Research paper : On a class of time-fractional differential equations

Abstract

In this paper we investigate Cauchy problem for a class of time-fractional differential equation D αtu(t) + c 1D β1tu(t)+ ... + c dD βdtu(t) = Au(t), t>0, u (j)(0) = x j, j= 0, ... ,m - 1, (0.1) where A is a closed densely defined linear operator in a Banach space X, a > β 1 > ... > β d > 0, c j are constants and m = [α]. A new type of resolvent family corresponding to well-posedness of (0.1) is introduced. We derive the generation theorems, algebraic equations and approximation theorems for such resolvent families. Moreover, we give the exact solution for a kind of generalized fractional telegraph equations. Some examples are given as illustrations. © 2012 Diogenes Co., Sofia.