The paper is devoted to the development of numerical methods for solving non-stationary problems of mathematical physics. The purpose of the work is to develop and analyze an algorithm for solving initial-boundary-value problems of mathematical physics by the method of lines (in the longitudinal and transverse variants) with the subsequent restoration of the approximate solution using interlination operators.

The paper considers the first initial-boundary-value problem for a one-dimensional inhomogeneous heat equation. To construct an approximate solution to this problem, the method of lines (in the longitudinal or transverse variants) is first applied, with the help of which a partial discretization of the problem is carried out by one of the variables. Accordingly, the original initial-boundary-value problem is reduced either to the Cauchy problem for a system of ordinary differential equations, or to a sequence of boundary-value problems for an ordinary differential equation. After obtaining the values of the approximate solution on the system of lines, the approximate solution of the original problem is restored using interlination operators. Thus, the main feature and scientific novelty of the proposed approach is the integration into the computational scheme of the method of lines the operators of polynomial interlination in the Lagrange form, which allows overcoming the discrete nature of the classical method of lines. The effectiveness of the proposed approach is demonstrated on a test problem and compared with the exact solution obtained by the Fourier method. Prospects for further research are associated with the transition to solving multidimensional non-stationary problems of mathematical physics using the constructive apparatus of the theory of R-functions.

[1] Evans L.C. Partial Differential Equations. American Mathematical Society, Providence, 2010.

[2] Kurpa L.V., Mazur O.S., Shmatko T.V. Application of the R-functions theory to the solution of nonlinear problems of multilayer plates dynamics. V dele, Kharkiv, 2016. (in Ukrainian)
[3] Lytvyn O.M. Interlination of functions. Osnova, Kharkiv, 1993. (in Ukrainian)

[4] Lytvyn O.M. Interlination of functions and some of its approaches. Osnova, Kharkiv, 2002. (in Ukrainian)

[5] Perestyuk M.O., Marynets V.V. The theory of mathematical physics equations. Lybid, Kyiv, 2006. (in Ukrainian)

[6] Schiesser W. E. The Numerical Method of Lines: Integration of Partial Differential Equations. Academic Press, San Diego, 1991.

[7] Schiesser W.E., Griffiths G.W. A Compendium of Partial Differential Equation Models: Method of Lines Analysis with Matlab. Cambridge University Press, Cambridge, 2009. https://doi.org/10.1017/CBO9780511576270

[8] Sergienko I.V., Lytvyn O.M. New information operators in mathematical modeling. Naukova dumka, Kyiv, 2018. (in Ukrainian)