Initial-Boundary Value Problem for Fractional Partial Differential Equations of Higher Order

Djumaklych Amanov,Allaberen Ashyralyev

doi:10.1155/2012/973102

Abstract

The initial-boundary value problem for partial differential equations of higher-order involving the Caputo fractional derivative is studied. Theorems on existence and uniqueness of a solution and its continuous dependence on the initial data and on the right-hand side of the equation are established.

Highlights

Many problems in viscoelasticity 1–3, dynamical processes in self-similar structures 4, biosciences 5, signal processing 6, system control theory 7, electrochemistry 8, diffusion processes 9, and linear time-invariant systems of any order with internal point delays 10 lead to differential equations of fractional order
We present some basic definitions and preliminary facts which are used throughout the paper
2.5 has a unique solution z t defined by the following formula: tztftλt − τ α−1Eα,α λ t − τ α f τ dτ, 2.6 where Eα,β z

Summary

Introduction

Many problems in viscoelasticity 1–3 , dynamical processes in self-similar structures 4 , biosciences 5 , signal processing 6 , system control theory 7 , electrochemistry 8 , diffusion processes 9 , and linear time-invariant systems of any order with internal point delays 10 lead to differential equations of fractional order. In the paper 44 , the initial-boundary value problem for heat conduction equation with the Caputo fractional derivative was studied. Abstract and Applied Analysis value problem for partial differential equations of higher order with the Caputo fractional derivative was studied in the case when the order of the fractional derivative belongs to the interval 0,1. In the paper 46 , the initial-boundary value problem in plane domain for partial differential equations of fourth order with the fractional derivative in the sense of Caputo was studied in the case when the order of fractional derivative belongs to the interval 1,2. The present paper generalizes results of 46 in the case of space domain for partial differential equations of higher order with a fractional derivative in the sense of Caputo.

Preliminaries

L2 Ω0 f

Conclusion