Automatic solution to systems of ordinary differential equations by the Tau Method

Ortiz and Samara's operational approach to the Tau Method is extended to the numerical solution of systems of linear and nonlinear ordinary differential equations (ODEs), together with initial or boundary conditions. They lead to accurate results through the use of simple algorithms. A Tau software called TAUSYS3 for mixed-order systems of ODEs was written based on this approach. In this paper we give a brief descriptions of the Tau Method, the structure of the Tau program, and the testing of the TAUSYS3. We consider several examples and report results of high accuracy. These include linear and nonlinear, stiff and singular perturbation problems for ordinary and systems of ordinary differential equations in which the solution may not be unique.