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In this article, we consider two inverse problems with a generalized fractional derivative. The first problem, IP1, is to reconstruct the function u based on its value and the value of its fractional derivative in the neighborhood of the final time. We prove the uniqueness of the solution to this problem. Afterwards, we investigate the IP2, which is to reconstruct a source term in an equation that generalizes fractional diffusion and wave equations, given measurements in a neighborhood of final time. The source to be determined depends on time and all space variables. The uniqueness is proved based on the results for IP1. Finally, we derive the explicit solution formulas to the IP1 and IP2 for some particular cases of the generalized fractional derivative.

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An inverse problem to recover a space-dependent factor of a source term and an order of a time derivative in a fractional diffusion equation from final data is considered. The uniqueness and stability of the solution to this problem is proved. A direct method to regularize the problem is proposed.

In this paper the method of Lavrent'ev for regularization of ill-posed problems with near-to-monotone operators is considered. An a priori rule of choice of the parameter of regularization is presented and the corresponding error estimate is derived. The theory is applied to the autoconvolution equation. Numerical examples are provided.

Inverse problems of identification of memory kernels in linear heat conduction are dealt with in case of weakly singular kernels in the space Lp and of continuous kernels with power singularity. The problems are reduced to nonlinear Volterra integral equations of convolution type for which by the method of contraction with weighted norms global existence and stability of solutions are proved.

An inverse problem to determine a space-dependent factor in a semilinear timefractional diffusion equation is considered. Additional data are given in the form of an integral with the Borel measure over the time. Uniqueness of the solution of the inverse problem is studied. The method uses a positivity principle of the corresponding differential equation that is also proved in the paper.

In this paper an inverse problem for identification of a memory kernel in heat conduction is dealt with where the kernel is represented by a finite sum of products of known spatiallydependent functions and unknown time-dependent functions. Using the Laplace transform method an existence and uniqueness theorem for the memory kernel is proved.