Geodesic
Divergent series and generalized mixed problem for a wave equation of the simplest type
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 22 (2022) no. 3, pp. 322-331 
A. P. Khromov. Divergent series and generalized mixed problem for a wave equation of the simplest type. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 22 (2022) no. 3, pp. 322-331. http://geodesic.mathdoc.fr/item/ISU_2022_22_3_a5/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Khromov A. P., Kornev V. V., “Divergent series in the Fourier method for the wave equation”, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 27, no. 4, 2021, 215–238 (in Russian) | DOI | MR

[2] Eiler L., Differential Calculus, GITTL, M.–L., 1949, 580 pp. (in Russian)

[3] Khardi G., Divergent Series, Foreign Languages Publishing House, M., 1951, 504 pp. (in Russian) | MR

[4] Khromov A. P., “Divergent series and generalized mixed problem for wave equation”, Proceedings of the 21st International Saratov Winter School, Contemporary Problems of Function Theory and Their Applications, 21, Saratov State University Publ, Sarartov, 2022, 319–324 (in Russian) (accessed 15 February 2022)

[5] Khromov A. P., “Necessary and sufficient conditions for the existence of a classical solution of the mixed problem for the homogeneous wave equation with an integrable potential”, Differential Equations, 55:5 (2019), 703–717 | DOI | DOI | MR | Zbl

[6] Natanson I. P., The Theory of Functions of a Real Variable, GITTL, M.–L., 1957, 552 pp. (in Russian) | MR

[7] Kornev V. V., Khromov A. P., “Convergence of a formal solution by the Fourier method in a mixed problem for the simplest inhomogeneous wave equation”, Mathematics. Mechanics, 19, Saratov State University Publ, Saratov, 2017, 41–44 (in Russian)