2014 , Volume 19, № 5, p.3-11

Goal: the development of method, algorithm and software package for numerical solution of the well-posed (in the sense of Hadamard) two- and three-dimensional problems for the systems of highly nonlinear (i. e., in general, not solved with respect to higher derivatives) integro-differential algebraic equations. Method: contrary to the known methods, based on the procedure of interpolation of the unknown function by algebraic polynomials, an absolutely original approach is applied, which is based on the procedure of interpolation of the derivative of the unknown function. As a result, the necessity is eliminated to use the grid nodes, which do not belong to the domain, where the integration is performed. Consequently, the process of solving highly nonlinear integro-differential algebraic equations can be maximally formalized: it is sufficient to enter the equations in the usual mathematical form and necessary numerical parameters into the computer memory. When solving the partial differential equations, the described procedure is used in the first direction first and then the interpolation procedure in the second direction is performed using the obtained results, etc. The results of the solution of test problems for highly nonlinear integro-differential algebraic equations, which are obtained using the software in automatic mode, i. e. using the minimum of initial data without additional programming, confirm the high accuracy of the suggested approach. Conclusion: the first time in the history of computational mathematics the method is suggested and the first test programs are developed for the automatic solution of the wellposed (in the sense of Hadamard) systems of highly nonlinear integro-differential algebraic equations. Contrary to the all known software packages, the developed software computes and outtypes the node values not only of functions, but also their derivatives and, among them, probably, mixed ones.