Numerical solution of first order initial value problem using quartic spline method

Abstract

Any first order initial value problem can be integrated numerically by discretizing the interval of integration into a number of subintervals, either with equally distributed grid points or non-equally distributed grid points. Hence, as the integration advances, the numerical solutions at the grid points are calculated and being known. However, the numerical solutions between the grid points remain unknown. This will form difficulty to individuals who wish to study a particular solution which may not fall on the grid points. Therefore, some sorts of interpolation techniques are needed to deal with such difficulty. Spline interpolation technique remains as a well known approach to approximate the numerical solution of a first order initial value problem, not only at the grid points but also everywhere between the grid points. In this short article, a new quartic spline method has been derived to obtain the numerical solution for first order initial value problem. The key idea of the derivation is to treat the third derivative of the quartic spline function to be a linear polynomial, and obtain the quartic spline function with undetermined coefficients after three integrations. The new quartic spline function is ready to be used when all unknown coefficients are found. We also described an algorithm for the new quartic spline method when used to obtain the numerical solution of any first order initial value problem. Two test problems have been used for numerical experimentations purposes. Numerical results seem to indicate that the new quartic spline method is reliable in solving first order initial value problem. We have compared the numerical results generated by the new quartic spline method with those obtained from an existing spline method. Both methods are found to have comparable accuracy.