Kernel determination problem in multi-dimensional time fractional diffusion equation in bounded domain

Аннотация

In this paper, we investigate the inverse problem on identification of the convolution kernel in an integral term of a multi–dimensional time–fractional diffusion equation, where the spatial operator is a uniformly elliptic operator in divergence form. The corresponding direct problem is formulated as an initial–boundary value problem for a multi–dimensional time–fractional integro–differential diffusion equation. We establish global existence and uniqueness theorems for the solution to the direct problem by using the Fourier spectral method. To determine the unknown kernel, we impose an additional integral–type overdetermination condition on the solution of the direct problem. The inverse problem is then transformed into an equivalent auxiliary problem. By applying fractional integration and differentiation techniques the inverse problem is reduced to a nonlinear Volterra integral equation of the second kind with convolution structure. Using the fixed point principle, we prove local existence and global uniqueness of the solution to the inverse problem. In addition, a stability estimate for the solution to the inverse problem is obtained.