Romanovski-Jacobi spectral schemes for high-order differential equations

We develop direct solution techniques for solving high -order differential equations with constant coefficients using the spectral tau method. The spatial approximation is based on RomanovskiJacobi polynomials { R-n(alpha,beta )(x)(n=0)(N )with alpha > -1, beta < -2N-alpha-1, x is an element of (0, infinity) and n is the polynomial degree. Then, a hybrid approach combining the Romanovski-Jacobi tau method with the Romanovski-Jacobi collocation technique is introduced for the numerical solution of high -order differential equations with variable coefficients. The main characteristic behind this approach is that it reduces the problems to those of solving systems of algebraic equations, which greatly simplifies the problem. Finally, the Romanovski-Jacobi collocation method is presented for solving nonlinear high -order initial value problems. Numerical examples are included to demonstrate the validity and applicability of the techniques, and comparisons are made with the existing results. The method is easy to implement and yields very accurate results.