthe inversion of the source coefficient based on the integralidentity (24)

Theorem 7 Suppose that the assumptions (A) (B) and (C1)are satisfied and (119888

119894 119902119894) (119894 = 1 2) are two pairs of solutions to

the inverse problem (2)ndash(6) corresponding to the data 119888119894

0and

119892119894(119894 = 1 2) and 120579

119894(119894 = 1 2) are the additional observations

Then for 119902 isin 119878119886119889

and 119908 isin 119882 there exists constant 119872 =

119872(119863 V Ω119879 1205792) such that

10038171003817100381710038171199021 minus 1199022

10038171003817100381710038172

le 119872(10038171003817100381710038171205791 minus 120579

2

10038171003817100381710038172+

100381710038171003817100381710038171198881

0minus 1198882

0

100381710038171003817100381710038172+

10038171003817100381710038171198921 minus 1198922

10038171003817100381710038172)

(40)

6 Abstract and Applied Analysis

Proof Firstly by assertion (1) of Theorem 1 there is

[120579 (119909)]2le [119888 (119909 119905)]

2le [1198880 (119909)]2

0 lt 119909 lt 119897 0 lt 119905 lt 119879

(41)

for any solution 119888(119909 119905) of the forward problem (2)ndash(5) withthe final function 120579(119909) = 119888(119909 119879) and the initial function1198880(119909) = 119888(119909 0) Then noting condition (A) we have for the

solution 1198882= 119888(119909 119905 119902

2)

1

1198720radic119879

le1

1003817100381710038171003817119888010038171003817100381710038172

radic119879

le1

1003817100381710038171003817119888210038171003817100381710038172

le1

1205792radic119879

le1

120576radic119879

(42)

Next by definition (39) and utilizing the integral identity (24)and Cauchy-Schwartz inequality we get

10038171003817100381710038171199021 minus 1199022

10038171003817100381710038172le

1

1205792radic119879

sdot 10038171003817100381710038171205791 minus 120579

2

10038171003817100381710038172sup119908isin119882 119908 = 0

1199082

1199082

+100381710038171003817100381710038171198881

0minus 1198882

0

100381710038171003817100381710038172sup119908isin119882 119908 = 0

1003817100381710038171003817120593(119909 0)10038171003817100381710038172

1199082

+10038171003817100381710038171198921 minus 119892

2

10038171003817100381710038172sup119908isin119882 119908 = 0

1003817100381710038171003817120593 (0 119905)10038171003817100381710038172

1199082

(43)

Also by the general theory of 1D parabolic equation we knowthat there exists a constant119872 gt 0 depending on119863 V and thedomain Ω

119879such that

1003817100381710038171003817120593(119909 0)10038171003817100381710038172

1003817100381710038171003817120593(0 119905)

10038171003817100381710038172le 1198721199082 (44)

Therefore by setting

119872 =1

1205792radic119879

max 1119872 (45)

and noting (43) it follows that (40) is valid The proof is over

Remark 8 This theorem shows a conditional Lipschitz sta-bility for the inverse problem (2)ndash(6) Furthermore by thistheorem we can see that the uniqueness in the meaning of 1198712for the inverse problem can be easily deduced by the stabilityestimate (40)

Conflict of Interests

The authors declare that there is no conflict of interests withany commercial identities regarding the publication of thispaper

Acknowledgments

The authors thank the anonymous referees for valuablecommentsThiswork is supported by the Basic andAdvancedResearch Program of Henan Science Committee (Research

on Some Problems in the Category of Quantic Lattices)and the Research Award for Teachers in Nanyang NormalUniversity (no nynu200749) and is partially supported by theNational Natural Science Foundation of China (nos 11071148and 11371231)

References

[1] P DuChateau ldquoMonotonicity and invertibility of coefficient-to-datamappings for parabolic inverse problemsrdquo SIAM Journal onMathematical Analysis vol 26 no 6 pp 1473ndash1487 1995

[2] P DuChateau ldquoAn inverse problem for the hydraulic propertiesof porous mediardquo SIAM Journal on Mathematical Analysis vol28 no 3 pp 611ndash632 1997

[3] P DuChateau ldquoAn adjoint method for proving identifiability ofcoefficients in parabolic equationsrdquo Journal of Inverse and Ill-Posed Problems vol 21 no 5 pp 639ndash663 2013

[4] A Hasanov P DuChateau and B Pektas ldquoAn adjoint problemapproach and coarse-fine mesh method for identification of thediffusion coefficient in a linear parabolic equationrdquo Journal ofInverse and Ill-Posed Problems vol 14 no 5 pp 435ndash463 2006

[5] Y H Ou A Hasanov and Z H Liu ldquoInverse coefficientproblems for nonlinear parabolic differential equationsrdquo ActaMathematica Sinica vol 24 no 10 pp 1617ndash1624 2008

[6] G Li D Yao and F Yang ldquoAn inverse problem of identifyingsource coefficient in solute transportationrdquo Journal of Inverseand Ill-Posed Problems vol 16 no 1 pp 51ndash63 2008

[7] N-Z Sun Mathematical Modeling of Groundwater PollutionSpringer New York NY USA 1996

[8] J R Cannon The One-Dimensional Heat Equation vol 23Addison-Wesley London UK 1984

[9] I Bushuyev ldquoGlobal uniqueness for inverse parabolic problemswith final observationrdquo Inverse Problems vol 11 no 4 pp L11ndashL16 1995

[10] M Choulli and M Yamamoto ldquoConditional stability in deter-mining a heat sourcerdquo Journal of Inverse and Ill-Posed Problemsvol 12 no 3 pp 233ndash243 2004

[11] V Isakov ldquoSome inverse problems for the diffusion equationrdquoInverse Problems vol 15 no 1 pp 3ndash10 1999

[12] E G Savateev and R Riganti ldquoInverse problem for thenonlinear heat equation with the final overdeterminationrdquoMathematical and Computer Modelling vol 22 no 1 pp 29ndash431995

[13] J R Cannon and P DuChateau ldquoStructural identification of anunknown source term in a heat equationrdquo Inverse Problems vol14 no 3 pp 535ndash551 1998

[14] G Li ldquoData compatibility and conditional stability for aninverse source problem in the heat equationrdquo Applied Mathe-matics and Computation vol 173 no 1 pp 566ndash581 2006

[15] R P SperbMaximumPrinciples andTheir Applications vol 157Academic Press New York NY USA 1981

[16] R Courant and D Hilbert Methods of Mathematical Physics IInterscience New York NY USA 1953

[17] L Gongsheng and M Yamamoto ldquoStability analysis for deter-mining a source term in a 1-D advection-dispersion equationrdquoJournal of Inverse and Ill-Posed Problems vol 14 no 2 pp 147ndash155 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

Proof Firstly by assertion (1) of Theorem 1 there is

[120579 (119909)]2le [119888 (119909 119905)]

2le [1198880 (119909)]2

0 lt 119909 lt 119897 0 lt 119905 lt 119879

(41)

for any solution 119888(119909 119905) of the forward problem (2)ndash(5) withthe final function 120579(119909) = 119888(119909 119879) and the initial function1198880(119909) = 119888(119909 0) Then noting condition (A) we have for the

solution 1198882= 119888(119909 119905 119902

2)

1

1198720radic119879

le1

1003817100381710038171003817119888010038171003817100381710038172

radic119879

le1

1003817100381710038171003817119888210038171003817100381710038172

le1

1205792radic119879

le1

120576radic119879

(42)

Next by definition (39) and utilizing the integral identity (24)and Cauchy-Schwartz inequality we get

10038171003817100381710038171199021 minus 1199022

10038171003817100381710038172le

1

1205792radic119879

sdot 10038171003817100381710038171205791 minus 120579

2

10038171003817100381710038172sup119908isin119882 119908 = 0

1199082

1199082

+100381710038171003817100381710038171198881

0minus 1198882

0

100381710038171003817100381710038172sup119908isin119882 119908 = 0

1003817100381710038171003817120593(119909 0)10038171003817100381710038172

1199082

+10038171003817100381710038171198921 minus 119892

2

10038171003817100381710038172sup119908isin119882 119908 = 0

1003817100381710038171003817120593 (0 119905)10038171003817100381710038172

1199082

(43)

Also by the general theory of 1D parabolic equation we knowthat there exists a constant119872 gt 0 depending on119863 V and thedomain Ω

119879such that

1003817100381710038171003817120593(119909 0)10038171003817100381710038172

1003817100381710038171003817120593(0 119905)

10038171003817100381710038172le 1198721199082 (44)

Therefore by setting

119872 =1

1205792radic119879

max 1119872 (45)

and noting (43) it follows that (40) is valid The proof is over

Remark 8 This theorem shows a conditional Lipschitz sta-bility for the inverse problem (2)ndash(6) Furthermore by thistheorem we can see that the uniqueness in the meaning of 1198712for the inverse problem can be easily deduced by the stabilityestimate (40)

Conflict of Interests

The authors declare that there is no conflict of interests withany commercial identities regarding the publication of thispaper

Acknowledgments

The authors thank the anonymous referees for valuablecommentsThiswork is supported by the Basic andAdvancedResearch Program of Henan Science Committee (Research

on Some Problems in the Category of Quantic Lattices)and the Research Award for Teachers in Nanyang NormalUniversity (no nynu200749) and is partially supported by theNational Natural Science Foundation of China (nos 11071148and 11371231)

References

[1] P DuChateau ldquoMonotonicity and invertibility of coefficient-to-datamappings for parabolic inverse problemsrdquo SIAM Journal onMathematical Analysis vol 26 no 6 pp 1473ndash1487 1995

[2] P DuChateau ldquoAn inverse problem for the hydraulic propertiesof porous mediardquo SIAM Journal on Mathematical Analysis vol28 no 3 pp 611ndash632 1997

[3] P DuChateau ldquoAn adjoint method for proving identifiability ofcoefficients in parabolic equationsrdquo Journal of Inverse and Ill-Posed Problems vol 21 no 5 pp 639ndash663 2013

[4] A Hasanov P DuChateau and B Pektas ldquoAn adjoint problemapproach and coarse-fine mesh method for identification of thediffusion coefficient in a linear parabolic equationrdquo Journal ofInverse and Ill-Posed Problems vol 14 no 5 pp 435ndash463 2006

[5] Y H Ou A Hasanov and Z H Liu ldquoInverse coefficientproblems for nonlinear parabolic differential equationsrdquo ActaMathematica Sinica vol 24 no 10 pp 1617ndash1624 2008

[6] G Li D Yao and F Yang ldquoAn inverse problem of identifyingsource coefficient in solute transportationrdquo Journal of Inverseand Ill-Posed Problems vol 16 no 1 pp 51ndash63 2008

[7] N-Z Sun Mathematical Modeling of Groundwater PollutionSpringer New York NY USA 1996

[8] J R Cannon The One-Dimensional Heat Equation vol 23Addison-Wesley London UK 1984

[9] I Bushuyev ldquoGlobal uniqueness for inverse parabolic problemswith final observationrdquo Inverse Problems vol 11 no 4 pp L11ndashL16 1995

[10] M Choulli and M Yamamoto ldquoConditional stability in deter-mining a heat sourcerdquo Journal of Inverse and Ill-Posed Problemsvol 12 no 3 pp 233ndash243 2004

[11] V Isakov ldquoSome inverse problems for the diffusion equationrdquoInverse Problems vol 15 no 1 pp 3ndash10 1999

[12] E G Savateev and R Riganti ldquoInverse problem for thenonlinear heat equation with the final overdeterminationrdquoMathematical and Computer Modelling vol 22 no 1 pp 29ndash431995

[13] J R Cannon and P DuChateau ldquoStructural identification of anunknown source term in a heat equationrdquo Inverse Problems vol14 no 3 pp 535ndash551 1998

[14] G Li ldquoData compatibility and conditional stability for aninverse source problem in the heat equationrdquo Applied Mathe-matics and Computation vol 173 no 1 pp 566ndash581 2006

[15] R P SperbMaximumPrinciples andTheir Applications vol 157Academic Press New York NY USA 1981

[16] R Courant and D Hilbert Methods of Mathematical Physics IInterscience New York NY USA 1953

[17] L Gongsheng and M Yamamoto ldquoStability analysis for deter-mining a source term in a 1-D advection-dispersion equationrdquoJournal of Inverse and Ill-Posed Problems vol 14 no 2 pp 147ndash155 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of