A New Modified Algorithm Based Upon Exponential and Polynomial Functions for Solving Initial Value Problems

doi:10.7176/mtm/11-6-01

Abstract

The main focus of this paper is to propose a fourth-order convergent modified numerical algorithm by using an exponential and polynomial type function for getting the approximate solution of initial value problems (IVPs) in ordinary differential equations (ODEs). The proposed algorithm has been tested and analyzed for the convergence, stability, consistency, of several numerical experiments that proved the superiority of the proposed algorithm against algorithms including the fourth-order arithmetic mean (AM), the fourth-order harmonic mean (HM), and the fourth-order Contra harmonic mean( ). Keywords: Modified numerical algorithm, Ordinary differential equation, Order of convergence, Initial value problems, Stability, and Consistency. DOI: 10.7176/MTM/11-6-01 Publication date: December 31 st 2021

Highlights

Many researchers are working to solve many real-world physical problems
The idea of stability may be taken in different conditions: it may be correlated with the specific numerical algorithm used, or the step size h used in numerical computations or with the particular problem being solved, one www.iiste.org of the popular way to apply the test problem for stability analysis of the proposed numerical algorithm is
Proposed numerical algorithm is more accurate than the fourth-order arithmetic mean (AM), The fourth-order harmonic mean (HM), and The fourthorder Contra harmonic mean (CvM) as we can see from Tables [1,2,3,4]

Summary

Introduction

Many researchers are working to solve many real-world physical problems. They are the ones that came up with the laws and theories that are related to natural physical problems. It is a known fact that several mathematical models arising from the real and physical life situations cannot be solved explicitly in most of the cases such as nonlinear lotka Volterra competition model and logistic equation in population dynamics, Lorenz system in meteorology, pendulum, and duffing equations in mechanical engineering, VanderPol equation in electrical engineering, Newton’s law of cooling in thermodynamics, geodesic equation in geology, radioactive decay in nuclear physics, the motion of a charged particle, Fermi–Ulam–Pasta Oscillator and many more In such situations, numerical methods of different characteristics and orders are needed mainly due to nonlinear terms involved in the practical problems.

Derivation of New modified algorithm

Consistency Analysis

NUMERICAL PROBLEMS

CONCLUSION