Automated differential equation solver based on the parametric approximation optimization

Abstract: The numerical methods for differential equation solution allow obtaining a discrete field that converges towards the solution if the method is applied to the correct problem. Nevertheless, the numerical methods have the restricted class of the equations, on which the convergence with a given parameter set or range is proved. Only a few "cheap and dirty" numerical methods converge on a wide class of equations without parameter tuning with the lower approximation order price. The article presents a method that uses an optimization algorithm to obtain a solution using the parameterized approximation. The result may not be as precise as an expert one. However, it allows solving the wide class of equations in an automated manner without the algorithm's parameters change.