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A new sixth-order algorithm for general secondorder ordinary differential equationsD.O. Awoyemi aa Department of Industrial Mathematics and Computer Science, Federal University ofTechnology, AkureVersion of record first published: 19 Mar 2007.

To cite this article: D.O. Awoyemi (2001): A new sixth-order algorithm for general second order ordinary differentialequations, International Journal of Computer Mathematics, 77:1, 117-124

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A NEW SIXTH-ORDER ALGORITHM FOR GENERAL SECOND ORDER

ORDINARY DIFFERENTIAL EQUATIONS

D. 0. AWOYEMI

Department of Industrial Mathematics and Computer Science, Federal University of Technology, Akure

(Received 17 November 1999)

An important factor among others that should be considered in developing a numerical integrator for the solution of ordinary differential equations is the prudent management of computer time, which depends essentially on the number of functions to be evaluated per iteration. Recognising the importance of this factor, this article therefore proposes a class of four-step methods each with three functions evaluation per iteration. The procedure which yields a system of equations for stepnumber k 2 4 is based on collocation of the differential system at selected grid points which ensures the symmetry of the methods. The methods are compared for accuracy and computer time with an existing four-step method containing five functions evaluation aer iteration (see Awovemi, 1999a). Three other Predictors are similarly . . proposed for use in &e main methbd.

Keywork Numerical Integrator; Collocation; Predictor; Computer time

Category: G1.7

1. INTRODUCTION

It is a common practice equation of the form

to solve a general nth order ordinary differen

y(n) = f ( x , y , y', . . . , y *-I) (1.1)

by reducing it into a system of fist order equations and solve the system by any method suitable for first order system (see Fatunla, 1988; Lambert, 1993, 1991 and Dahlquist, 1978).

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118 D. 0. AWOYEMI

This approach is burdensome especially when we consider time implication and effort in developing a computer program for the method. Apart from the main program, there is always a need to develop separate sub-programs for the starting values and functions arising from the system of equations. At this stage many promising numerical methods are aban- doned especially by novices for lack of adequate knowledge and courage to produce suitable programs to test the accuracy of their methods.

Furthermore, a lot of efforts by prominent scholars have been put on numerical methods for solving special ordinary differential equations of the form

Fatunla (1984, 1985 and 1988); Lambert (1993, 1991); Jeftschi (1978); Jain et al. (1994); Lambert and Watson (1976) and Dahlquist (1978) were some of the authors who have contributed tremendously to solve such problems. The procedure they adopted for this class of methods is such that the resultant methods are not continuous, and therefore it is impossible to find the first and higher order derivative of y with respect to x; thus the scope of this class of methods is limited in application.

In this article we solve problem (1.1) for n = 2 directly without reducing it to a system of first order equation. This approach reduces the problem to

This is possible because of the continuity of the method which allows derivatives of y to any desired order to be computed.

Some attention has also been given to problem (1.2) by some eminent scholars. Henrici (1962) and Lambert (1973) discussed the theory of direct finite difference method for solving (1.2). Hairer and Wanner (1976) pro- posed Nystrom type method in which order conditions for the determina- tion of the parameters of the method were discussed. Henrici (1962); Gear (1971); Chawla and Sharma (1985) and Hairer (1977) developed indepen- dently explicit and implicit Runge Kutta-Nystron method for the nu- merical solution of (1.2). There are several other authors on this subject whose works cannot be accommodated in this article for lack of space. But little seems to have been done in using continuous linear multistep col- location methods to solve directly initial and boundary value problems of special and general second order ordinary differential equations of the types (1.2) and (1.3).

Thus in Awoyemi (1999b) we propose a four-step collocation procedure which leads to a class of symmetric continuous schemes for special ordinary

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ODE SOLVE 119

differential equations. Extension of that method to solve general equations of the form (1.3) is hereby proposed.

2. DESCRIPTION OF THE METHOD

In Onumanyi and Taiwo (1991) and Awoyemi (1993), the polynomial given

by

$ j (x ) = x i - j ( j - 1 ) $ j - 2 ( ~ ) , j = 0 , l

was constructed as a basis function for y " = f (x,y). A similar basis function

$,(x) = x j - j+,-1 - j ( j - l )+ j -2 (x ) , j = 0, 1, (2.1)

is constructed for Eq. (1.3). Thus we propose an approximate solution to (1.3) in the form

From (1.2) and (2.3), we get

Collocation of (2.4) at x,+,, j = 0, 2 and 4 and interpolating of (2.2) at x , + ~ , j = 0,1,2,3 for k = 4 lead to a system of equations given as:

Solving (2.5) for ajs and substituting their values into (2.2) yield a scheme expressed in the form D

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[T

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Bri

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Col

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t 03:

56 2

4 Fe

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013

120 D. 0. AWOYEMI

where

fn+2j = f (xn+2j I Yn+2j 7 Y ;+zj)

Also let

t = ( X - xn+3)lh1 (2.7)

we obtain the coefficients o f (2.6) and its first order derivatives as follows:

1 ~ ( t ) = - [3t5 + 15t4 - lo? - 90? - 68t],

150 1

al ( t ) = - - [25t6 + 207t5 + 410t4 - 690$ - 2835? - 1917t], 450

h2 P4(t) = - [25t6 + 234t5 + 770t4 + 1020t3 + 4053 - 54t],

2 1600 1

( t ) = - [ I 5t4 + 60? - 30% - l8Ot - 681, l5Oh

1 a',(t) = - - [150tS + 1035t4 + 1640? - 20703 - 5670t - 19171,

450h 1

a i ( t ) = - [300$ + 1935t4 + 2740? - 3870? - 9720t - 36721, 450h

h Ph ( t ) = - [l 50tS + 90t4 - 1240? - 1803 + 2970t + 12421, 2 1 600

h &(t ) = - [3450tS + 20970t4 + 254808 - 44640F - 88290t - 265141,

10800

Evaluation o f (2.6) using (2.8) when t = 1 gives a special symmetric case as follows:

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ODE SOLVE 121

The order and stability interval of (2.9a) are given as (P= 6, C8 = 0.068254) and ( - 6.4, 0) respectively (see Awoyemi, 1999b). Furthermore, noting that in (2.7)

we obtain the corresponding first order derivative of (2.9a) as

where,

is computed from

3. THE PREDICTORS

The predictors used to calculate y, + 2 and y b+,, y, + and y h+, , and y, + and y ;+, respectively are listed in Awoyemi (1998a and 1999b).

4. NUMERICAL EXAMPLES

In thls section, the accuracy of the new four-step method (2.9) is compared with that of the previously developed four-step methods in Awoyemi (1999a). For ease of reference, the discrete method and its first order derivative are reproduced as follows:

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122 .D. 0. AWOYEMI

Each of the following examples is considered for step size h = 11320. The errors arising from computed results and theoretical solutions at selected points are shown in Tables 4.1-4.3 respectively.

Example 4.1

Theoretical Solution: y = sin2x.

TABLE 4.1 Computed results to Example 4.1

x Awoyemi (1999~) for k = 4 New method (2.9)

1.1 0.9512401504D - 06 0.4692146182D - 06 1.2 0.8426500347D - 06 0.4080286853D - 06 1.3 0.5988632682D - 06 6.2289737586D - 06 1.4 0.2091234934D - 06 . 0.8128718132D - 07 1.5 0.3251187510D - 06 0.5244721664D - 06 1.6 0.9893958631D - 06 0.1089743773D - 05 1.7 0.1756384212D - 05 0.1753725421D - 05 1.8 0.2587040796D - 05 0.2481480677D - 05 1.9 0.3432669163D - 05 0.3228415464D - 05 2.0 0.4237819460D - 05 0.3943014561D - 05

TABLE 4.2 Computed results to Example 4.2 - -

Awoyemi (1999~) for k = 4

0.2238494723D - 08 0.6123078045D - 08 0.1287143436D - 07 0.2437828339D - 07 0.4300405609D - 07 0.7089294085D - 07 0.1 143550554D - 06 0.1795141937D - 06 0.2768979277D - 06 0.4233120228D - 06

New method (2.9)

TABLE 4.3 Computed results to Example 4.3

x Awoyemi (1999~) for k= 4 New method (2.9)

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ODE SOLVE

Example 4.2

Theoretical Solution: y = 1 + (112) ln((2 + x)/(2 - x)).

Example 4.3

Theoretical Solution: y = 2xcosx + 4xsinx + (1/2)xex.

5. CONCLUSION

A collocation approach which produces a family of order six continuous methods has been described for the approximate solution of problem (1.3). Three test examples have been solved to compare the accuracy of the new method with our previously proposed order five continuous method (see Awoyemi, 1999a), which contains five functions evaluation per iteration instead of three contained by the new method (2.9).

A look at Tables 4.1 -4.3 clearly shows that the new method (2.9) is better in accuracy than Awoyemi (1999a) for k=4. Furthermore, it has less functions to be evaluated per iteration than in Awoyemi (1999a). Thus the new method consumes less computer time, since consumption of computer time depends mainly on the number of functions to be evaluated per iteration in a numerical method.

The new method is therefore recommended for use on non-stiff general problems because of its obvious advantages over our previous method, Awoyemi (1999a) for k = 4.

References

Awoyemi, D. 0 . (1993) On some continuous linear multistep methods for initial value problems, Ph.D. Thesis (Unpublished), University of Ilorin, Nigeria.

Awoyemi, D. 0. (1999a) A class of continuous methods for general second order initial value problems in ordinary differential equations, International Journal of Computer and Mathemarics, 72, 29 - 37.

Awoyemi, D. 0. (1999b) A New Sixth-Order Method for the continuous solution of Special Ordinary Differential Equations. To appear in Science Forum, Journal of Pure and Applied Sciences (ATBU).

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Fatunla, S. 0 . (1984) One Leg Multistep Method for Second-Order Differential Equations, Computer and Mathematics with Application, 10(1), 1-4.

Fatunla, S. 0. (1985) One Leg Hybrid Formulas for Second-Order Differential Equations, Computer and Mathematics with Application, 11(4), 329- 333.

Fatunla, S. 0. (1988) Numerical Methods for Initial Value Problems in Ordinary Differential Equations, Academic Press.

Gautschi, W. (1961) Numerical Integration of Ordinary Differential Equations Based on Trigonometric Polynomials, Numerical Methodr in Mathematics, 3, 381 -397.

Hairer, E. (1976) Unconditionally Stable Methods for Second Order Differential Equations, Numerische Mathematik, 32, 373 - 379.

Henrici, P. (1962) Discrete Variable Methods for Ordinary Differential Equations, New York: John Wiley & Sons.

Jain, R. K., Kambo, N. S. and Rakesh Goel (1984) A Sixth-Order P-Stable Symmetric Multistep Method for Periodic Initial Value Problems of Second-Order Differential Equations, IMA Journal of Numerical Analysis, 4 , 117- 125.

Jeftsch, R. (1978) Complete characterization of Multistep Methods with an Interval of Periodicity for Solving y" = f(x, y), Mathematics of Computation, 32, 1108- 11 14.

Lambert, J. D. (1973) Computational Methods for Ordinary Differential Equations, John Wiley & Sons, New York.

Lambert, J. D. and Watson, A. (1976) Symmetric Multistep Method for Periodic Initial Value Problems, Journal of the Institute of Mathematics Applications, 18, 189 - 202.

Lambert, J. D. (1991) Numerical Methods Ordinary Differential Systems-The initial value problems, John Wiley & Sons, New York, Chapters 3 and 4.

Stiefel, E. and Bettis, D. G. (1969) Stabilization of Cowell's Methods, Numerical Methodr in Mathematics, 13, 154- 175.

Taiwo, 0. A. and Onumanyi, P. (1991) A Collocation Approximation of Singularly Perturbed Second Order Ordinary Differential Equations, International Journal of Computer and Mathematics, 39, 205 -21 1.

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