Non-polynomial Spline Approximations for the Solution of Singularly-perturbed Boundary Value Problems

We consider the self-adjoint singularly perturbed two-point boundary value problems. We know that the numerical methods for solution of such problems based on nonpolynomial spline in grid points, can produce the fourth order method only. So that we look for an alternative to obtain higher order methods. We develop the non-polynomial spline in off-step points to rise the order of accuracy. Based on such spline, the purposed new methods are fourth, sixth and eighth-order accurate. These methods are applicable to problems both in singular and non-singular cases. The convergence analysis of the new eight-order method is proved. We applied the presented methods to test problems which have been solved by other existing methods in our references, for comparison of our methods with the existing methods. Numerical results are given to illustrate the efficiency of our methods.