Use of partial derivatives to derive a convergent numerical scheme with its error estimates

Abstract

Using the idea of the partial derivative with respect to the ordinate of a given mathematical function, a new numerical scheme having third order convergence has been devised for solving initial value problems in ordinary differential equations. Such problems are deemed to be indispensable in diverse fields of science, medical and engineering and are most often required to be solved by the numerical schemes. In view of this, the proposed numerical scheme is found to be efficient in solving both autonomous and non-autonomous type of problems as supported by some numerical experiments in the present study. Using the Taylor expansion for the slopes involved in the scheme, the leading term of the local truncation error is shown to have contained O(h 4) which proves third order accuracy of the scheme. In addition to this, consistency and linear stability analysis of the proposed scheme has extensively been discussed. Numerical experiments show better performance of the proposed numerical scheme when compared with existing numerical schemes of the same order as that of the scheme proposed. CPU time (seconds), maximum absolute relative error and the absolute relative error, computed at the last grid point of the integration interval for the associated initial value problem, are the parameters to test the performance of the proposed numerical scheme. MATLAB Version: 9.4.0.813654 (R2018a) in double-precision on a personal computer equipped with a Processor Intel (R) Core(TM) i3-4500U CPU@ 1.70 GHz running under the Windows 10 operating system has been employed in order to carry out all the required numerical computations.

Development of an Explicit Iterative Numerical Scheme Over the Modified Euler’s Method

“…Local truncation error can be found by taking the difference of exact Taylor series for v(µ + β) and Taylor series generated from composed scheme v i+1 . Taylor series for v(µ + β) is expanding as follow [12,14,19].…”

A four-stage multiderivative explicit Runge-Kutta method for the solution of first order ordinary differential equations

“…Among these standard numerical techniques, there have been growing interest in multiderivative methods to deal with these models that are in form of initial value problems. The list of different multiderivative methods can be found in [1], [2], [3], [5], [8], [10], [12], [13], [17] to mention a few. In this paper, our motivation is drawn from the research works of Goeken [10], Akanbi [3] and Wusu [17] to derive a 4-stage multiderivative method that will compete well with some 4-stage methods.…”