Numerical galerkin method using algorithm of evolutionary search the preferred solution

Purpose. In solving boundary value problems by Galerkin method should set basic functions. Set of basic functions in analytical form does not always correspond to the total solution. Attractive set basis functions is graphically. But while the challenge is estimating the parameters of the basic functions that may include a description not linear. The aim of this work is to develop a general approach for solving boundary value tasks by Galerkin method, when the basis function is given not analytically but in geometric form. Methodology. It was proposed а new general approach for solving boundary value problems by Galerkin method. In this method basis functions were given in geometric form but not in analytic form. To search the parameters of the basis functions which were defined in geometric form is used the evolutionary algorithm. Was constructed numerical algorithm Galerkin method with the help of evolutionary algorithm random search of the most attractive solutions and graphical representations of the basis functions. Findings. It was built general scheme of numerical Galerkin method with the use an evolutionary algorithm to find the most attractive solutions. As an example, were given the results of numerous solution to the boundary problem of heat conduction in the body of the two-dimensional temperature field. Were given the results of numerous calculation when setting the basic functions in graphic form a comparison with the exact solution. It was received a satisfactory coincidence of results. Originality. It was suggested to use not analytical but graphical representation of the basis functions for solving boundary value problems by Galerkin method. The unknown parameters of the expression solution using basis functions may include non-linear. It was suggested to use an evolutionary algorithm to search for the unknown parameters of the expression of the general solution. There is provided an evolutionary algorithm with the adaptation of search terms that match the desired solution with probability 1. Practical value. It was suggested that in solving boundary value problems by Galerkin method to use a graphical representation of basis functions, allowing objects to investigate when an unknown type of analytical functions. To use Galerkin method in solving problems when the basis functions given not analytically but in geometric form will expand class solutions boundary problems, including the possibility of formulating complex relationships selection to find solutions.