On shifted Jacobi spectral method for high-order multi-point boundary value problems

This paper reports a spectral tau method for numerically solving multi-point boundary value problems (BVPs) of linear high-order ordinary differential equations. The construction of the shifted Jacobi tau approximation is based on conventional differentiation. This use of differentiation allows the imposition of the governing equation at the whole set of grid points and the straight forward implementation of multiple boundary conditions. Extension of the tau method for high-order multi-point BVPs with variable coefficients is treated using the shifted Jacobi Gauss–Lobatto quadrature. Shifted Jacobi collocation method is developed for solving nonlinear high-order multi-point BVPs. The performance of the proposed methods is investigated by considering several examples. Accurate results and high convergence rates are achieved.

► A shifted Jacobi tau method is introduced for linear high-order multi-point boundary value problems (BVPs).
► Extension of the tau method for multi-point BVPs with variable coefficients is treated using Gauss quadrature.
► The nonlinear multi-point BVPs using Jacobi–Gauss collocation method is reduced to nonlinear algebraic equations.
► The Jacobi–Gauss collocation method has been easily implemented and gives very accurate results.
► Magnificent numerical results are obtained by selecting few collocation points.