#### For citation:

Khromov A. P. Divergent series and generalized mixed problem for a wave equation of the simplest type. *Izvestiya of Saratov University. Mathematics. Mechanics. Informatics*, 2022, vol. 22, iss. 3, pp. 322-331. DOI: 10.18500/1816-9791-2022-22-3-322-331, EDN: PTNPTE

# Divergent series and generalized mixed problem for a wave equation of the simplest type

With the use of the operation of integrating the divergent series of a formal solution in the separating variables method, there are obtained the results concerning a generalized mixed problem (homogeneous and non-homogeneous) for the wave equation. The key moment consists in finding the sum of the divergent series which corresponds to the simplest mixed problem with a summable initial function. This result helps to get the solution of the generalized mixed problem for a non-homogeneous equation under the assumption that non-homogeneity is characterized by a locally summable function. As an application, the mixed problem with a non-zero potential is considered, in which the differential equation is treated quite formally but the mixed problem itself is no longer a generalized one: instead of the formal solution of the separating variables method we get an integral equation which can be solved by the successive substitutions method. Thus we essentially simplify the arguments.

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