Numerical Solution of Boundary and Initial Boundary-Value Problems Using Polynomial And Non-Polynomial Spline Functions Method

Abstract

The boundary and initial boundary value problems have always played a vital character in the fields of science and technology. Different numerical techniques are used to obtain numerical approximations of such problems. We present and illustrate novel numerical techniques for the numerical approximations of higher order boundary and initial boundary value problems. The numerical techniques derived in this research work are based upon the fact of employing polynomial cubic spline (PCS) scheme and non polynomial cubic spline (NPCS) scheme in conjunction with the decomposition procedure. In the case for ordinary differential equations, the decomposition procedure is used to reduce the higher order boundary value problems (BVPs) into the corresponding system of second order boundary value problems. Then PCS and NPCS schemes are constructed for each second order ordinary differential equation. The first order derivatives are approximated by the central finite differences of (ℎ ). For partial differential equations, the second order time derivatives are decomposed into the first order derivatives. The process of decomposition generates a linear system of partial differential equations, where the first order time derivatives are approximated by the central finite differences. The performance of the new derived schemes is illustrated by numerical tests that involve comparing numerical approximations with analytical solutions on a collection of carefully selected problems from the literature. These problems range from those involving higher order ordinary differential equations, for example, fifth, sixth, seventh, twelfth, and thirteenth order ordinary differential equations and partial differential equations, like fourth order parabolic equations, one dimensional hyperbolic telegraph equations, and one dimensional wave equations. In addition, Adomian decomposition method is used to construct the boundary conditions for the solution of fourth order parabolic equations.