Construction and implementation of fourth-order Runge-Kutta method based on Contra-harmonic mean

CONSTRUCTION AND IMPLEMENTATION OF FOURTH-ORDER RUNGE-KUTTA METHOD BASED ON CONTRA-HARMONIC MEAN

ABSTRACT

In this project “Construction and implementation of fourth-order Runge-Kutta method based on Contra-harmonic mean” new Runge-Kutta formulae of order four based on the contra- harmonic mean was derived. Numerical evidence confirming the accuracy of the new method and comparison with alternative Runge-Kutta formula based on arithmetic, geometric and harmonic means are also presented.

Contents

Certification………………………………………………………………………………………….. i

Abstract……………………………………………………………………………………………… ii

Dedication………………………………………………………………………………………….. iii

Acknowledgement……………………………………………………………………………….. iv

INTRODUCTION 2
BACKGROUND OF STUDY………………………………………………………….. 2
PRELIMINARIES………………………………………………………………. 5
STATEMENT OF PROBLEM………………………………………………………… 7
OBJECTIVE OF THE STUDY……………………………………………………. 8
SIGNIFICANCE OF STUDY…………………………………………………………… 8
LITERATURE REVIEW 10
INTRODUCTION……………………………………………………………………….. 10
The necessity of a solution to a differential equation………………… 10
History of Ordinary Differential Equations……………………………………….. 11
Overview………………………………………………………………………. 11
Evolution of the methods of solving ordinary Differential Equations as seen in literature………………………………………. 14

Methodology 19
INTRODUCTION……………………………………………………………………….. 19
NEW FOURTH ORDER CONTRA-HARMONIC MEAN C_oM 21
NUMERICAL EXAMPLE 27
CONCLUSION 29

Reference 30

Chapter 1 INTRODUCTION

This chapter focuses on the background of the study, statement of the problem, objectives of the study and significant of the study. It also highlights the key preliminaries in the study and outlines how the study will be carried out.

BACKGROUND OF STUDY

A differential equation is an equation that relates a function to its derivatives. The unknown is the function. A differential equation is said to be ordinary if the function is univariate and more precisely if it’s domain is a connected subset of R.

Ordinary differential equations arise in many different contexts. These different contexts include fundamental laws of physics, mechanism, electricity, thermodynamics and also population and growth modelling. Ordinary differential equations (ODEs) arise in many contexts of mathematics and social and natural sciences, for example, in physics, the Legendre DE, which is a self-adjoint ODE, arises in the solutions of hydrogen atom wave functions and angular momentum in single particle quantum mechanics. Their solution form the polar angle part of the spherical harmonics basis for the multi pole expansion, which is used in both electromagnetic and gravitational statics. In engineering, for example, many difficult problems in the field of static and dynamic mechanics can be solved by computing the solutions self-adjoint Bessel equations.

Mathematical descriptions of change use differentials and derivatives. Various differential derivatives and functions become related via equations, such that a differential is a result that describes dynamically changing phenomenal, evolution and variation. Often quantities are define as the rate of change of other quantities (for example, derivatives of displacement with respect to time), or gradients of quantities, which is how they enter differential equations.

It is a common truth that differential equations are among the most important mathematical tools used in producing models in the engineering, mathematics, physics, aeronautics, elasticity, astronomy, dynamics, biology, chemistry, environmental sciences, social sciences, banking and many other areas.

During the past few decades, the development of non-linear analysis, dynamical systems and their applications to science and engineering has stimulated renewed enthusiasm for the theory of ordinary differential equations (ODE). Differential equation and mathematical modelling can be used to study a wide range of social issues. Among the topics that have a natural fit with the mathematics in a course on ordinary differential equations are all aspects of populations problems; growth of population, overpopulation, carrying capacity of an ecosystem, the effect of harvesting, such as hunting or fishing, on a population and how over-harvesting can lead to species extinction, interactions between multiple species populations, such as predator-prey, co- operative and competitive species.

Ordinary differential equations are ubiquitous in science and engineering, in geometry and mechanics, in chemical reaction kinetics, molecular dynamics and many more application areas. They also arise after semi-discretization in space, in the numerical treatment of time-dependent partial differential equations, which are even more impressively omnipresent in our technologically developed and financially controlled world.

The most common specific fields that require modelling in terms of differential equations include geometry and analytical mechanics. Scientific fields include much of physics and astronomy (celestial mechanics), metrology (weather modelling), chemistry (reaction rates), biology (infections diseases, genetic variation), ecology and population modelling (population competition), eco- nomics (stock trends, interest rates and market equilibrium price changes). Finding solution to differential equations allow me to make reasonable pre- diction about most natural phenomena, the problem here is what method do I use to solve an ODE? Which is the simplest method? and how accurate is the method? All these questions have to be taken into account before solving an ODE and this research will help to answer these questions by classifying the different available methods according to different case usages comparing their relative efficiency so that a novice mathematician or other researchers find it simple to choose a method to use when they encounter a differential equation.

To determine the solution of differential equations, there are different analytical methods available.in certain cases, however, analytical methods are not capable of solving such complicated or complex differential equations. for this reason approximate solutions through numerical methods are important. Numerical methods are used to achieve the solution to the complicated differential equations. Among the existing numerical methods, Runge-Kutta method is one of the best commonly used numerical method for solving an initial value problem (IVP). It is indeed a great taste to analyze and modify Runge-kutta (RK) methods.

This is mainly because of it’s nature of efficiency, flexibility and accuracy. During the last few decades, there has been a growing interest in problem solving systems based on the Runge-kutta methods. In the development of methods for solving ordinary differential equations it is not clear that the arithmetic mean is always the best choice (Sanugi B, 1986). Naturally the arithmetic mean formulae are the most convenient to use but are not necessarily the most accurate formulae to use for all types of problems. This has been shown in recent years by researchers. There are, however, several other types of ’mean’ which have also produced consistent approximations. [Evans and Yaacob, 1995], constructed a new 4th order Runge-kutta method based on the Heronia formular in which they used Heronian mean in the derivation. They went further to compare the scheme with several Runge-Kutta methods of 4th order based on variety of mean. A new 4th order embedded method based on harmonic mean was constructed by [Yaacob and Sanugi B, 1995] where harmonic mean was embedded in the arithmetic mean viewed Runge-kutta methods. The proposed method was found accurate and cost effective compared to the classical methods.

PRELIMINARIES

In [5], it was shown that the standard fourth order arithmetic mean (AM) Runge-kutta formula for solving IVPs of the form y^′ = f (x, y) maybe written as:

y = y

+ h k₁ + k₂ + k₂ + k₃ + k₃ + k₄ (1.1)

where,

n+1 n 3 2 2 2

k₁ = f (x_n, y_n)

k₂ = f (x_n + a₁h, y_n + ha₁k₁)

k₃ = f (x_n + (a₂ + a₃)h, y_n + ha₂k₁ + ha₃k₂) (1.2)

k₄ = f (x_n + (a₄ + a₅ + a₆)h, y_n + ha₄k₁ + ha₅k₂ + ha₆k₃)

A fourth order accuracy is obtained through the standard procedure of adjustment of parameters a_i, 1 ≤ i ≤ 6 for formula above where

1 1

a₁ = 2 , a₂ = 0, a₃ = 2 , a₄ = 0, a₅ = 0, a₆ = 1. (1.3)

+ k

In [3], a new fourth order Runge-kutta formula based on the concept of averaging the harmonic mean functional is given in the form:

n+1

y = y

+ 2h k₁k₂ + k₂k₃ + k₃k₄ (1.4)

The improved accuracy of the formula was achieved by adjusting the parameters a₁, 1 ≤ i ≤ 6 in (1.2), where

1 1 5 1 7 9

a₁ = 2 , a₂ = −8 , a₃ = 8 , a₄ = −4 , a₅ = 20 , a₆ = 6 (1.5)

The geometric mean (GM) Runge-kutta formula was developed in [1] in the form

y = y + |k k | + |k k | + |k k | (1.6)

n+1

h √ √ √

by replacing the arithmetic mean (AM) of the functional values in (1.1) by the average of the geometric mean (GM) values and adjusting the parameters a₁, 1 ≤ i ≤ 6 in (1.2) to give

1 1 9 1 5 11

a₁ = 2 , a₂ = −16 , a₃ = 16 , a₄ = −8 , a₅ = 24 , a₆ = 12 (1.7)

The heronian mean (HeM) Runge-kutta formula was developed in [1] the form

n+1

y = y

+ h k

+ 2(k

+ k ) + k

+ √|k k | + √|k k | + √|k k | (1.8)

obtained by replacing the (AM) in (1.1) by the related heronian mean (HeM)

defined by

HeM =

(2(AM ) + (GM ))

(1.9)

caution must be exercised when using the (HeM) and (GM) formulas because both have the same type of deficiency.

A less deficient formula may be developed by using the root-mean square

(RM) defined by

RM = √2(AM )2 − (GM )2 (1.10)

Replacing (AM) in (1.1) by (RM) of the geometric mean formula yields the (RM) formula

y_n₊₁ = y_n +

h √ K² + K² √ K² + K² √ K² + K²

(1.11)

1 2

+ 2 3

+ 3 4

3 2 2 2

The centroid mean (CeM) Runge-kutta formula was developed in [1] in the

form

2 √K² + K₁K₂ + K²

√K² + K₂K₃ + K²

√K² + K₃K₄ + K²

y_n₊₁ = y_n+

1 2 + 2 3 + 3 4

9 k₁ + k₂

k₂ + k₃

k₃ + k₄ (1.12)

by replacing the (AM) in (1.1) by the centroidal mean

−

4(AM )² (GM )²

CeM =

3(AM )

(1.13)

given by

RM = √2(AM )2 − (GM )2 (1.14)

A preliminary test reveals that this formula requires more calculations at each computational step. However, this is normally not a serious difficulty unless f (x, y) is very complicated.

In this project, my concern is to establish a new fourth order Runge-kutta formula based on the concept of averaging the contra-harmonic mean (CoM) functional values. The establishment of the new method is identical to the above discussion with emphasis on the numerical comparison.

The numerical results are very encouraging and accurate when compared to it’s equivalents. This demonstrates improved performance and better accuracy compared to other well-known method present in the literature.

STATEMENT OF PROBLEM

Many studies have been devoted in order to find the solution of ordinary differential equations. In the case where the equation is linear, it is not a major problem since it can be solve by analytical methods. Unfortunately, most of the interesting differential equations that got after modelling real world problems are non-linear and it causes a major problem to solve that equation by analytical methods. Thus, numerical methods have been developed and have proved really helpful to solve those ordinary differential equations. Furthermore, there is much computer software that has been developed to help the user to solve those equations.

In as much as ordinary differential equations frequently occurs as mathematical models in many branches of science, engineering and economics. It is unfortunate that these equations seldom have solutions that can be expressed in closed form, so it is common to seek approximate solutions by means of numerical methods; These research will compare the various methods for solving problems on variety of means, and constructing and implementing a RK4 method based on CoM mean and illustrating their error.

OBJECTIVE OF THE STUDY

To derive an RK4 method based on
To implement the result with some
To determine the absolute

SIGNIFICANCE OF STUDY

Differential equations are commonly used for mathematical modelling in science and engineering. Many problems of mathematical physics can be stated in the form of differential equations. These equations also occur as reformations of other mathematical problems such as ordinary differential equations and partial differential equations. In most real life situations, the differential equation that models the problem is too complicated to solve exactly, and one of the approaches is taken to approximate the solution.

The first approach is to simplify the differential equation to one that can be solved exactly and then use the solution of the simplified equation to ap- proximate the solution of original problem. This is the approach that is most commonly taken since the approximation methods give more accurate results and realistic error information. Numerical methods are generally used for solving mathematical problems that are formulated in science and engineering where it is difficult or even impossible to obtain exact solutions. Only a limited number of differential equations can be solved analytically. There are many analytical methods for finding the solution of ordinary differential equations but even then there exist a large number of ordinary differential equations whose solutions cannot be obtained in closed form by using the well-known analytical methods to get the approximate solution of a differential equation under the prescribed initial conditions. There are many types of practical numerical methods for solving initial value problems for ordinary differential equations.

In this project, I present, analyze and compare Runge-kutta methods for solving initial value problems of ordinary differential equations based on variety of means.

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