On the Numerical Solution of Fractional Parabolic Partial Differential Equations with the Dirichlet Condition

Zafer Cakir,Allaberen Ashyralyev

doi:10.1155/2012/696179

Abstract

The first and second order of accuracy stable difference schemes for the numerical solution of the mixed problem for the fractional parabolic equation are presented. Stability and almost coercive stability estimates for the solution of these difference schemes are obtained. A procedure of modified Gauss elimination method is used for solving these difference schemes in the case of one-dimensional fractional parabolic partial differential equations.

Highlights

IntroductionIt is known that various problems in fluid mechanics dynamics, elasticity and other areas of physics lead to fractional partial differential equations
A procedure of modified Gauss elimination method is used for solving these difference schemes in the case of one-dimensional fractional parabolic partial differential equations
It is known that various problems in fluid mechanics dynamics, elasticity and other areas of physics lead to fractional partial differential equations

Summary

Introduction

It is known that various problems in fluid mechanics dynamics, elasticity and other areas of physics lead to fractional partial differential equations. Methods of solutions of problems for fractional differential equations have been studied extensively by many researchers see, e.g., 1–28 and the references therein. The role played by stability inequalities well posedness in the study of boundary value problems for parabolic partial differential equations is well known see, e.g., 29–34. The mixed boundary value problem for the fractional parabolic equation. The first and second order of accuracy in t and second orders of accuracy in space variables difference schemes for the approximate solution of problem 1.1 are presented. A procedure of modified Gauss elimination method is used for solving these difference schemes in the case of one-dimensional fractional parabolic partial differential equations

Difference Schemes and Stability Estimates

Numerical Results

Conclusion