Application of spline to approximate the solution of singularly perturbed boundary-value problems

We develop a class of new methods based on modification of polynomial spline function for the numerical solution of singularly perturbed boundary-value problems. The modified spline contains the exponential terms and named by tension spline, which is infinity smooth. Tension spline contain parameter, by choosing arbitrary values of such parameters the various classes of spline can be obtained. The proposed methods are accurate for solution of linear and non-linear singularly perturbed boundary-value problems. Boundary formulas are developed to associate with spline methods. These methods are converging. The analysis of convergence is shown to yield up to O(h(8)) approximation to the solution of singularly perturbed boundary-value problems. Comparison are carried out, numerical examples are given to showing the efficiency of our methods.