Solution of the Nonlinear Problems of Mathematical Physics by the Collocation Method

Abstract

The work objective is to develop a collocation method for numerical solving nonlinear boundary value problems of mathematical physics. A feature of the being testing numerical method, the direct collocation method, is the irregular arrangement of the collocation nodes in the solution domain. That can drastically increase the accuracy of the numerical solution by improving the quality of the linear algebraic equations system that the boundary value problem leads to. Using various systems of basis functions, we present a numerical solution in the form of a polynomial, trigonometric series, and a series of local basis functions. The proposed method allows us to obtain an approximate solution of boundary value problems for a wide range of elliptic, parabolic, wave linear and nonlinear equations in an analytical form. To confirm the effectiveness of the studied numerical methods, two-dimensional and three-dimensional boundary value problems were solved for linear and nonlinear elliptic and parabolic equations with known solutions. The dependences of the numerical solution error on the number of linear equations in the resulting system are obtained. It is shown that even with a small number of equations in the system, the achieved solution accuracy is higher than the accuracy of alternative numerical methods. The investigated numerical method allows us to significantly expand the application area of traditional numerical methods in solving applied problems of modeling various physical fields, described by linear and nonlinear elliptic and parabolic equations. The results obtained in this paper show the high potential capabilities of the direct collocation method, which are based on the universality of the method and the high accuracy of numerical solutions. These qualities of the method indicate the prospects of its use in solving a wide range of applied problems.