Construction of an Exponentially-Fitted Multiderivative Milne-Simpson Method

Introduction: Application of classical methods to oscillatory or periodic problems is significantly hindered due to the fact that very small step size is required with corresponding decrease in performance, especially in terms of efficiency. Aims: To overcome this limitation, the construction of a class of two-step exponentially-fitted Milne--Simpson's methods involving first and second derivatives is presented in this work. Materials and Methods: This construction is based on the six-step flow chart described in the literature. In this work, a classical multi--derivative Milne--Simpson's method is constructed and fitted exponentially to allow for easy application to oscillatory or periodic problems. Results: In this work, we extended the classical two-step fourth-order Milne-Simpson to involve the second derivative and hence increasing the attainable order of the method, the extended method is fitted exponentially. Conclusion:The constructed class of methods is shown to be of order of six (6) and well suited for oscillatory or periodic problems.