A HYBRID METHOD FOR GENERAL THIRD ORDER ORDINARY DIFFERENTIAL EQUATIONS.

Abstract

This work proposes numerical method for solving directly general third order ordinary differential equations by collocation at the grid points x = X n+i'i =0(l)k and at an off grid point x = Xn+uand interpolation of the approximate solution at the grid points x = X n+ i, i = o( 1) k - 1. k is the step-number of the method, u is an arbitrary rational , number in (xn,xn+k). A predictor of order 2k-l is also proposed to cater for Yn+kin the main method. Taylor series expansion is employed for the calculation of Yn+1,Yn+2 and Yn+u,and their higher derivatives. Evaluation of the resulting method at x = Xn+kf or any value of u in the specified open interval yields a particular discrete scheme as a special case of the method. At u=3/2 the interval of absolute stability of the resulting discrete method is located at the origin which makes it to be of little or no practical applications. Finally the efficiency of the method is tested on some general initial value problems of third order ordinary differential equations.