New Numerical Schemes for the Solution of Slightly Stiff Second Order Ordinary Differential Equations

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1 American Journal of Computational and Applied Mathematics 24, 4(6): DOI:.5923/j.ajcam New Numerical Schemes for the Solution of Slightly Stiff Second Order Ordinary Differential Equations Obayomi A. A., Olabode B. T. 2,* Department of Mathematical Sciences, Ekiti-State University, Ado-Ekiti, Nigeria 2 Department of Mathematical Sciences, Federal University of Technology, Akure, Nigeria Abstract This paper proposes new schemes for the numerical treatment of slightly stiff second order ordinary differential equations. There has been a considerable effort in the recent years to construct numerical procedures which correctly replicate the properties of the original dynamical system by using the Non-Standard finite difference method. In this work, we eliminate the problems that can arise with standard numerical integration schemes and construct Non-Standard schemes by adopting some proven guidelines for discretization. A set of discrete models is obtained for each IVP, and the properties of the resulting schemes are compared with the known behavioral pattern of the original equation. The results compare favorably with some earlier results and the analytic solution. Keywords Finite Difference Scheme, Ordinary Differential Equations (ODEs), Initial Value Problems (IVPs), Non-Standard Finite Difference Scheme (NSFDS), Stiff problems. Introduction One of the major problems encountered in discrete modeling is the computational problem for stiff equations. Stiff equations do not only provide computational difficulties but consumes so much time and memory resources of the computer. It is worthy of note that constructing schemes that are qualitatively stable for such equations, is not trivial. Our major aim here is to produce computationally reliable numerical schemes for some special second order stiff equations. Non-Standard Finite Difference Schemes (NSFD) have emerged as an alternative method for solving a wide range of problems whose mathematical models involve algebraic, differential and biological models as well as chaotic systems (see [7]). Furthermore, [5-7] introduced certain rules for obtaining the best difference equations, one of the most important of which is that the nonlinear terms of f (t, y(t)) are approximated in a non-local form: yy 2 2 aayy nn yy nn+ + byy nn, a + b =, aa, bbϵr () yy 3 yy 2 3 nn yy nn+ + byy nn, bϵr. (2) For such systems, some relevant techniques for constructing Non-Standard or unconventional integration methods has been developed in various works by [5, 2, 4, 8] and other researchers. It is the latter set of techniques that are * Corresponding author: olabodebola@yahoo.com (Olabode B. T.) Published online at Copyright 24 Scientific & Academic Publishing. All Rights Reserved outlined here. This paper therefore, proposes the use of Non-standard finite difference approach for the solution of the second order initial value problems of ordinary differential equations of the form: yy = ff(tt, yy, yy ), yy(aa) = ηη, yy (aa) = ηη 2. (3) In the construction of these models, we will particularly employ a combination of rule 2 and rule 3 of the Non-standard modeling rules as proposed by [5] and extended by [2]. 2. Construction of the Schemes We will apply rule2 in [5] to each of the components of the equations as shown below where where yy yy kk+ 2yy kk+ yy kk (h) h 2 + (h 4 )aaaah (4) y (yy kk+ y k ), (h) h + (h 2 )aaaah (5) y (yy kk+ ββ y k ), where (h) h + (h 2 ), ββ(h) aaaah (6) The examples of these functions areas follows:,

2 24 Obayomi A. A. et al.: New Numerical Schemes for the Solution of Slightly Stiff Second Order Ordinary Differential Equations Problem. = 4ssssss 2 ( h 2 ), ssssss2 (h), ϵ R = ssssss(h), ee h ββ = cccccc(h)., λϵ R etc, yy = yy, yy() =, yy () =, yy(xx) = ( ee xx ). Some Standard numerical schemes have been designed for this particular problem by some researchers (see [,, 3]) in which collocation approach were adopted. In this paper, we employ Non-standard Finite Difference Schemes approach and accuracy comparison is made with the existing schemes. Scheme yy kk+ 2yy kk + yy kk yy kk+ = 2yy kk + yy kk+ = yy kk+ yy kk yy kk yy kk yy kk+ = yy kk 2 yy kk yy kk+ = 2 yy kk yy kk ( ) Scheme 2 Scheme 3 = 4ssssss 2 ( h 2 ) = ee h, λϵ R etc. yy kk+ 2yy kk + yy kk yy kk+ = 2yy kk + yy kk+ = yy kk+ ββyy kk yy kk yy kk yy kk+ = 2 yy kk yy kk 2 yy kk yy kk yy kk+ = = 4ssssss 2 ( h 2 ), = ssssss(h), ββ = cccccc(h). yy kk+ = 2 yy kk yy kk = 4ssssss 2 ( h 2 ), = ee h, λϵ R, ββ = cccccc(h). Table shows that, the new Non-standard schemes performed better in accuracy than the Standard schemes developed by [, 3, ] for problem, as the errors (ER), (ER2), (ER3) are smaller than errors in the existing schemes. Table. Error of Deviation from Analytic Solution for New and Existing Schemes Adesanya (2) Anake (2) Yahaya & Badmus (29) Error in the New Scheme (ER) Error in the New Scheme2 (ER2) Error in the New Scheme 3 (ER3) E E E E E E-4.49E-8.492E E E E E E-5.492E E E E E E E E E E E E E E E-6.999E E E-3.328E E E E E E E E E E E E E E E E E E E E E E E E E E Numerical Experiment The schemes were tested for varying value of h (.,.) and the resulting solution is here presented in 3D graphs. We also obtain error of approximation and compare them with the existing results presented by some researchers:

3 American Journal of Computational and Applied Mathematics 24, 4(6): y(t k ) YNEW YNEW3 Figure. Graph of the scheme of yy = yy, yy() =, yy () = Deviation from Analytic solution for h= ER ER2 ER3 Adesanya (2) Anake 2 Figure 2. Graph of error of the schemes for yy = yy, yy() =, yy () = Graph of schemes for h=. y(t k ) YNEW YNEW3 t k Figure 3. Graph of the scheme for yy = yy, yy() =, yy () =

4 242 Obayomi A. A. et al.: New Numerical Schemes for the Solution of Slightly Stiff Second Order Ordinary Differential Equations Problem 2 Consider the equation yy = yy, yy() =, yy(xx) = 3 2 eexx 2 ee xx. Using the same transformation as in equations (4), (5) and (6), we have Scheme yy kk+ 2yy kk + yy kk = aaaa kk+ + bbyy kk Scheme 2 yy kk+ = 2yy kk + aaaaaa kk+ + bbbbyy kk yy kk yy kk+ ( aaaa) = yy kk (2 + bbbb) yy kk yy kk+ = (2 + bbbb)yy kk yy kk ( aaaa) = 4ssssss 2 ( h 2 ), = ssssss(h). yy kk+ = (2 + bbbb)yy kk yy kk ( aaaa) = 4ssssss 2 ( h 2 ), = ee h, λϵ R. Deviation from Analytic solution for h= ER ER2 ER Figure 4. Graph of the Errors in the scheme for yy = yy, yy() =, yy () = Numerical Experiment. y(t k ) t k YNEW Figure 5. Graph of the schemes for yy = yy, yy() =

5 American Journal of Computational and Applied Mathematics 24, 4(6): Deviation from Analytic solution for h= E E2 Figure 6. Graph of Error of the schemes for yy = yy, yy() = Graph of schemes for h= YNEW Figure 7. Graph of Error in Scheme 2 for yy = yy, yy() = Deviation from Analytic solution for h= E E2 Figure 8. Graph of Error in Scheme 2 for yy = yy, yy() =

6 244 Obayomi A. A. et al.: New Numerical Schemes for the Solution of Slightly Stiff Second Order Ordinary Differential Equations Problem 3 Consider the equation yy + yyyy =, yy() =, yy () =, y(xx) = tttttt ππ xx ππ < xx < 3ππ Using the same transformation as in equations (4), (5) and (6), we have Scheme yy kk+ 2yy kk +yy kk = yy kk ( yy kk+ yy kk ), yy kk+ = 2yy kk yy kkyy kk+ + yy kk 2 yy kk yy kk+ + yy kk = yy kk 2 + yy kk yy kk yy kk+ = 2 + yy kk yy kk yy kk + yy kk = 4ssssss 2 ( h 2 ), = ee h, λϵ R, ββ = cccccc(h). Scheme 2 yy kk+ 2yy kk + yy kk yy kk+ = 2yy kk yy kkyy kk+ = yy kk ( yy kk+ ββββ kk ) + yy kk 2 yy kk yy kk+ + yy kk = yy kk 2 + ββββ kk yy kk yy kk+ = 2 + yy kk yy kk yy kk + yy kk = 4ssssss 2 ( h 2 ), = ssssss(h), ββ = cccccc(h) Scheme 3 yy kk+ = 2 + yy kk yy kk yy kk + yy kk = 4ssssss 2 ( h 2 ), = ee h, λϵ R, ββ = cccccc(h). Numerical Experiment YNEW -2 YNEW3 Figure 9. Graph of yy + yyyy =, yy() =, yy () =.8 Error of deviation for h= E E2 E Figure. Graph of error foryy + yyyy =, yy() =, yy () =

7 American Journal of Computational and Applied Mathematics 24, 4(6): Graph of schemes for h= YNEW YNEW3 Figure. Graph of yy + yyyy =, yy() =, yy () = Error of deviation for h= E E2 E3 Figure 2. Graph of Errors in the schemes for yy + yyyy =, yy() =, yy () = 3. Summary and Conclusions In this paper we have been able to develop some Non-standard schemes that replicates the dynamics of the original equations for the selected equations. We observed that the remodeling of the first derivative using y = (yy kk+ ββy k ), where, (h) h + (h 2 ), ββ(h), aaaah is better than y = (yy kk+ y k ), Where (h) h + (h 2 )aaaah see figures 4, and 2 for error of deviation for E compare with E2 and E3 for the different schemes and the method in which the first derivates are modeled. We also observed that the use of the denominator functions = ee h, λ ϵ R instead of = ssssss(h) results in a better scheme interms of error of approximation see E3 and E in figure. These supports the authors earlier work on non-standard schemes for stiff equation. This denominator gives the opportunity to vary the value of λ in the numerical experiment so that we can obtain a minimum error of approximation (see [8]). We also observed that the non-standard schemes performed better than the standard schemes developed by [, 3, ]. (see figure 2 and Table ) REFERENCES [] A. O. Adesanya (2): Block Methods for Direct solutions of general Higher Order Initial Value problems of Ordinary Differential Equations. Ph.D Thesis, Federal University of Technology, Akure. Nigeria (In Press). [2] R. Anguelov, J.M.S Lubuma (23). Non-Standard finite difference method by nonlocal approximation, Mathematics and Computers in simulation 6: [3] T.A Anake (2): Continuous Implicit Hybrid One-step Methods for the solution of Initial value Initial Value problems of General Second order Ordinary Differential Equations. Ph.D Thesis, Covenant University, Ota, Ogun state Nigeria (in Press). [4] E. A. Ibijola, A. A. Obayomi, and B. TOlabode (23) New Non-standard Finite Difference Schemes for Ordinary

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