In this work, an explicit solution of the initial-boundary value problem for a multidimensional time-fractional differential equation is constructed. The possibility of obtaining this equation from an integro-differential wave equation with a Mittag–Leffler–type memory kernel is shown. An explicit solution to the problem under consideration is obtained using the Laplace and Fourier transforms, the properties of the Fox H-functions and the convolution theorem.

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In the paper we prove the theorems of conditional stability of the solution of the inverse problem of determining the potential a (x) of the Schrödinger equation iut + ∆u + a (x) u = 0, in the multidimensional case x ∈ R n , in non-stationary and spectral formulations. The method of proof in the non-stationary formulation is based on the Carleman type a priori weight estimates with operator coefficients. The stability of the solution of the inverse problem in the spectral formulation is investigated using the connection of this formulation with the corresponding formulation of the inverse dynamic problem.

We present new numerical algorithms for solving the structural inverse gravimetry problem for the case of multiple surfaces. The inverse problem of finding the multiple surfaces that divide the constant density layers is an ill‐posed one described by a nonlinear integral equation of the first kind. To solve it, it is necessary to apply the regularization ideas. The new regularized variants of the gradient type methods with the weighting factors are constructed, namely, the steepest descent and conjugate gradient method. We suggest the empirical rule for choosing the regularization parameters. On the basis of the constructed methods, we elaborate the parallel algorithms and implement them in the multicore CPU using the OpenMP technology. A set of experiments with the disturbed data is performed to test the gradient algorithms and study performance of the developed code. For the test problems with quasi‐real data, these new regularized algorithms increase the accuracy and speed up computation in comparison with the unregularized ones. By using the 8‐core CPU, we achieve the speedup of 8 times.