Numerical solution of mathematical physics problems by the collocation method

Abstract

A modified collocation method for the numerical solving boundary value problems of mathematical physics is proposed. The irregular arrangement of collocation nodes in the problem solving domain can sharply increase the accuracy of the numerical solution by improving the quality of the linear algebraic equations system, to which the solved boundary value problem leads. Various basis functions systems are considered. The proposed method allows one to obtain an approximate solution of boundary value problems for a wide range of linear and nonlinear elliptic, parabolic and wave equations in an analytical form. This numerical method makes it possible to significantly expand the application field of traditional numerical methods when solving applied problems for modelling fields of various physical natures, described by linear and nonlinear equations of mathematical physics. The developed method is used to solve a quantum-mechanical problem for a hydrogen molecule ion. The results obtained in this work show the high potentialities of the complete collocation method, which are based on the universality of the method and high accuracy of numerical solutions. The energy of the ion ground state calculated with the minimum number of collocation nodes differs from the experimentally obtained value by 13%.