Abstract
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.
Highlights
Hyperbolic differential equations are a powerful instrument of mathematical modelling of different processes and phenomena which modern science collides with
A wide range of numerical methods is used for approximate solution of hyperbolic equations:
In the paper we propose a numerical method for approximate solution of nonlinear hyperbolic differential equations
Summary
Hyperbolic differential equations are a powerful instrument of mathematical modelling of different processes and phenomena which modern science collides with. At present time there are many results in the field of development of numerical methods for solution of linear and nonlinear hyperbolic equations [1,2,3,4,5,6,7]. By means of the general solution formula the original problem is replaced by a nonlinear integral equation. This equation is solved with the help of continuous operator method. Let us introduce a brief description of continuous operator method following the paper [13]. Suppose it is required to find a solution of the operator equation Let us introduce a brief description of continuous operator method following the paper [13]. A solution of the problem (1.2)–(1.3) converges to the solution y∗ of the equation (1.1)
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