Numerical Schemes Based on Non-Standard Methods for Initial Value Problems in Ordinary Differential Equations

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1 Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 3 (): Scholarlink Research Institute Journals, 22 (ISSN: 24-76) jeteas.scholarlinkresearch.org Numerical Schemes Based on Non-Standard Methods for Initial Value Problems in Ordinary Differential Equations Ibijola, E. A and Obayomi, A. A. Department of Mathematical Sciences, Ekiti State University, Ado-Ekiti. Nigeria. Corresponding Author: Obayomi, A. A. Abstract In this paper we present new methods of constructing Non-Standard schemes. Two new logical ways for constructing the denominator functions for the discrete derivatives were introduced and schemes were constructed based on Non Local Approximation. The schemes were applied on sampled IVPs in ordinary differential equations to verify whether the schemes are computationally reliable. The results obtained show that the schemes are suitable for problems for which they were proposed. Keywords: local approximations, denominator functions (type A and B), modeling, non-standard method, IVP (Initial Value Problem), rock. INTRODUCTION In this paper we developed computationally reliable Non-Standard difference schemes that support the qualitative properties of the considered initial value problems. Standard Finite difference schemes for differential equations exhibit a level of numerical instability (Mickens 994). Mickens (994, 2), gave valuable reasons for numerical instabilities in some particular investigated cases. Thus, the preservation of the qualitative properties of the considered differential equation with respect to these schemes is of great significance. He proposed a new method of construction of discrete models whose solution have the same qualitative properties as that of the corresponding differential equation for all step size and thus eliminate the elementary numerical instabilities that can arise in finite difference models of differential equations. The major consequence of this result is that such scheme does not allow numerical instabilities to occur (Mickens 994). Anguelov and Lubuma (23) based their works on Mickens (994, 2) and deduced a set of nonstandard modeling techniques using the two rules given below i. The denominator can be replaced by a function φ (h) h + o (h 2 ) as h o and o< φ (h) <, h>o which can be considered as the renormalization of step size h. ii. That the non-linear terms must be approximated non-locally on the computational grid by a suitable function of several points of the mesh 49 Other works of Ibijola and Sunday (2), Sunday, Ibijola and Skwame (2), discussed extensively on Exact Non-standard schemes while further works of Ibijola, Lubuma and Ade-Ibijola 2 present the theory of nonstandard finite difference schemes which can be used to solve some initial value problems in ordinary differential equations. Methods of construction and mode of implementation of these methods were also discussed. The motivation for these work is that of Anguelov & Lubuma (23) and Ibijola & Sunday (2). Throughout this work we shall consider problems of the form y f(t,y), y(t o ) y o () Which occur most often in Biological, Chemical, Physical, Management sciences and Engineering. The aim of this research work is to construct discrete models whose solution have the same qualitative properties as that of the corresponding differential equation for all step size and thus eliminate the elementary numerical instabilities that can arise in finite difference models. Exact Finite-Difference Scheme Definition (Mickens 994) An exact finite-difference scheme is one which the solution to the difference equation has the same general solution as the associated differential equation. Definition 2: Type A denominators (increment functions) are those constructed with the knowledge of analytic solution. Type B denominators are those constructed otherwise. We will exploit the point of intersection between the exact finite difference schemes and the expected

2 solution of the original differential equation and use Mickens rules [Mickens (994)] to develop new Non-Standard finite difference schemes. Non-Standard Finite Difference Modeling Rules A finite difference scheme is called non-standard finite difference method, if at least one of the following conditions is met (Anguelov and Lubuma 23): i. In the discrete derivative, the traditional denominator is replaced by a non-negative function φ such that, φ (h) h + o(h 2 ), as h. ii. Non-linear terms that occur in the differential equation are approximated in a non-local way i.e. by a suitable function of several points of the mesh. Derivation of the New Method The discrete solution of a finite difference scheme can be written as y αy (2) where, : α g(y(x ), y(x )) α is a real valued function. A suitable candidate for the function g is y(x n) y(x n ) which is the exact Scheme (Ibijola and Sunday 2), where y(x )is is the analytic solution at x. i. Consider the non-standard finite difference scheme below for any dimensionless equation of the form (.) given by y y + φf(x y ) or f(x () y ) (3) ii. Consider also the exact scheme in the form Take y n y(x n) y(x n ) α ( ) ( ) y n ; g(y(x ), y(x )) If we compare these schemes in (i) and scheme (ii) above y y + φf(x y ) y(x n) y ( ) ( ) f (x y ) y(x n ) y n Then φ can be chosen as φ ( ) ( ) y ( ) ( ) ( ) y (4) Hence, we have a method of choosing φ y y + φ f(x, y ), φ () (, ) (5) 5 We can consider the point where the expected analytical and the numerical solution coincides i.e y(x ) y then the scheme in (5) can be written as: () ( ) ( ) φ (, ) y x k+ y(x k ) h (6) h (, ) Then taking limit as h approaches zero φ f (x, y ) (, ) φ h (x k, y k ) (7) f(x k, y k ) Where f (x, y) is the derivative of f(x, y) and f (x k, y k ) is the value f (x,y) at point x h may be replaced with higher powers of h to have φ (h) h + o (h 2 ) as h o (Mickens994) Using the following normalization (φ ) functions we will test behavior of each scheme to be developed : ) φ sin h 2) φ e 3) y y ( ) ( ) 4) φ ( ) (, ) solution exact scheme Derived from 5) φ h, (,) Derived from taking (,) derivative Construction of the New Non Standard Schemes Problem : Consider the IVP y 3y; y () 5; (8) with exact solution y(t) 5e a) At the points x x and x x kh; y 5e () and y 5e () Let ( ) ( ) Then, the Exact finite differences scheme is y y e() e () y (9)

3 b) Using 3y 3{ay + ( a)y } a Є R and (3) in (8) we obtainy y 3φ{ay + ( a)y } () Then y ( () Comparing this with the exact Scheme y y we have y y ( and φ ( ) (2) c) Using (5) φ () (, ) we have φ () (3) d) Using (7) φ h( (, ) ) we have (, ) f (x, y ) 3 φ h( (4) Or φ h to protect the fact that φ (h) h + o (h 2 ) as h o e) Using (4) in y y + φf(x y ) Then y 3h + y (5) Problem 2 Consider the IVP y y y() 2 (6) with analytic solution y (x) + e At the points x x hk and x x (k + )h ; y + e, y + e () (7) The Exact finite difference scheme is therefore given by y k e(k)h e kh y k where () (8) a) Let y ay + ( a)y then from (3) and (6) We obtain ay + ( a) y Then a Non-standard scheme is y n (aφ h )y n φ h ) (aφ h φ h ) From (2) and (9) y y φ h n (a a)y n Hence a non standard scheme (9) ( ) ) and y n (aφ h)y n φ h ) (aφ h φ h ) y n (a a)y n b) Using (5) φ () Using (7) (, ) φ h (α)y y (2) (2) c) φ h( (, ) ) (, ) f (x, y ) φ h( φ applied ) to (9). (22) Problem 3 Consider the IVP y'.2y; y () ; (23) with analytic y(t) e. b) At the points x x and x x ; y e.(), y 5e.() The Exact finite difference scheme is y y e.2(k)h e.2(k)h y k (24) b) Using.2y.2{ay + ( a)y } a Є R and (3) in (23) we have y y.2φ{ay + ( a)y }. Then y ( )y.. (25) Comparing this with the Exact finite difference scheme y y then y y ( φ.( ). )y.. (26) and c) Using (5) φ () (, ) we have φ (). (27) d) Using (7) φ h( (, ) (, ), f (x, y ).2 and φ h(.. (28) Or φ h to protect the fact that φ (h) h + o (h 2 ) as h o e) Using (28) in y y + φf(x y ) Then y.2h + y (29) Problem 4 Consider the IVP y λy( y), λ, y() (3) This is a form of logistic equation, the analytic solution is given by y(t) + 9e 5

4 and the Exact finite difference scheme is given by y y (3) Where [ + 9e + 9e () ] a) Let y ay + ( a)y y Then y - y y (ay + ( a)y y ) y (() ) From (() ) Then the new scheme becomes y (() ) ( ) ( ()) b) Using (5) φ () (, ) y (α) y (32), φ (33) (34) c) Using(7) φ h( (, ) ) (, ) f (x, y ) 2y φ h( applied ) () to (32) () Group of scheme 2 for Problem 4 Problem 4 can be modeled into another set of different schemes by representing the non-linear term as given below a) Let y ay y + ( a)y () ay y + ( a)y y using ( ) ( ) φ ( ) ( ) Then the new scheme becomes y ( ) ( ) φ ( ) y (36) y, where (37) a. Using (5) φ () (α) y (, ) (38) b) Using φ h( (, ) ) (, ) f (x, y ) 2y φ h( ) applied to (36) (39) () Graphical Representation of the Numerical Result y 3y, y() 5 h. a phi h/y Fig y 3y, y() 5 h. a phi h Derivative ( 4 ) Exact ( 9 ) DExact ( 3 ) Fig 2 y y, y() 2 a.5 h. phi h/y Derivative ( 4 ) Exact ( 9 ) DExact ( 3 ) ( 2 ) Derivative (22) Exact (8) DExact (2) (2) Fig 3 y y, y() 2 a.5 h. phi h

5 Derivative (22) Derivative () Exact (8) DExact (2) 5 Exact (3) DExact (34) (2) Phi Sine (33) Fig 4 y.2y, y() h. a.5 phi h/y Derivative (28) Fig 8 y y( y), y() ; Group Scheme a.5 h. Phi h 5 5 Exact (24) DExact (27) (25) Fig 6 y.2y, y() h. a.5 phi h Derivative () Exact (3) DExact (34) (33) Derivative (28) Exact (24) DExact (27) (25) Phi Sine Fig 9 y y( y), y() ; Group 2 Schemes a.5 h. Phi h( () ) Fig 7 y y( y), y() ; Group Scheme a.5 h. Phi h( () ) 53

6 y y( y), y() ; Group 2 Schemes a.5 h. phi h 5 5 Fig Group 2 Schemes 5 5 Fig 8 8 Derivative (39) Exact (3) DExact (38) (37) a.5 h. phi h Derivative (39) Exact (3) DExact (38) (37) SUMMARY AND CONCLUSION We have been able to construct two groups or family of schemes for the last three IVP s and one group each for the first two. The schemes have been tested numerically in terms of their consistency with the known behavior of the analytic solution of the particular initial value problem. The last equation is a form of the logistic equation. The Non-standard schemes developed are not consistent with the analytic solution for a>. It can be seen that the schemes oscillate close to the fixed point (y ). The result is expected to be different if the equation is represented in its dimensionless form. The methods based on type B denominators will behave better if it is modified to satisfy the non-standard modeling rules. This is because the phi tends to produce a linear scheme most especially when they are applied directly to the non-standard Finite difference scheme(3). The advantage of the last technique is that the analytic solution does not have to be known provided it exist. The construction of phi to satisfy the Mickens rules is sacrosanct to the validity of this new techniques. An area of improvement is to develop a method of modifying phi accordingly to satisfy the requirement of Non-standard schemes.the numerical experiment has shown that: The denominator function can be constructed in the following ways (in the tested cases) φ sin(h), e i.e using classical functions (Mickens 994) φ () (,), α g [y(x ), y(x )], y(x)is the known analytic solution. φ (,) (,) n. All the schemes have been found to be consistent with literature and compared favorably in a many cases. However the schemes ( and 39) do not exhibit the same qualitative property of the corresponding IVP. even where phi(φ) is an exponential or sine function. REFERENCES Anguelov R, Lubuma JMS (2). On Mathematical Foundation of the Nonstandard finite Differential Method, in Fiedler B, Grooger K, Sprekels J (Eds), Proceeding of the International Conference on Differential Equations, EQUADIFF 99 Word Scientific, Singapore, Anguelov R, Lubuma JMS (2). Nonstandard finite difference method,by Non-local Approximation Keynote address at the South Africa Mathematical Society, Pretoria, South Africa 6-8 October, Notices of S. A. Math. Soc. 3: Anguelov R, Lubuma JMS (23). Nonstandard finite difference method by nonlocal approximation, Mathematics and Computers in simulation 6: Ibijola, E. A., and Sunday J. (2). A comparative study of Standard and Exact Finite Difference Schemes for Numerical of ODEs Emanating fro the Radioactive decay of Substances. Australian Journal of Basic and Applied Sciences, 4(4): Lambert J. D. (99). Numerical Methods for Ordinary Differential System; Wiley, New York. Mickens R.E (994). Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore. Pp -5, Mickens R.E. (2). Applications of Nonstandard Methods for initial value problems, World Scientific, Singapore. 54

7 Simmons G. F. (98): Differential Equations with Applications and Historical Notes. T M H edition McGraw-hill publishing. Sunday J.,Ibijola E. A., and Skwame Y. (2) On the Theory and Application of Non-standard Finite Difference Methods for Singular ODEs. Journal of emerging Trends in Engineering and Applied Sciences UK, 294): Waiter W (998). Ordinary Differential Equations, Springer, Berlin. Zill D G, Cullen R M (25) Differential Equations with boundary value problems (sixth Edition) Brooks /Cole Thompson Learning Academic Resource Center. Pp 4-2,