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Fourier method in an initial-boundary value problem for a first-order partial differential equation with involution

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Abstract

The Fourier method is used to obtain a classical solution of an initial-boundary value problem for a first-order partial differential equation with involution in the function and its derivative. The series Σ produced by the Fourier method as a formal solution of the problem is represented as Σ = S 0 + (Σ − Σ0), where Σ0 is the formal solution of a special reference problem for which the sum S 0 can be explicitly calculated. Refined asymptotic formulas for the solution of the Dirac system are used to show that the series Σ − Σ0 and the series obtained from it by termwise differentiation converge uniformly. Minimal smoothness assumptions are imposed on the initial data of the problem.

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Authors and Affiliations

  1. Voronezh State University, Universitetskaya pl. 1, Voronezh, 394006, Russia

    M. Sh. Burlutskaya

  2. Saratov State University, ul. Astrakhanskaya 83, Saratov, 410026, Russia

    A. P. Khromov

Authors
  1. M. Sh. Burlutskaya
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  2. A. P. Khromov
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Correspondence to M. Sh. Burlutskaya.

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Original Russian Text © M.Sh. Burlutskaya, A.P. Khromov, 2011, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2011, Vol. 51, No. 12, pp. 2233–2246.

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Burlutskaya, M.S., Khromov, A.P. Fourier method in an initial-boundary value problem for a first-order partial differential equation with involution. Comput. Math. and Math. Phys. 51, 2102–2114 (2011). https://doi.org/10.1134/S0965542511120086

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  • DOI: https://doi.org/10.1134/S0965542511120086

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