A spectral Rayleigh–Ritz scheme for nonlinear partial differential systems of first order

Nonlinear systems of partial differential problems of first order with Dirichlet boundary conditions is considered. ultraspherical integral zero- boundary (UIZB) method is combined with Rayleigh–Ritz method to approximate the unknowns. The approach converts the problem to be a multi-objective constrained optimization problem which is easier to solve. Accurate results can be obtained by selecting a limited number of collocation points. Numerical examples are included to demonstrate the accuracy and efficiency of the propped method.