Open Access Research

Well-posedness of fractional parabolic equations

Allaberen Ashyralyev

Author Affiliations

Department of Mathematics, Fatih University, Istanbul, 34500, Turkey

Boundary Value Problems 2013, 2013:31 doi:10.1186/1687-2770-2013-31

Published: 18 February 2013

Abstract

In the present paper, we consider the abstract Cauchy problem for the fractional differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M1">View MathML</a>

(1)

in an arbitrary Banach space E with the strongly positive operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/31/mathml/M2">View MathML</a>. The well-posedness of this problem in spaces of smooth functions is established. The coercive stability estimates for the solution of problems for 2mth 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 2mth order multidimensional fractional parabolic equation are obtained.

MSC: 65M12, 65N12.

Keywords:
fractional parabolic equation; Basset problem; well-posedness; coercive stability