Numerical Solution and Distinguishability in Time Fractional Parabolic Equation

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 D(t)(alpha)u(x, t) = (k(x)u(x))(x) + r(t)F(x, t), 0 < alpha = 1, with mixed boundary conditions u(0, t) = psi(0)(t), u(x)(1, t) = psi(1)(t). By defining the input-output mappings Phi[center dot] : kappa -> C-1[0, T] and psi[center dot] : kappa -> C[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 Phi[center dot] and psi[center dot]. 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 Phi[center dot] : kappa -> C-1[0, T] and psi[center dot] : kappa -> C[0, T] can be described explicitly, where Phi[r] = k(x)u(x)(x, t; r)vertical bar(x= 0) and psi[r] = u(x, t; r)vertical bar(x= 1). Also, numerical tests using finite difference scheme combined with an iterative method are presented.