Optimal B-spline collocation method for self-adjoint singularly perturbed boundary value problems

We present a B-spline collocation method of higher order for a class of self-adjoint singularly perturbed boundary value problems. The essential idea in this method is to divide the domain of the differential equation into three non-overlapping subdomains and solve the regular problems obtained by transforming the differential equation with respective boundary conditions on these subdomains using the present higher order B-spline collocation method. The boundary conditions at the transition points are obtained by the asymptotic approximation of order zero to the solution of the problem. The convergence analysis is given and the method is shown to have optimal order convergence; by collocating the perturbed differential equation, which is satisfied by a special cubic spline interpolate of the true solution. Numerical experiments are conducted to demonstrate the efficiency of the method.