Abstract
This paper is devoted to approximate methods of solving direct and inverse problems for parabolic equations. An approximate method to solve the initial problem of a multidimensional nonlinear parabolic equation has been proposed. It is based on reducing the initial problem to a nonlinear multidimensional Fredholm intergral equation of the second kind, which is approximated by a system of nonlinear algebraic equatiions using a method of mechanical quadratures. In constructing a computational scheme, the points of local splines have been applied for optimal with respect to order approximation of a functional class that contains the solutions of parabolic equations. For the numerical implementation of the computational scheme, we have used the generalization of a continuous method of solving nonlinear operator equations that is described in the paper. In addition, the inverse problem of a parabolic equation with a fractional order derivative with respect to a time variable has been studied. Approximate methods of determining the fractional order of the time derivative and a coefficient at a spatial derivative have been proposed.
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Boykov, I.V., Ryazantsev, V.A. On an Iterative Method of Solving Direct and Inverse Problems for Parabolic Equations. Tech. Phys. 68, 250–263 (2023). https://doi.org/10.1134/S1063784223700160
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DOI: https://doi.org/10.1134/S1063784223700160