On Third Order Stable Difference Scheme for Hyperbolic Multipoint Nonlocal Boundary Value Problem

Meltem Uzun,Ozgur Yildirim

doi:10.1155/2015/121437

Meltem Uzun, Ozgur Yildirim

Open Access

https://doi.org/10.1155/2015/121437

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Abstract

This paper presents a third order of accuracy stable difference scheme for the approximate solution of multipoint nonlocal boundary value problem of the hyperbolic type in a Hilbert space with self-adjoint positive definite operator. Stability estimates for solution of the difference scheme are obtained. Some results of numerical experiments that support theoretical statements are presented.

Highlights

Hyperbolic partial differential equations (PDEs) are of growing interest in several areas of engineering and natural sciences such as acoustic, electromagnetic, hydrodynamic, elasticity, fluid mechanics, and other areas of physics
It is known that various multipoint nonlocal boundary value problems (NBVPs) for hyperbolic equations can be reduced to the problem d2u (t) dt2
We introduce the Hilbert space L2(Ω) of all square integrable functions defined on Ω, equipped with the norm

Summary

Introduction

Hyperbolic partial differential equations (PDEs) are of growing interest in several areas of engineering and natural sciences such as acoustic, electromagnetic, hydrodynamic, elasticity, fluid mechanics, and other areas of physics (see, e.g., [1,2,3,4,5] and the references given therein). Many researchers study nonlocal boundary value problems (NBVPs) for hyperbolic and mixed types of partial differential equations. There are many results (see [6,7,8,9,10]) on integral inequalities with two dependent limits to theory of integral-differential equations of the hyperbolic type and of difference schemes for the approximate solution of these problems. An appropriate model for the analysis of stability is provided by a suitable unconditionally stable difference scheme with an unbounded operator. It is known (see [29, 30]) that various multipoint NBVPs for hyperbolic equations can be reduced to the problem d2u (t) dt. Note that many scientists have studied the solutions of boundary value problems such as parabolic equations, elliptic equations, and equations of mixed types extensively (see, e.g., [6,7,8,9,10, 21,22,23,24,25,26,27,28,29,30,31] and the references therein)

Third-Order Accurate Stable Difference Scheme

A3τ5 36

Numerical Results

Conclusion