A fourth-order orthogonal spline collocation method to fourth-order boundary value problems

Abstract

In this article, we use orthogonal spline collocation methods (OSCM) for the fourth-order linear and nonlinear boundary value problems (BVPs). Cubic monomial basis functions and piecewise Hermite cubic basis functions are used to approximate the solution. We establish the existence and uniqueness solution to the discrete problem. We discuss dynamics of the stationary Swift–Hohenberg equation for different values of α. Finally, we perform some numerical experiments and using grid refinement analysis, we compute the order of convergence of the numerical method. Comparative to existing methods, we show that the OSCM gives optimal order of convergence for norms and superconvergent result for -norm at the knots with minimal computational cost.