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