SOLVING THE PROBLEM FOR THE FOURTH-ORDER NONSTRICTLY HYPERBOLIC EQUATION WITH DOUBLE CHARACTERISTICS

This article is concerned with studying the classical solutions of boudary problems for the fourth-order nonstrictly hyperbolic equation with double characteristics. A classical solution is understood as a function that is defined everywhere in the domain closure and has all classical derivatives entering the equation and the problem conditions. The classical solution is built in analytical form for higher-order equations of interest for computational mathematics in testing numerical algorithms. Note that the correct formulation of mixed problems for hyperbolic equations not only depends on the number of characteristics, but also on their location. The operator appearing in the equation involves a composition of first-order differential operators. The equation is defined in the half-band of two independent variables. There are Cauchy’s conditions on the domain bottom and Dirichlet’s conditions and Neumann’s conditions on other boundary. Using the method of characteristics, the analytic solution of the considered problem is written. The uniqueness of the solutions is proved. In addition, it states: under what conditions a linear differential equation with constant fourth-order coefficients can be represented in the form of the nonstrictly hyperbolic equation considered in the article.

1. Korzyuk V.I., Cheb E.S. Mixed problem for the fourth order hyperbolic equation. Izvestiia Natsionalnoi akademii nauk Belarusi. Ser. fiziko-matematicheskikh nauk [Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series], 2004, no 2, pp. 9–13. (In Russian).

2. Korzyuk V.I., Nguen Van Vin’. Classical solutions of mixed problems for the one-dimensional biwave equation. Izvestiia Natsionalnoi akademii nauk Belarusi. Ser. fiziko-matematicheskikh nauk [Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series], 2016, no 1, pp. 69–79. (In Russian).

3. Korzyuk V.I., Nguen Van Vin’. Classical solution of a problem with an integral condition for the one-dimensional biwave equation. Izvestiia Natsionalnoi akademii nauk Belarusi. Ser. fiziko-matematicheskikh nauk [Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series], 2016, no 3. pp. 16–29. (In Russian).

4. Korzyuk V.I., Vinh N.V. Cauchy problem for some fourth-order nonstrictly hyperbolic equations. Nanosystems: Physics, Chemistry, Mathematics, 2016, vol. 7, no. 5, pp. 869–879. Doi: 10.17586/2220-8054-2016-7-5-869-879

5. Korzyuk V.I., Kozlovskaya I.S. Solution of the Cauchy problem for a hyperbolic equation with constant coefficients in the case of two independent variables. Differential equations, 2012, vol. 48, no. 5, pp. 1–10. Doi: 10.1134/S0012266112050096.