New Modification on Heun’s Method Based on Contraharmonic Mean for Solving Initial Value Problems with High Efficiency

Abushet Hayalu Workie,Jia-Bao Liu

doi:10.1155/2020/6650855

Abstract

In this paper, small modification on Improved Euler’s method (Heun’s method) is proposed to improve the efficiency so as to solve ordinary differential equations with initial condition by assuming the tangent slope as an average of the arithmetic mean and contra-harmonic mean. In order to validate the conclusion, the stability, consistency, and accuracy of the system were evaluated and numerical results were presented, and it was recognized that the proposed method is more stable, consistent, and accurate with high performance.

Highlights

Differential equations, either ordinary derivatives or partial derivatives, are equations which contain derivatives
An ordinary differential equation together with initial condition is an initial value problem (IVP) which specifies the value of the unknown function at a given point in the domain. ere are a lot of physical problems in science and engineering which exist in the form of differential equations and are commonly used in physics, chemistry, biology, and economics [1]
Numerical methods are used to achieve the solution to the complicated differential equations [2, 3]

Summary

Introduction

Differential equations, either ordinary derivatives or partial derivatives, are equations which contain derivatives. Numerical methods are used to achieve the solution to the complicated differential equations [2, 3]. With the help of computer programming, numerical methods are very valuable tools for solving complex problems very quickly. Numerous numerical methods for solving ordinary differential equations (ODEs) with initial value problems (IVPs) have been developed by many researchers. Many authors have attempted to solve initial value problems (IVP) to obtain high accuracy rapidly by using a numerous methods, such as Euler’s method and Heun’s method, and some other methods. Some improvements have been made from time to time in numerical methods to get better performance according to our needs. Heun’s method and its modification are applied for solving ordinary differential equation in initial value problems. Is demonstrates improved performance and better accuracy compared to other well-known methods of second order present in the literature

Methodology of Research of the Proposed Method

Consistency Analysis

Numerical Results and Discussion