Abstract
In this paper, the boundary value problems for nonlinear third order differential equations are treated. A generic approach based on nonpolynomial quintic spline is developed to solve such boundary value problem. We show that the approximate solutions of such problems obtained by the numerical algorithm developed using nonpolynomial quintic spline functions are better than those produced by other numerical methods. The algorithm is tested on a problem to demonstrate the practical usefulness of the approach.
Highlights
IntroductionEngineering problems that are time-dependent are often described in terms of differential equations with conditions imposed at single point (initial/final value problems); while engineering problems that are position dependent are often described in terms of differential equations with conditions imposed at more than one point (boundary value problems)
We show that the approximate solutions of such problems obtained by the numerical algorithm developed using nonpolynomial quintic spline functions are better than those produced by other numerical methods
Engineering problems that are time-dependent are often described in terms of differential equations with conditions imposed at single point; while engineering problems that are position dependent are often described in terms of differential equations with conditions imposed at more than one point
Summary
Engineering problems that are time-dependent are often described in terms of differential equations with conditions imposed at single point (initial/final value problems); while engineering problems that are position dependent are often described in terms of differential equations with conditions imposed at more than one point (boundary value problems). In this paper Nonpolynomial quintic spline functions are applied to obtain a numerical solution of the following nonlinear third order two-point boundary value problems y f x, y, y , y g x y2 , x 0,1. The existence theorems for the solution of (3) subjected to boundary condition (4) are derived by Xueqin Li and Minggen Cui [5]. Our main objective is to apply non-polynomial quintic spline function [12,13,14] that has a polynomial and trigonometric parts to develop a new numerical method for obtaining smooth approximations to the solution of nonlinear third-order differential equations of the system of form (3) subjected to (4).
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