Lifting Scheme for the Numerical Solution of Fisher’s

Copyright © 2020 by Modern Scientific Press Company, Florida, USA

International Journal of Modern Mathematical Sciences, 2020, 18(1): 11-30

International Journal of Modern Mathematical Sciences

Journal homepage: www.ModernScientificPress.com/Journals/ijmms.aspx

ISSN: 2166-286X

Florida, USA

Article

Lifting Scheme for the Numerical Solution of Fisher’s Equations

Using Different Wavelet Filter Coefficients

S. C. Shiralashettia*, L. M. Angadib, A. B. Deshic

a Department of Mathematics, Karnatak University Dharwad – 580003, India

b Department of Mathematics, Govt. First Grade College, Chikodi – 591201, India

c Department of Mathematics, KLE CET, Chikodi – 591201, India

*Author to whom correspondence should be addressed; E-Mail:[email protected]

Article history: Received 19 October 2019; Revised 1 February 2020; Accepted 12 February 2020;

Published 20 February 2020.

Abstract: Nonlinear partial differential equations appear in a wide variety of scientific

applications such as plasma physics, solid state physics, optical fibers, biology, fluid

dynamics and chemical kinetics. In this paper, we proposed Lifting scheme for the

numerical solution of Fisher’s equations by different wavelet filter coefficients. The

obtained numerical results using this scheme are compared with the exact solution to reveal the

accuracy and also speed up convergence in lesser computational time as compared with

existing scheme. Some test problems are presented for the applicability and validity of the

schemes.

Keywords: Fisher’s equations; Haar, Daubechies and Biorthogonal wavelet filter

coefficients; Lifting scheme.

AMS Subject Classification: 65T60, 97N40, 46G15, 49K20

1. Introduction

Various phenomena’s in engineering, biology, fluid mechanics and other sciences can be

modeled as both linear and nonlinear partial differential equations (PDEs) [1]. In last decade there has

been much interest for finding the numerical solution of such type of partial differential equations

(PDEs). In general PDEs with or without reaction terms are used as a primary tool to model a broad

mailto:[email protected]

Int. J. Modern Math. Sci. 2020, 18(1): 11-30


12

class of phenomena occurring in physical and biological sciences, such as Fisher’s equation. This

equation proposed by Fisher in the year 1930 as a model for the propagation of a mutantgene and this

combines diffussion with logistic nonlinearty. Moreover, the same equation occurs in logestic

population growth models, flame propagation, autocatalytic chemical reactions and branching

Brownian motion processes.

In recent times, a number of the iterative methods are used for the numerical and analytical

solutions of linear and nonlinear partial differential equations. Some of them are viz., Adomian

decomposition method [2], He’s variational iteration technique [3] and the homotopy perturbation

method [4], etc. Mathematical models of basic flow equations which express unsteady transport

problems are governed by a single or a system of nonlinear PDEs.

Wavelet analysis assumed significance due to successful applications in signal and image

processing during the 1980s. The smooth orthonormal basis obtained by the translation and dilation of

a single function in a hierarchical fashion proved very valuable to build up compression algorithms for

signals and images up to a selected threshold of relevant amplitudes. Some of the most important

contributors to this theory are: multiresolution signal processing used in computer vision; sub band

coding, developed for speech and image compression; and wavelet series expansion, developed in

applied mathematics.

Some of the previous works on wavelet based methods can be establish in [5-6]. In recent

times a few of the researchers have developed various wavelet based multigrid methods for example:

Daubechies Wavelet based Multigrid and Full Approximation Scheme [7], Biorthogonal wavelet based

full approximation scheme [8] etc. The wavelet based full approximation scheme (WFAS) has

exposed to be a very competent and constructive method for numerous problems associated to

computational science and engineering fields [9]. These methods can be either used as an iterative

solver or as a preconditioning methods, contribution in many cases a better performance than some of

the most new and existing FAS algorithms. Due to the competence and potentiality of WFAS,

researches further have been conceded out for its improvement. In order to understand this task, work

build that is orthogonal/biorthogonal discrete wavelet transform by lifting scheme [10]. Sweldens is

introduced wavelet based lifting technique [11], which permits several improvements on the properties

of existing wavelet transforms. The technique has some numerical benefits as a reduced number of

operations which are essential in the context of the iterative solvers. Obviously all attempts to shorten

the wavelet solutions for PDE are welcome. In PDE, matrices arising from system are intense with

non-smooth diagonal and smooth away from the diagonal. This efficiency of the matrix transforms into

compactness using wavelet transform and it leads to propose the effective lifting scheme using

differernt wavelets.



13

Lifting scheme is a new approach to build the so-called second generation wavelets that are not

necessarily translations and dilations of one function. The latter we refer to as a first generation

wavelets or classical methods. The lifting scheme has some additional advantages in association with

the classical wavelets. This transform works for signals of an arbitrary size with accurate treatment of

boundaries. A further characteristic of the lifting scheme is that all constructions are resulting in the

spatial domain. This is in distinction to the conventional approach, which relies deeply on the

frequency domain. The two most important advantageous are:

i. It leads to a more intuitively appealing treatment improved suited to those in attracted in

applications than mathematical foundations.

ii. It makes a computational time most favorable and from time to time increasing the speed of

calculations.

The lifting scheme starts with a set of well-known filters, subsequently lifting steps are used an

effort to improve (lift) the properties of an equivalent wavelet decomposition. This process has a few

mathematical benefits as a condensed number of operations which are crucial in the circumstance of

the iterative solvers. In addition to this, the current paper illustrates that the relevance of the lifting

scheme using different wavelets for the numerical solution of Fisher’s equations.

The present paper is organized as follows: In section 2, Preliminaries of wavelet filter coefficients and

lifting scheme. The method of solution describes in section 3. In section 4 provides numerical

simulation of test problems and finally, in section 5 conclusion of the proposed work is given.

2. Preliminaries of Wavelet Filter Coefficients and Lifting Scheme

The lifting scheme starts with a set of familiar filters; thereafter lifting steps are used in attempt

to advance the properties of corresponding wavelet decomposition.

Now, we have discussed about different wavelet filters as follows:

2.1. Haar Wavelet Filter Coefficients

We are familiar with that of low pass filter coefficients 0 1

1 1, ,

2 2

T

Ta a

and high pass

filter coefficients 0 1

1 1, ,

2 2

T

Tb b

play a significant role in decomposition. Thus it is natural to

wonder that it achievable to model the decomposition in terms of linear transformations i.e. matrices.

Furthermore, while digital signals and images are composed of discrete data, we want to discrete

analog of the decomposition algorithm so that we can process signal and image data.



14

2.2. Daubechies Wavelet Filter Coefficients

Daubechies introduced scaling functions having the shortest feasible support. The scaling

function N has support 0, 1N , while the equivalent wavelet

N has carry in the

interval 1 / 2, / 2N N .

For Daubechies wavelets, the low pass filter coefficients

0 1 2 3

1 3 3 3 3 3 1 3

4 2 4 2 4 2 4 2, , , , , ,

T

Ta a a a

and high pass filter coefficients 0 1 2 3

1 3 3 3 3 3 1 3

4 2 4 2 4 2 4 2, , , , , ,

T

Tb b b b

2.3. Biorthogonal (CDF (2,2)) Wavelets

Let’s consider the (5, 3) biorthogonal spline wavelet filter pair, the low pass filter pair are

1 0 1

1 1 1( , , ) , ,

2 2 2 2 2a a a

and 2 1 0 1 2

1 1 3 1 1( , , , , ) , , , ,

4 2 2 2 2 2 2 2 4 2a a a a a

.

But, we have 1 1( 1) and ( 1)k k

k k k kb a b a ,

the high pass filter pair are

0 1 2

1 1 1, ,

2 2 2 2 2b b b

&

1 0 1 2 3

1 1 3 1 1, , , ,

4 2 2 2 2 2 2 2 4 2b b b b b

2.4. Foundations of Lifting Scheme

Consider to numbers a, b as two neighbouring samples of a sequence and then these have

some correlation which we would like to take advantage. The simple linear transform which replaces a

and b by average s and difference d i.e.

&2 2

a b a bs d

The idea is that if a and b are highly correlated, the expected absolute value of their difference

d will be small and can be represented with fever bits. In case that a = b, the difference is simply zero.

We have not lost any information because we can always recover a and b from the gives s and d as:

&2 2

d da s b s

Finally, a wavelet transform built through lifting consists of three steps: split. Predict and

update as given in the Figure 1 [12]



15

Fig. 1. Steps in lifting scheme

Split: Splitting the signal into two disjoint sets of samples.

Predict: If the signal contains some structure, then we can guess correlation between a sample and its

nearest neighbors. i. e. 1 1 1odd P(even )j j jd

Update: Given an even entry, we have predicted that the next odd entry has the identical value, and

stored the difference. Afterward, we update our even entry to reflect our knowledge of the signal. i.e.

1 1 1even U( )j j js d

Detailed algorithms by different wavelets are given in section 3. The general lifting stages for

decomposition and reconstruction of a signal are given in Figure 2.

Fig. 2. Lifting wavelet algorithm

3. Method of Solution

Consider the Fisher’s equation of the form,

(1 ), 0 1 & 0t xxu u u u x t (3.1)

Where is any constant.

Now, we discretizing the equation (3.1) through the finite difference method (FDM) and get the

system of algebraic equations as

Au b (3.2)

where A is N N coefficient matrix, b is N N matrix and u is N N matrix to be determined.

Also 2JN , N is the number of grid points and J is the level of resolution.



16

By solving Eq. (3.2) through the iterative method, we obtain approximate solution.

Approximate solution contains some error, so required solution equals to sum of approximate solution

and error. There are various methods to decrease such error to get the precise solution. A few of them

are HWLS, DWLS BWLS etc. Now we are using the superior procedure called as lifting schemes

using different wavelets. Recently, lifting schemes are helpful in the signal analysis and image

processing in the area of science and engineering. But at present it extends to approximations in the

numerical analysis [9].

Now, we are discussing the algorithm of the lifting schemes as follows.

3.1. Haar Wavelet Lifting Scheme (HWLS)

In Daubechies and Sweldens have shown that every wavelet filter can be decomposed into

lifting steps [10]. Further, details of the advantages as well as other significant structural advantages of

the lifting technique can be existing in [11]. The demonstration of Lifting scheme using Haar wavelet

is presented as:

Decomposition:

Assume the approximate solution jS P similar to as signal and then we apply the HWLS

decomposition (finer to coarser) procedure as,

1 1 1

2 2 1 2 1

1 1

1

1, ,

2

12 and

2

j j jd S S s S d

S s D d

(3.3)

Here this step to conclude, the latest approximation as,

1[ ]S S D . (3.4)

Reconstruction:

Consider Eq. (3.2) and then, we applied the HWLS reconstruction (coarser to finer) procedure as,

1 1

1

1 1 1

2 1 2 2 1

12 , ,

2

1and

2j j j

d D s S

S s d S d S

. (3.5)

This is the required solution of the given equation.

3.2. Daubechies Wavelet Lifting Scheme (DWLS)

As discussed in the above section 3.1, we follow the similar procedure using Daubechies 4th

order wavelet coefficient. The DWLS procedure is as follows;



17

Decomposition:

1

2 1 2

1 1 1

2 1

2 1 1

1

2

1

1

3 ,

3 3 2,

4 4

,

3 1and

2

3 1

2

j j

j

j

j

s S S

d S s s

s s d

S s

D d

(3.6)

Here, we obtain new approximation as,

1[ ]S S D . (3.7)

Reconstruction:

Consider Eq. (3.5) and then, we apply the DWLS reconstruction (coarser to finer) procedure as,

1

2

1

2 1

1 1

1 1

2 1 1

1

2 1 2

2,

3 1

2,

3 1

,

3 3 2and

4 4

3

j j

j j

j

j j

d D

s S

s s d

S d s s

S s S

(3.8)


3.3. Biorthogonal Wavelet Lifting Scheme (BWLS)

As discussed in the earlier sections 3.1 and 3.2, we follow the similar procedure using

biorthogonal wavelet (CDF(2,2)). The BWLS procedure is as follows;

Decomposition:

1

2 2 1 2 2

1 1 1

2 1 1

1

1

1

1,

2

1,

4

1,

2

2

j j j

j j

d S S S

s S d d

D d

S s

(3.9)

At this stage finally, we obtain new signal as,

1[ ]S S D . (3.10)



18

Reconstruction:

Consider Eqn. (3.10) and then, we apply the DWLS reconstruction (coarser to finer) procedure as

1

1

1

1 1 1

2 1 1

1

2 2 1 2 2

1,

2

2 ,

1

4

1) ,

2

j j

j j j

s S

d D

S s d d

S d S S

(3.11)


The coefficients 1

js and

1

jd are the average and thorough coefficients respectively of the

approximate solution au . The new approaches are tested through a variety of problems and the results

are revealed in section 4.

4. Numerical Simulation

In this section, we applied Lifting scheme for the numerical solution of Fisher’s equation and

also show the applicability and competence of HWLS, DWLS and BWLS. The error is computed by

maxma x e au uE , where eu and au are exact and approximate solution respectively.

Test Problem 4.1: Now, we consider the Fisher equation

6 (1 ), 0 1 & 0t xxu u u u x t (4.1.1)

subject to the I.C.: 21

10,xe

xu

(4.1.2)

and B.C.s: 251

1,0te

tu

, 2)51(1

1,1te

tu

(4.1.3)

Which has the exact solution 2)5(1

1,txe

txu

[13].

Now discretizing the Eq. (4.1.1) by using finite difference scheme,

1

1 1

2

26 1

j j jj ji i i j ji i

i i

u u uu uu u

h h

(4.1.4)

1 2

1 12 6 1j j j j j j j

i i i i i i ih u u u u u h u u

, , 1, 2,....i j N .

1

1 1 1 6 2 0j j j j j

i i i i iu u hu u h h hu

(4.1.5)

, 0F i j (4.1.6)



19

Where 1, 1 6 2

1 1jj j j j

F i j u u hu u h h hui i ii i

Which is the system of nonlinear equations, have N N equations with N N unknowns. Solving

Eq. (4.1.6) through the Gauss Seidel iterative method for 4N , we get approximate solution v for u

i.e.

0.486 0.730 0.876 0.948

0.435 0.685 0.848 0.934

0.380 0.639 0.822 0.922

0.319 0.589 0.797 0.913

u

The wavelet based numerical solutions of Eq. (4.1.1) are obtained as per the procedure

explained in section 3 and are as follows,

Assume S u , then apply the HWLS as explained in section 3.1 as,

Decomposition:

1

16 15d S S ,

10.243 0.072 0.250 0.085 0.259 0.100 0.270 0.115d .

1

1

152

ds S ,

10.608 0.912 0.560 0.891 0.509 0.872 0.454 0.855s .

1

1 2S s .

1 0.860 1.290 0.792 1.260 0.720 1.233 0.642 1.209S .

11

2D d .

0.172 0.051 0.177 0.060 0.183 0.071 0.191 0.081D

Reconstruction:

Then apply the HWLS reconstruction procedure as,

12d D ,

10.243 0.072 0.250 0.085 0.259 0.100 0.270 0.115d .

1

1

1

2s S ,

10.608 0.912 0.560 0.891 0.509 0.872 0.454 0.855s .

1 1

15

1

2S s d ,

15 0.486 0.876 0.435 0.848 0.380 0.822 0.319 0.797S .

1

16 15S d S ,

16 0.730 0.948 0.685 0.934 0.639 0.922 0.589 0.913S .



20

0.486 0.730 0.876 0.948

0.435 0.685 0.848 0.934

0.380 0.639 0.822 0.922

0.319 0.589 0.797 0.913

S

This is the required HWLS solution of the given equation.

Similarly, DWLS as discussed in the section 3.2, we follow the same procedure as,

Decomposition:

1

15 163s S S ,

11.750 2.518 1.622 2.465 1.486 2.419 1 .338 2.378s .

1 1 0

16

3 3 2

4 4d S s s

,

10.131 -0.025 0.152 -0.025 0.160 -0.026 0.171 -0.027d .

2 1 1s s d , 2

1.776 2.366 1 .647 2.305 1 .512 2.248 1 .366 2.247s .

2

1

3 1

2S s

,

1 0.919 1 .225 0.853 1 .193 0.783 1 .163 0.707 1 .163S .

13 1

2D d

,

0.253 -0.049 0.293 -0.049 0.310 -0.050 0.331 -0.053D .

Reconstruction:

Then apply the DWLS reconstruction procedure as,

1 2

3 1d D

,

10.131 -0.025 0.152 -0.025 0.160 -0.026 0.171 -0.027d

.

2

1

2

3 1s S

,

11.776 2.366 1 .647 2.305 1 .512 2.248 1 .366 2.247s .

0 2 1s s d , 0

1.750 2.518 1 .622 2.465 1 .486 2.419 1 .338 2.378s

1 2 0

16

3 3 2

4 4S d s s

,

16 0.730 0.948 0.685 0.934 0.639 0.922 0.589 0.913S

.

1

15 163S s S ,



21

15 0.486 0.876 0.435 0.848 0.380 0.822 0.319 0.797S

.

0.486 0.730 0.876 0.948

0.435 0.685 0.848 0.934

0.380 0.639 0.822 0.922

0.319 0.589 0.797 0.913

S

This is the required DWLS solution of the given equation.

And also, BWLS is explained in the section 3.3, we follow the similar procedure as follows

Decomposition:

1

16 15 17

1,

2d S S S

10.144 0.043 0.148 0.049 0.154 0.054 0.064 0.040d

. 1 1 1

15 0

1

4s S d d ,

10.532 0.923 0.483 0.897 0.431 0.874 0.348 0.824s .

11

2D d ,

0.102 0.030 0.105 0.034 0.109 0.039 0.046 0.028D .

1

1 2S s ,

1 0.753 1 .305 0.683 1 .269 0.609 1 .236 0.493 1 .165S

Reconstruction:

Then apply the BWLS reconstruction procedure as,

1

1

1

2s S ,

10.532 0.923 0.483 0.897 0.431 0.874 0.348 0.824s .

12d D ,

10.144 0.043 0.148 0.049 0.154 0.054 0.064 0.040d

.

1 1 1

15 0

1

4S s d d ,

15 0.486 0.876 0.435 0.848 0.380 0.822 0.319 0.797S .

1

16 15 18

1

4S d S S ,

16 0.730 0.948 0.685 0.934 0.639 0.922 0.589 0.913S .

0.486 0.730 0.876 0.948

0.435 0.685 0.848 0.934

0.380 0.639 0.822 0.922

0.319 0.589 0.797 0.913

S

This is the required BWLS solution of the given equation.

The obtained numerical solutions and compared with exact solution are presented in table 1 and

figure 1. The maximum absolute errors with CPU time of the methods are presented in table 2.



22

Table 1. Comparison of numerical solutions with exact solution of the test problem 4.1.

00.5

1

0

0.5

10

0.5

1

xt

HW

LS

00.5

1

0

0.5

10

0.5

1

xt

DW

LS

00.5

1

0

0.5

10

0.5

1

xt

BW

LS

00.5

1

0

0.5

10

0.5

1

xt

Exact

00.5

1

0

0.5

10

0.5

1

xt

HW

LS

00.5

1

0

0.5

10

0.5

1

xt

DW

LS

00.5

1

0

0.5

10

0.5

1

xt

BW

LS

00.5

1

0

0.5

10

0.5

1

xt

Exact

(a) N N = 8 8 (b) N N = 16 16

Fig. 1. Comparison of numerical solutions with exact solution of the test problem 4.1 for

(a) N N =8 8 & (b) 16 16N N

x t FDM HWLS DWLS BWLS EXACT

0.2 0.2 0.4863946 0.4863946 0.4863946 0.4863946 0.4760647

0.4 0.2 0.7297687 0.7297687 0.7297687 0.7297687 0.7364196

0.6 0.2 0.8761914 0.8761914 0.8761914 0.8761914 0.8886377

0.8 0.2 0.9477997 0.9477997 0.9477997 0.9477997 0.9567162

0.2 0.4 0.4351551 0.4351551 0.4351551 0.4351551 0.4168721

0.4 0.4 0.6851006 0.6851006 0.6851006 0.6851006 0.6922546

0.6 0.4 0.8482495 0.8482495 0.8482495 0.8482495 0.8665033

0.8 0.4 0.9337046 0.9337046 0.9337046 0.9337046 0.9475134

0.2 0.6 0.3797588 0.3797588 0.3797588 0.3797588 0.3584269

0.4 0.6 0.6386589 0.6386589 0.6386589 0.6386589 0.6434990

0.6 0.6 0.8220533 0.8220533 0.8220533 0.8220533 0.8405723

0.8 0.6 0.9218497 0.9218497 0.9218497 0.9218497 0.9364521

0.2 0.8 0.3186782 0.3186782 0.3186782 0.3186782 0.3023174

0.4 0.8 0.5886171 0.5886171 0.5886171 0.5886171 0.5906303

0.6 0.8 0.7974316 0.7974316 0.7974316 0.7974316 0.8104492

0.8 0.8 0.9126654 0.9126654 0.9126654 0.9126654 0.9232025



23

Table 2. Maximum error and CPU time (in seconds) of the methods of the test problem 4.1

Test Problem 4.2: Next, we consider another Fisher equation

1t xxu u u u (4.2.1)

subject to the I.C.: ,0u x ( is parameter) (4.2.2)

and B.C.s: 0,1

t

t

eu t

e

, 1,

1

t

t

eu t

e

(4.2.3)

Which has the exact solution ,1

t

t

eu x t

e[14]. By applying the methods explained in the

section 3 and in problem 4.4.1, we obtain the numerical solutions for 1 & 1 and compared

with exact solution are presented in table 3 & 5 and figure 2 & 3. The maximum absolute errors with

CPU time of the methods are presented in table 4 & 6.

Case (i): For 1 , By taking 0.99

N N Method maxE Setup time Running time Total time

4 4

FDM 2.1332e-02 2.9144 0.0004 2.9148

HWLS 2.1332e-02 0.0011 0.0017 0.0028

DWLS 2.1332e-02 0.0010 0.0128 0.0138

BWLS 2.1332e-02 0.0009 0.0044 0.0053

16 16

FDM 8.7386e-03 3.2582 0.0015 3.2597

HWLS 8.7386e-03 0.0011 0.0017 0.0028

DWLS 8.7386e-03 0.0010 0.0109 0.0119

BWLS 8.7386e-03 0.0005 0.0027 0.0032

64 64

FDM 2.6256e-03 10.1280 0.0034 10.1314

HWLS 2.6256e-03 0.0007 0.0012 0.0019

DWLS 2.6256e-03 0.0006 0.0083 0.0089

BWLS 2.6256e-03 0.0005 0.0029 0.0034



24

Table 3. Comparison of numerical solutions with exact solution of the test problem 4.2.

(a) N N = 8 8 (b) N N = 16 16


(a) N N =8 8 & (b) 16 16N N


0.2 0.2 0.99172 0.99172 0.99172 0.99172 0.99810

0.4 0.2 0.99171 0.99171 0.99171 0.99171 0.99810

0.6 0.2 0.99170 0.99170 0.99170 0.99171 0.99810

0.8 0.2 0.99174 0.99174 0.99174 0.99174 0.99810

0.2 0.4 0.99311 0.99311 0.99311 0.99311 0.99327

0.4 0.4 0.99307 0.99307 0.99307 0.99307 0.99327

0.6 0.4 0.99310 0.99310 0.99310 0.99310 0.99327

0.8 0.4 0.99318 0.99318 0.99318 0.99318 0.99327

0.2 0.6 0.99421 0.99421 0.99421 0.99421 0.99449

0.4 0.6 0.99414 0.99414 0.99414 0.99414 0.99449

0.6 0.6 0.99421 0.99421 0.99421 0.99421 0.99449

0.8 0.6 0.99435 0.99435 0.99435 0.99435 0.99449

0.2 0.8 0.99508 0.99508 0.99508 0.99508 0.99548

0.4 0.8 0.99498 0.99498 0.99498 0.99498 0.99548

0.6 0.8 0.99509 0.99509 0.99509 0.99509 0.99548

0.8 0.8 0.99529 0.99529 0.99529 0.99529 0.99548



25

Table 4. Maximum error and CPU time (in seconds) of the methods of the test problem 4.2.

Case (ii): For 1 , By taking 1.1

Table 5. Comparison of numerical solutions with exact solution of test problem 4.2


4 4

FDM 4.9756e-04 2.7199 0.0009 2.7208

HWLS 4.9756e-04 0.0115 0.0040 0.0155

DWLS 4.9756e-04 0.0021 0.0703 0.0724

BWLS 4.9756e-04 0.0038 0.0157 0.0195

16 16

FDM 6.3353e-05 3.9130 0.0021 3.9151

HWLS 6.3353e-05 0.0024 0.0038 0.0062

DWLS 6.3353e-05 0.0020 0.0212 0.0232

BWLS 6.3353e-05 0.0018 0.0092 0.0110

64 64

FDM 2.3252e-05 1.0344 0.0065 1.0409

HWLS 2.3252e-05 0.0024 0.0040 0.0064

DWLS 2.3252e-05 0.0021 0.0212 0.0233

BWLS 2.3252e-05 0.0018 0.0094 0.0112


0.2 0.2 1.0809 1.0809 1.0809 1.0809 1.0804

0.4 0.2 1.0812 1.0812 1.0812 1.0812 1.0804

0.6 0.2 1.0812 1.0812 1.0812 1.0812 1.0804

0.8 0.2 1.0810 1.0810 1.0810 1.0810 1.0804

0.2 0.4 1.0654 1.0654 1.0654 1.0654 1.0649

0.4 0.4 1.0656 1.0656 1.0656 1.0656 1.0649

0.6 0.4 1.0657 1.0657 1.0657 1.0657 1.0649

0.8 0.4 1.0654 1.0654 1.0654 1.0654 1.0649

0.2 0.6 1.0528 1.0528 1.0528 1.0528 1.0525

0.4 0.6 1.0530 1.0530 1.0530 1.0530 1.0525

0.6 0.6 1.0531 1.0531 1.0531 1.0531 1.0525

0.8 0.6 1.0529 1.0529 1.0529 1.0529 1.0525

0.2 0.8 1.0427 1.0427 1.0427 1.0427 1.0426

0.4 0.8 1.0428 1.0428 1.0428 1.0428 1.0426

0.6 0.8 1.0429 1.0429 1.0429 1.0429 1.0426

0.8 0.8 1.0429 1.0429 1.0429 1.0429 1.0426



26

(a) N N = 8 8 (b) N N = 16 16


(a) N N =8 8 & (b) 16 16N N .

Table 6. Maximum error and CPU time (in seconds) of the methods of the test problem 4.2

Test Problem 4.3: Finally, we consider another Fisher equation of the form

2 1 , 0 1t x xu u u u x (4.3.1)

subject to the I.C.: 2

1,0

1x

u x

e

(4.3.2)


4 4

FDM 7.8248e-04 2.5436 0.0010 2.5446

HWLS 7.8248e-04 0.0025 0.0040 0.0065

DWLS 7.8248e-04 0.0022 0.0211 0.0233

BWLS 7.8248e-04 0.0019 0.0096 0.0115

16 16

FDM 2.9996e-04 3.3152 0.0021 3.3173

HWLS 2.9996e-04 0.0025 0.0040 0.0065

DWLS 2.9996e-04 0.0022 0.0208 0.0230

BWLS 2.9996e-04 0.0017 0.0089 0.0106

64 64

FDM 8.2737e-05 1.0463 0.0066 1.0529

HWLS 8.2737e-05 0.0023 0.0039 0.0062

DWLS 8.2737e-05 0.0020 0.0208 0.0228

BWLS 8.2737e-05 0.0017 0.0093 0.0110



27

and B.C.s: 2

10,

1t

u t

e

, 1 1

12 2

11,

1t

u t

e

(4.3.3)

Which has the exact solution 1 1

2 2

1( , )

1x t

u x t

e

[15]. By applying the methods explained in

the section 3 and in problem 4.4.1, we obtain the numerical solutions and compared with exact solution

are presented in table 7 and figure 4. The maximum absolute errors with CPU time of the methods are

presented in table 8.

Table 7. Comparison of numerical solutions with exact solution of test problem 4.3.


0.2 0.2 0.489638 0.489638 0.489638 0.489638 0.489646

0.4 0.2 0.454428 0.454428 0.454428 0.454428 0.454416

0.6 0.2 0.419675 0.419675 0.419675 0.419675 0.419637

0.8 0.2 0.385680 0.385680 0.385680 0.385680 0.385639

0.2 0.4 0.514525 0.514525 0.514525 0.514525 0.514640

0.4 0.4 0.479183 0.479183 0.479183 0.479183 0.479301

0.6 0.4 0.444108 0.444108 0.444108 0.444108 0.444168

0.8 0.4 0.409581 0.409581 0.409581 0.409581 0.409584

0.2 0.6 0.539280 0.539280 0.539280 0.539280 0.539561

0.4 0.6 0.503957 0.503957 0.503957 0.503957 0.504288

0.6 0.6 0.468741 0.468741 0.468741 0.468741 0.468973

0.8 0.6 0.433876 0.433876 0.433876 0.433876 0.433967

0.2 0.8 0.563834 0.563834 0.563834 0.563834 0.564285

0.4 0.8 0.528696 0.528696 0.528696 0.528696 0.529255

0.6 0.8 0.493512 0.493512 0.493512 0.493512 0.493933

0.8 0.8 0.458482 0.458482 0.458482 0.458482 0.458673



28

00.5

1

0

0.5

10

0.5

1

xt

HW

LS

00.5

1

0

0.5

10

0.5

1

xt

DW

LS

00.5

1

0

0.5

10

0.5

1

xt

BW

LS

00.5

1

0

0.5

10

0.5

1

xt

Exact

00.5

1

0

0.5

10

0.5

1

xt

HW

LS

00.5

1

0

0.5

10

0.5

1

xt

DW

LS

00.5

1

0

0.5

10

0.5

1

xt

BW

LS

00.5

1

0

0.5

10

0.5

1

xt

Exact

(a) N N = 8 8 (b) N N = 16 16


(a) N N =8 8 & (b) 16 16N N

Table 8. Maximum error and CPU time (in seconds) of the methods of the test problem 4.3.


4 4

FDM 5.5899e-04 2.7258 0.0003 2.7261

HWLS 5.5899e-04 0.0007 0.0010 0.0017

DWLS 5.5899e-04 0.0008 0.0127 0.0135

BWLS 5.5899e-04 0.0007 0.0043 0.0050

16 16

FDM 6.7437e-05 2.7272 0.0015 2.7287

HWLS 6.7437e-05 0.0010 0.0016 0.0026

DWLS 6.7437e-05 0.0008 0.0089 0.0097

BWLS 6.7437e-05 0.0005 0.0027 0.0032

64 64

FDM 2.8217e-05 10.0350 0.0035 10.0385

HWLS 2.8217e-05 0.0007 0.0011 0.0018

DWLS 2.8217e-05 0.0005 0.0084 0.0089

BWLS 2.8217e-05 0.0005 0.0029 0.0034



29

5. Conclusions

In this paper, we applied the Lifting scheme for the numerical solution of Fisher’s equations

using different wavelet filters coefficients. From the figures we observed that the numerical solutions

obtained by different Lifting schemes are agrees with the exact solution. Furthermore, in the tables the

convergence of the presented schemes i.e. the error decreases when the level of resolution N increases.

In addition to this, the calculations involved in Lifting schemes are simple, straight forward and low

computation cost compared to classical method i.e. FDM and FAS. Hence the presented Lifting

schemes in particular HWLS & BWLS are very effective for solving non-linear partial differential

equations.

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Lifting Scheme for the Numerical Solution of Fisher’s