The classical solution of arbitrary smoothness for the first mixed problem for the Klein – Gordon – Fock type equation
https://doi.org/10.29235/1561-2430-2022-58-1-34-47
Abstract
In this paper, we consider the first mixed problem for the one-dimensional Klein – Gordon – Fock type equation in a half-strip. Meanwhile, the existence and uniqueness of a solution of arbitrary smoothness is researched. While solving this problem using the method of characteristics, equivalent second type Volterra integral equations appear. The existence of a unique solution in the class of n times continuously differentiable functions is proven for these equations when initial functions are smooth enough. Moreover, it is shown that for the smoothness of the solution of the initial problem it is necessary and sufficient that the matching conditions for the given functions be fulfilled if they are sufficiently smooth. The method of characteristics, used for problem analysis, is reduced to separating the total area of the solution on subdomains in each of them so that the solution of the subproblem is constructed with the help of the initial and boundary conditions. Then, the obtained solutions are glued in common points, and the received glued conditions are the matching conditions. This approach permits to construct both exact and approximate solutions. The exact solutions can be found when it is possible to solve the equivalent Volterra integral equations. Otherwise, one can find an approximate solution of the problem either in analytical or numerical form. Along with this, when constructing an approximate solution, the matching conditions turn out to be essential, which must be taken into account when using numerical methods for solving the problem.
About the Authors
V. I. Korzyuk
Institute of Mathematics of the National Academy of Sciences of Belarus; Belarusian State University
Belarus
Belarus
Viktor I. Korzyuk – Academician of the National Academy of Sciences of Belarus, Dr. Sc. (Physics and Mathematics), Professor
11, Surganov Str., 220072, Minsk
4, Nezavisimosti Ave., 220030, Minsk
I. I. Stolyarchuk
Institute of Mathematics of the National Academy of Sciences of Belarus
Belarus
Belarus
Ivan I. Stolyarchuk – Ph. D. (Physics and Mathematics)
11, Surganov Str., 220072, Minsk
References
1. Korzyuk V. I., Stolyarchuk I. I. Classical solution of the first mixed problem for the Klein – Gordon – Fock equation in a half-strip. Differential Equations, 2014, vol. 50, no. 8, pp. 1098–1111. https://doi.org/10.1134/S0374064114080081
2. Korzyuk V. I., Stolyarchuk I. I. Classical Solution of the First Mixed Problem for Second-Order Hyperbolic Equation in Curvilinear Half-Strip with Variable Coefficients. Differential Equations, 2017, vol. 53, no. 1, pp. 74–85. https://doi.org/10.1134/S0012266117010074
3. Korzyuk V. I., Dzhenaliev M. T., Kozlovskaya I. S. Boundary problems for a weakly loaded operator of the second order hyperbolic equation in th cylindrical area. Doklady Natsionalꞌnoi akademii nauk Belarusi = Doklady of the National Academy of Sciences of Belarus, 2015, vol. 59, no. 6, pp. 33–39 (in Russian).
4. Ladyzhenskaya O. A. On the Solvability of Mixed Boundary Value Problems for hyperbolic equations. Izvestia Akademii nauk SSSR Matemticheskay seriya [Proceedings of the Academy of Sciences of USSR. Mathematical series], 1951, no. 15, pp. 545–562 (in Russian).
5. Moiseev E. I., Kholomeeva A. A. Solvability of the mixed problem for the wave equation with a dynamic boundary condition. Differential Equations, 2012, vol. 48, no. 10, pp. 1392–1397. https://doi.org/10.1134/S0012266112100096
6. Korzyuk V. I., Stolyarchuk I. I. Classical solution of the first mixed problem for the wave equation in the cylyndrical domain. Doklady Natsionalꞌnoi akademii nauk Belarusi = Doklady of the National Academy of Sciences of Belarus, 2021, vol. 65, no. 2, pp. 135–138 (in Russian). https://doi.org/10.29235/1561-8323-2021-65-2-135-138
7. Korzyuk V. I. Classical solution to the mixed problem for the Kleina – Gordona – Foka equation with the unlocal conditions. Trudy Instituta matematiki = Proceedings of the Institute of Mathematics, 2018. vol. 26, no. 1, pp. 56–72 (in Russian).
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