Determination of the unknown source function in time fractional parabolic equation with Dirichlet boundary conditions

Abstract

This article deals with the mathematical analysis of the inverse problem of identifying the distinguishability of input-output mappings in the linear time fractional inhomogeneous parabolic equation Dt ? u(x, t)=(k(x)ux)x+r(t)F(x, t) 0 < ? ? 1, with Dirichlet boundary conditions u(0, t) = ?0(t), u(1, t) = ?1(t). By defining the input-output mappings ?[·]: K ?C1[0,T ] and ?[·]: K ? C1[0,T] the inverse problem is reduced to the problem of their invertibility. Hence, the main purpose of this study is to investigate the distinguishability of the input-output mappings ?[·] and ?[·]. Moreover, the measured output data f (t) and h(t) can be determined analytically by a series representation, which implies that the input-output mappings ? [·] :K ? C1[0,T] and ?[·] :K ? C1[0,T] can be described explicitly. © 2016 NSP Natural Sciences Publishing Cor.