A CLASS OF CONTINUOUS TRIGONOMETRICALLY-FITTED METHODS FOR SOLVING SECOND ORDER OSCILLATORY ORDINARY DIFFERENTIAL EQUATIONS

Abstract

In this work, a class of Continuous Trigonometrically-Fitted Methods (CTFMs) for the solution of second order oscillatory Ordinary Differential Equations (ODEs) is considered. To develop the CTFMs, the assumed approximate solution from the basis function for the problem are interpolated at some selected grid points while collocation of the differential system arising from the basis function was done at all grid points. The coefficients of the methods developed depend on the frequency of the oscillatory component of each the continuous methods. This frequency is a function of the stepsize of the methods. The unknown parameters in the system of linear equations arising from the collocation and interpolation functions were determined and the values substituted in the approximate solution to the problem. The result was simplified to obtained each of the required continuous methods. Discrete Trigonometrically-Fitted Methods (DTFMs) were recovered from the CTFMs as a by product. Stability and other properties of the discrete methods were examined to confirm the accuracy and usability of the methods.The new methods were applied to solve linear and non-linear oscillatory problems in ODEs. The results were found to be better in accuracy than any of the methods considered in literature that solved the same problems.