IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 13, Issue 2 Ver. I (Mar. - Apr. 2017), PP 102-106 www.iosrjournals.org
Numerical Treatment of a Singularly Perturbed Boundary Value Problem Using Polynomial and Nonpolynomial Splines *Aminu Audu1, A. A. Shalangwa1, David John1 1
Gombe State University, Nigeria.
Abstract: In this study, two polynomial splines are developed and used to obtain the numerical solution of singularly perturbed two-point boundary value problems. The two polynomial splines developed are linear and non-linear polynomial splines. The applications of these splines to singularly perturbed two-point boundary value problems resulted to linear algebraic system of equations which are then solved by Gaussian elimination method to obtain the unknown constants arising from the splines used. Three singularly perturbed boundary value problems are solved. Keywords: singularly perturbation, polynomial and non-polynomial Spline, boundary value problems and Absolute Error.
I. Introduction Singularly perturbed boundary value problems are widespread in nature. Typically these problems are in various fields of applied mathematics such as fluid mechanics, elasticity, quantum mechanics, optimal control, chemical reactor theory, aerodynamics, reaction diffusion process, geophysics and many other areas. Equations of this type typically exhibit solutions with layers; that is, the domain of the differential equation contains narrow regions where the solution derivatives are extremely large. The numerical treatment of singularly perturbed differential equations gives major computational difficulties due to the presence of boundary or interior layers. Wide verities of papers and books have been published in the recent years, describing various methods for solving singularly perturbed two-point boundary value problems.
II. Spline Methods The formulation of a Spline function approximation and the development of some methods for numerical solution of second-order singularly perturbed boundary value problems. We derived the methods to solve a boundary value problem in the finite interval đ?‘Ž, đ?‘? , we partition the interval using equally spaced knotsđ?‘Ľđ?‘– = đ?‘Ž + đ?‘–â„Ž, đ?‘– = 0, 1, 2, ‌ , đ?‘ , đ?‘Ľ0 = đ?‘Ž, đ?‘Ľđ?‘ = đ?‘?, â„Ž = positive integer. For each sub-interval đ?‘Ľđ?‘– , đ?‘Ľđ?‘–+1 for đ?‘– = 0, 1, 2, ‌ , đ?‘ − 1.
đ?‘?−đ?‘Ž đ?‘
, where N is an arbitrary
III. Proposed Linear Polynomial Spline Function The proposed polynomial Spline function has the form: đ?‘†đ?‘– đ?‘Ľ = đ?‘Žđ?‘– + đ?‘?đ?‘– đ?‘Ľ − đ?‘Ľđ?‘– + đ?‘?đ?‘– (đ?‘Ľ − đ?‘Ľđ?‘– )2 + đ?‘‘đ?‘– (đ?‘Ľ − đ?‘Ľđ?‘– )3 , đ?‘– = 0, 1, 2, ‌ , đ?‘ − 1 (1) Where đ?‘Žđ?‘– đ?‘?đ?‘– đ?‘?đ?‘– đ?‘Žđ?‘›đ?‘‘ đ?‘‘đ?‘– are constants. A polynomial function đ?‘†(đ?‘Ľ) of class đ??ś 2 đ?‘Ž, đ?‘? interpolates đ?‘Ś(đ?‘Ľ) at the grid point đ?‘Ľđ?‘– for đ?‘– = 0, 1, 2, ‌ , đ?‘ . Let đ?‘Śđ?‘– be an approximation to đ?‘Ś(đ?‘Ľđ?‘– ), obtained by the polynomial spline S passing through the points đ?‘Ľđ?‘– , đ?‘Śđ?‘– and (đ?‘Ľđ?‘–+1 , đ?‘Śđ?‘–+1 ). The spline function (1) is not only required to satisfy the given differential equation and the associated boundary conditions at đ?‘Ľđ?‘– and đ?‘Ľđ?‘–+1 , but it must also satisfy the continuity of first derivatives at the common nodes (đ?‘Ľđ?‘– , đ?‘Śđ?‘– ). We derived an expression for the coefficients of (1) in-terms of đ?‘Śđ?‘– , đ?‘Śđ?‘–+1 , đ?‘€đ?‘– đ?‘Žđ?‘›đ?‘‘ đ?‘€đ?‘–+1 thus, đ?‘†(đ?‘Ľ đ?‘– ) = đ?‘Śđ?‘– , đ?‘†(đ?‘Ľ đ?‘–+1 ) = đ?‘Śđ?‘–+1 , đ?‘† ′′ đ?‘Ľđ?‘– = đ?‘€đ?‘– , đ?‘† ′′ đ?‘Ľđ?‘–+1 = đ?‘€đ?‘–+1 (2) From the algebraic manipulation, we obtain as from (2) (đ?‘Ś −đ?‘Ś ) â„Ž (đ?‘€đ?‘–+1 +2đ?‘€đ?‘– ) đ?‘€ đ?‘€ −đ?‘€ đ?‘Žđ?‘– = đ?‘Śđ?‘– , đ?‘?đ?‘– = đ?‘–+1 đ?‘– − , đ?‘?đ?‘– = đ?‘– , đ?‘‘đ?‘– = đ?‘–+1 đ?‘– (3) â„Ž 6 2 6â„Ž Where đ?‘– = 0, 1, 2, ‌ , đ?‘ − 1 One sided limit of the derivative of đ?‘†(đ?‘Ľ) are obtain as 1 â„Ž đ?‘† ′ đ?‘Ľđ?‘–− = đ?‘Śđ?‘– − đ?‘Śđ?‘–−1 + đ?‘€đ?‘– − 2đ?‘€đ?‘–−1 , đ?‘– = 0, 1, 2, ‌ , đ?‘ (4) â„Ž 6 And 1 â„Ž đ?‘† ′ đ?‘Ľđ?‘–+ = đ?‘Śđ?‘–+1 − đ?‘Śđ?‘– − đ?‘€đ?‘–+1 + 2đ?‘€đ?‘– , đ?‘– = 0, 1, 2, ‌ , đ?‘ − 1 (5) â„Ž 6 Using the continuity condition of the first derivatives at đ?‘Ľđ?‘– , đ?‘Śđ?‘– , that is ′ đ?‘†đ?‘–−1 đ?‘Ľđ?‘– = đ?‘†đ?‘–′ đ?‘Ľđ?‘– , We obtain the following consistency relation: DOI: 10.9790/5728-130201102106
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