Well-posedness of fractional parabolic equations

Allaberen Ashyralyev

doi:10.1186/1687-2770-2013-31

Abstract

In the present paper, we consider the abstract Cauchy problem for the fractional differential equation 1 in an arbitrary Banach space E with the strongly positive operators . The well-posedness of this problem in spaces of smooth functions is established. The coercive stability estimates for the solution of problems for 2m th order multidimensional fractional parabolic equations and one-dimensional fractional parabolic equations with nonlocal boundary conditions in a space variable are obtained. The stable difference scheme for the approximate solution of this problem is presented. The well-posedness of the difference scheme in difference analogues of spaces of smooth functions is established. In practice, the coercive stability estimates for the solution of difference schemes for the fractional parabolic equation with nonlocal boundary conditions in a space variable and the 2m th order multidimensional fractional parabolic equation are obtained. MSC:65M12, 65N12.

Highlights

It is known that differential equations involving derivatives of noninteger order have shown to be adequate models for various physical phenomena in areas like rheology, damping laws, diffusion processes, etc
With the help of A(t), we introduce the fractional spaces Eα(E, A(t)), < α
Problem ( ) has a unique smooth solution. This allows us to reduce problem ( ) to the abstract Cauchy problem ( ) in a Banach space E = Cμ(Rn) of all continuous bounded functions defined on Rn satisfying the Hölder condition with the indicator μ ∈ (, ) with a strongly positive operator At,x = Bt,x + δI defined by ( )

Summary

Introduction

It is known that differential equations involving derivatives of noninteger order have shown to be adequate models for various physical phenomena in areas like rheology, damping laws, diffusion processes, etc. For the solution u(t) in C(E) of initial value problem ( ), the following stability inequality holds:

Results

Conclusion