A method and a software package for numerical solution of the systems of nonlinear ordinary integro-differential-algebraic equations

A method, an algorithm and a software package for automatically solving the ordinary nonlinear integro-differential-algebraic equations (IDAEs) of a sufficiently general form are described. The author understands an automatic solution as obtaining a result without carrying out the stages of selecting a method, programming, and program checking. Both initial and boundary value problems for such equations are solved. It is assumed that the complete set of boundary and initial conditions at the beginning of the integration interval are given. By performing differentiation, the system of IDAEs can be modified, in general, into a system of ordinary nonlinear differential equations (IDEs). The problem of finding the solution of the above-mentioned system on the uniform grid on the integration interval is posed in two forms: as solving the system of IDAEs and as solving the appropriate system of IDEs, where the developed program is to be used. In order to reduce the system of IDAEs and the system of IDEs to the systems of ordinary nonlinear algebraic equations, at every stage of the algorithm the integration and differentiation formulas obtained earlier by N.G. Bandurin are used. Systems similar to those test systems of both nonlinear IDAEs and IDEs considered in this investigation are solved by using the computer programs. It is evident that the coincidence of the results for one and the same system of equations in its different forms can serve as good evidence of the correctness of the obtained results.