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INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 1, ISSUE 9, OCTOBER 2012

ISSN 2277-8616

On The Error Analysis of The New Formulation of One Step Method Into Linear Multi Step Method For The Solution of Ordinary Differential Equations Fadugba S. Emmanuel Okunlola J. Temitayo Abstract-This paper presents error analysis of the new formulation of one step method into linear multistep method for the solution of ordinary differential equations. Error analysis in one step method is considerably more difficult than linear multistep method due to the loss of linearity in the method, especially for Runge Kutta Method that makes no mention of the function which defines the differential equation, makes it impossible to define the order of the method, independently of the differential equation. The new formulation of one step method into linear multistep method helps us in overcoming the shortcoming of the method. The error constant, zero stability, consistency and convergence of one step method were determined, through a similar process like linear multistep methods. Keywords: Convergence, Error, Linear Multistep Method, One Step Method, Runge Kutta Method, Zero Stability ————————————————————

j = 1,2,..., r , the real parameters ci , k i , a ji define the method and r is the stage number.

1 INTRODUCTION

Where for

The problem of solving ordinary differential equations is classified into initial value and boundary value problems, depending on the conditions specified at the end points of the domain. There are numerous one step methods for the solution of initial value problems in ordinary differential equations such as Euler's method which was the oldest and simplest method originated by Leonhard Euler in 1768, Improved Euler's method and Runge Kutta methods described by Carl Runge and Martin Kutta in 1895 and 1905 respectively. There are many excellent and exhaustive texts on this subject that may be consulted, such as [2], [5] and [6] just to mention few. The initial problem for a system of first order ordinary differential equations is given by

According to [7], if a matrix is strictly lower triangular i.e. the internal stages can be calculated without depending on later stages, and then the method is called an explicit method otherwise it is called an implicit method. The importance of an implicit method is due to its high orders of accuracy which is superior to the explicit methods. This makes it more suitable for solving stiff problems. In this work we present error analysis of the new formulation of one step method (Runge Kutta Method) into linear multistep method for the solution of ordinary differential equations.

The general linear multistep method is defined by

x ′ = f (t , x), x(t 0 ) = x0 , t ∈ [a, b] We shall consider the Runge Kutta method of the form

x n +1 = x n + hω (t n , x n , h)

(1)

r

k

k

j =0

j =0

stage of

∑α j yn +1 = h∑ β j f n + j

(2)

Where

(5)

α j and β j are constants, h is the step size.

2.1 ANALYSIS OF BASIC PROPERTIES OF THE LINEAR MULTISTEP METHOD The basic properties of numerical method are of order of accuracy, consistency, zero stability and convergence. These are examined for the derived method.

Where r

ω (t n , x n , h) = ∑ b j k j

2 LINEAR MULTISTEP METHOD

(3)

j =1

r

k j = f (t n + ci h, x n + h∑ a ji k i )

(4)

i =1

———————————————— • Fadugba S. Emmanuel is currently a Lecturer in the Department of Mathematical Sciences, Ekiti State University, Ado Ekiti, Nigeria, PH-08067032044. E-mail: emmasfad2006@yahoo.com • Okunlola J. Temitayo is currently a Lecturer in the Department of Mathematical and Physical Sciences, Afe Babalola University, Ado Ekiti, Nigeria, PH-08052751164. E-mail: tayookunlola@hotmail.com

2.1.1 ORDER AND ERROR TERM A characteristic of discretization method is that errors are generated when they are adopted for the solution of ordinary differential equations. Consequently, these errors have to be examined to ensure the approximate value does not deviate significantly from the exact solution as the iteration progresses. The magnitude of these errors is referred to as global error denoted by

en = y ( xn ) − yn

(6)

35 IJSTR©2012 www.ijstr.org


INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 1, ISSUE 9, OCTOBER 2012

y ( xn ) and yn are the exact and computed solution

Where at

xn respectively.

This is supposed to be small since the

magnitude of the error determines the accuracy of the scheme. The linear multistep method (5) is said to be of order p if c0 = c1 = ... = c p = 0 but c p +1 ≠ 0 and c p +1 is called the error constant, where k

cq = ∑ α j j =0

k jq j ( q −1) − ∑βj q! j = 0 (q − 1)!

k

∑α j =0 k

iii.

∑ j =0

iv. v.

j

k1 = f c1 , k2 = f c 21 ,..., k s = f c s From equation (11), we have

(12)

c1 = 0, c2 = c3 =

1 and 2 have

2

 1 yn +1 = yn +  f 0 + 4 f 1 + f1  6 2 

(13)

2.2.1 RUNGE KUTTA METHOD OF ORDER FIVE The well-known fifth order Runge Kutta method is given by [1]

=0 k

jα j = ∑ β j

(8)

yn +1 = yn +

j =0

P(1) = 0 P′(1) = σ (1)

2.1.4 CONVERGENCE The necessary and sufficient conditions for a linear multistep to convergence are zero stability and consistency [3] with the addition of the hypothesis

 η h −η 0 lim h 

  = ηˆ 

k1 = f ( xn , yn ) , k2 = f ( xn + 13 h, yn + 13 k1 ) , (15) k3 = f ( xn + 52 h, yn + 25h (4k1 + 6k2 )) , k4 = f ( xn + h, yn + h4 (k1 − 12k2 + 15k3 )) ,

k5 = f ( xn + 23 h, yn + 81h (6k1 + 90k2 − 50k3 + 8k4 )) k6 = f ( xn + 54 h, yn + 75h (6k1 + 36k2 + 10k3 + 8k4 )) From equation (15), we have

1 2 2 c1 = 0, c2 = , c3 = , c4 = 1, c5 = and 3 5 3

c6 =

4 then, 5

applying (12), we have

(9)

k1 = f 0 , k2 = f 1 , k3 = f 2 , k4 = f1 , k5 = f 2 and k6 = f 4 ,

2.2 THE NEW FORMULATION We shall consider the classical Runge Kutta method of order four, order five and sixth order implicit Runge Kutta method as follows: 2.2.1 CLASSICAL RUNGE KUTTA METHOD OF ORDER FOUR [4] The Runge Kutta method of order four is given by

h (k1 + 2k2 + 2k3 + k4 ) 6

h (23k1 + 125k3 − 81k5 + 126k6 ) (14) 192

Where

2.1.3 ZERO STABILITY According to [5, 6] a linear multistep method is said to be zero stable if no root of the first characteristic polynomial P(ν ) has modulus greater than one and if every root of modulus one is simple.

yn +1 = yn +

We shall let

applying (12), we c4 = 1 then, k1 = f 0 , k2 = k3 = f 1 and k4 = f1 , thus (10) yields

(7)

2.1.2 CONSISTENCY [5, 6] Equation (5) is said to be consistent if it can satisfy the following conditions: i. The order of the scheme must be greater than or equal to one ii.

ISSN 2277-8616

(10)

Where

k1 = f ( xn , yn ) , k2 = f ( xn + 0.5h, yn + 0.5k1 ) , k3 = f ( xn + 0.5h, yn + 0.5k2 ) , (11) k4 = f ( xn + h, yn + k3 ) ,

3

3

3

5

thus (14) yields

yn +1 = yn +

 h   + − + 23 f 125 f 81 f 125 f 0 2 2 4  192  5 3 5

(16)

3 THE ERROR ANALYSIS OF THE NEW FORMULATION Now we shall investigate the error analysis of the new formulation of the fourth and fifth orders Runge Kutta method by considering the order, error constant and zero stability [1]. From equations (5) and (7), we have for the classical Runge Kutta method of order four in equation (13)

c1 = c2 = c3 = c4 = 0 , but c p +1 = c5 = 3.472 × 10−4 . 36 IJSTR©2012 www.ijstr.org


INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 1, ISSUE 9, OCTOBER 2012

ISSN 2277-8616

Also we have for new fifth order Runge Kutta method on equation (16) that

c1 = c2 = c3 = c4 = c5 = 0 , but c p +1

= c6 = 1.852 × 10−5 .

Hence from the definitions in 2.1.3 and 2.1.4, equations (13) and (16) are zero stable and convergent.

4 CONCLUSION This approach reduces computational efforts in determining the order and convergence of Runge Kutta method. It also allows arbitrary higher order to be formulated. The new formulation of Runge Kutta method into linear multistep method is consistent, zero stable and convergent.

REFERENCES [1] Z. A Adegboye and Y. A. Yahaya, Reformulation of Runge Kutta Method into Linear Multistep Method for Error and Convergence Analysis, Pacific Journal of Science and Technology, 13(2012), pp. 238-243. [2] J. C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley and Sons, New York, NY, 2003. [3] G. Dahiquist, Convergence and Stability in the Numerical Integration of Ordinary Differential Equations, Bill, 18(1956), pp. 133-136 [4] S. E. Fadugba, R. B. Ogunrinde and J. T. Okunlola, On Some Numerical Methods for Solving Initial Value Problems in Ordinary Differential Equations, International Organization of Scientific Research, Journal of Mathematics, India, 3(2012), pp. 25-31. [5] J. D. Lambert, Computational Methods in Ordinary differential Equations, John Wiley and Sons, New York, N.Y, 1973. [6] J. D. Lambert, Numerical Methods for Ordinary differential Equations, John Wiley and Sons, New York, N.Y, 1991. [7] G. Yu. Kulikov, Symmetric Runge Kutta Method and their Stability, Russ J. Numeric Analyze and Maths. Modelling, 18(2003), pp. 13-41.

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On-The-Error-Analysis-Of-The-New-Formulation-Of-One-Step-Method-Into-Linear-Multistep-Method-For-The