Solving Multi-Point Boundary Value Problems Using Sinc-Derivative Interpolation

Kenzu Abdella,Jeet Trivedi

doi:10.3390/math8122104

Abstract

In this paper, the Sinc-derivative collocation method is used to solve linear and nonlinear multi-point boundary value problems. This is done by interpolating the first derivative of the unknown variable via Sinc numerical methods and obtaining the desired solution through numerical integration of the interpolation and all higher order derivatives through successive differentiation of the interpolation. Non-homogeneous boundary conditions are reduced to homogeneous using suitable transformations. The efficiency and the accuracy of the method are tested using illustrative examples previously considered by other researchers who used different approaches. The results show the excellent performance of the Sinc-derivative collocation method.

Highlights

Due to their numerous applications in science and engineering, multi-point boundary value problems (MPBVPs) continue to attract considerable research interests [1,2,3,4]
MPBVPs constitute boundary value problems consisting of differential equations which are subjected to boundary conditions involving the boundary data of the subdomains of the overall domain under consideration
In terms of their applications, MPBVPs have been used to model a wide range of real world problems including, the optimal design of large size bridges which are contrived with multi-point supports [5], the vibration of a guy wire of uniform cross-section with multiple parts of varying densities [6], the flow of fluids through ground layers in which each layer can be considered a subdomain [7], the deformation of beams and plate deflection theory [5,7,8] and many problems involving the theory of elastic stability [9]

Summary

Introduction

Due to their numerous applications in science and engineering, multi-point boundary value problems (MPBVPs) continue to attract considerable research interests [1,2,3,4]. MPBVPs constitute boundary value problems consisting of differential equations which are subjected to boundary conditions involving the boundary data of the subdomains of the overall domain under consideration. In this paper we use the Snc-derivative collocation numerical method to solve the multi-point boundary value problem of the form y(n) = g( x, y, y0 , · · · , y(n−1) ), Mathematics 2020, 8, 2104; doi:10.3390/math8122104 x ∈ [ a, b]. We use the Sinc-derivative collocation method to obtain numerical solution of second order MPBVPs in which nonhomogeneous boundary conditions are treated via appropriate transformations which reduces them to homogeneous ones. Illustrative examples that demonstrate the accuracy of the proposed method and brief concluding remarks are presented in Section 4 and Section 5 respectively

Sinc Preliminaries

The Sinc-Derivative Method for MPBVPs

Numerical Illustrations

Conclusions