Multistage pseudo-spectral method (method of collocations) for the approximate solution of an ordinary differential equation of the first order

The classical pseudospectral collocation method based on the expansion of solution in a basis Chebyshev polynomials is considered. A new approach to constructing systems linear algebraic equations for solving ordinary differential with variable coefficients and initial (and/or boundary) conditions makes possible significant simplification structure matrices, reducing it diagonal form. system reduced multiplying matrix values selected grid by vector function describing given derivative at points. subsequent multiplication obtained two-diagonal spectral matrix, inverse respect differentiation yields all sought except first one. This coefficient determined second stage condition. novelty select class (set) functions that satisfy equation, using stable computationally simple interpolation (collocation) future solution. Then (except one) are terms calculated integration matrix. Finally, from this set solutions only those correspond selected.