Classical solution of the mixed problem for the Klein – Gordon – Fock type equation in the half-strip with curve derivatives at boundary conditions

The mixed problem for the one-dimensional Klein – Gordon – Fock type equation with curve derivatives at boundary conditions is considered in the half-strip. The solution of this problem is reduced to solving the second-type Volterra integral equations. Theorems of existence and uniqueness of the solution in the class of twice continuously differentiable functions were proven for these equations when initial functions are smooth enough. It is proven that the fulfillment of the matching conditions on the given functions is necessary and sufficient for the existence of the unique smooth solution when initial functions are smooth enough. The method of characteristics is used for the problem analysis. This method is reduced to splitting the original area of definition to the subdomains. The solution of the subproblem can be constructed in each subdomain with the help of the initial and boundary conditions. Then, the obtained solutions are glued in common points, and the obtained glued conditions are the matching conditions. This approach can be used in constructing as an analytical solution when a solution of the integral equation can be found in an explicit way, so an approximate solution. Moreover, approximate solutions can be constructed in numerical or analytical form. When a numerical solution is built, the matching conditions are essential and they need to be considered while developing numerical methods.

Klein – Gordon – Fock equation, characteristics method, curve derivatives, classical solution, mixed problem, matching conditions

1. Bogolyubov N. N., Shirkov D. V. The Quantum Fields. Moscow, Fizmatlit Publ., 2005. 384 p. (in Russian).

2. Ivanenko D. D., Sokolov A. A. Classical Field Theory (New Problems). Moscow, Leningrad, Gostekhteoretizdat Publ., 1951. 479 p. (in Russian).

3. Baranovskaya S. N., Yurchuk N. I. Mixed problem for the string vibration equation with a time-dependent oblique derivative in the boundary condition. Differential Equations, 2009, vol. 45, no. 8, pp. 1212–1215. https://doi.org/10.1134/ s0012266109080126

4. Lomovtsev F. E., Novikov E. N. Necessary and sufficient conditions for the vibrations of a bounded string with directional derivatives in the boundary conditions. Differential Equations, 2014, vol. 50, no. 1, pp. 128–131. https://doi.org/10.1134/ S0374064114010178

5. Korzyuk V. I., Stolyarchuk I. I. Classical solution to the mixed problem for the wave equation with the integral condition. Doklady Natsional’noi akademii nauk Belarusi = Doklady of the National Academy of Sciences of Belarus, 2016, vol. 60, no. 6, pp. 22–27 (in Russian).

6. 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

7. Mikhlin S. G. Course of Mathematical Physics. 2nd ed. Saint Petersburg, Lan’ Pybl., 2002. 575 p. (in Russian).

8. Korzyuk V. I., Stolyarchuk I. I. Classical solution to the mixed problem for the Klein-Gordon-Fock equation with the unlocal conditions. Doklady Natsional’noi akademii nauk Belarusi = Doklady of the National Academy of Sciences of Belarus, 2017, vol. 61, no. 6, pp. 20–27 (in Russian).

9. Korzyuk V. I., Stolyarchuk I. I. Classical solution to the mixed problem for the Klein – Gordon – Fock equation with the unlocal conditions. Trudy Instituta matematiki = Proceedings of the Institute of Mathematics, 2018. vol. 26, no. 1, pp. 56–72 (in Russian).

10. Kamke E. Gewöhnliche Differentialgleichungen. St. Petersburg, Lan’ Publ., 2003, 589 p. (in Russian).