On an iterative method for solution of direct problem for nonlinear hyperbolic differential equations
I.V. Boykov1, V.A. Ryazantsev2
| Аннотация | An iterative method for solution of Cauchy problem for one-dimensional nonlinear hyperbolic differential equation is proposed in this paper. The method is based on continuous method for solution of nonlinear operator equations. The keystone idea of the method consists in transition from the original problem to a nonlinear integral equation and its successive solution via construction of an auxiliary system of nonlinear differential equations that can be solved with the help of different numerical methods. The result is presented as a mesh function that consists of approximate values of the solution of stated problem and is constructed on a uniform mesh in a bounded domain of two-dimensional space. The advantages of the method are its simplicity and also its universality in the sense that the method can be applied for solving problems with a wide range of nonlinearities. Finally it should be mentioned that one of the important advantages of the proposed method is its stability to perturbations of initial data that is substantiated by methods for analysis of stability of solutions of systems of ordinary differential equations. Solving several model problems shows effectiveness of the proposed method. |
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| Ключевые слова | nonlinear hyperbolic equation, nonlinear integral equation, continuous operator method, system of differential equations, stability theory |
1Il’ya V. Boykov, Head of Higher and Applied Mathematics Chair, Penza State University (40 Krasnaya St., Penza 440026, Russia), http://orcid.org/0000-0002-6980-933X, i.v.boykov@gmail.com
2Vladimir A. Ryazantsev, Assistan Professor of Higher and Applied Mathematics Chair, Penza State University (40, Krasnaya St., Penza 440026, Russia), http://orcid.org/0000-0003-0875-9823, ryazantsevv@mail.ru
Цитирование: Boykov I. V., Ryazantsev V. A. On an iterative method for solution of direct problem for nonlinear hyperbolic differential equations // Журнал Средневолжского математического общества. 2020. Т. 22, № 2. С. 155–163.
DOI 10.15507/2079-6900.22.202002.155-163
