construction of some implicit linear multi-step …

Construction of Some Implicit Linear Multi-Step………….. Ademola M Badmus, Mohammed S Mohammed

Corresponding Author: [email protected] and [email protected] website: www.academyjsekad.edu.ng

CONSTRUCTION OF SOME IMPLICIT LINEAR MULTI-STEP METHODS INTO

RUNGE-KUTTA TYPES METHODS FOR INITIAL VALUE PROBLEMS (IVP) OF FIRST

ORDER DIFFERENTIAL EQUATIONS

Ademola M Badmus and Mohammed S Mohammed

Department of Mathematics Nigerian Defence Academy, Kaduna Nigeria

Abstract

Two implicit Hybrid block methods at step length 𝑘 = 2 and 3 were derived through collocation

procedures. The two derived block methods also reconstructed to an equivalent S stage Runge-Kutta

type methods for the solution of 𝑦 ′ = 𝑓(𝑥, 𝑦). Both methods derived were implemented on the same

numerical experiments but Runge-Kutta type methods (RKTM) show its superiority over its

equivalent Linear Multi-step Methods of the same value of step length 𝑘

Keywords: Implicit, Hybrid block methods, Runge-Kutta type methods, Step length

1.0 INTRODUCTION:

Numerical analysis is the area of mathematics

and computer science that creates, analyzes

and implements algorithms for solving

numerically, the problems of continuous

mathematics (using differential equations,

which are functions that describe the rate of

change of one using another derived function).

With the advent of Computer, the numerical

solutions of differential equations have

become easy to obtain. Hence in this paper,

we present some methods of numerical

solution for ordinary differential equations. No

doubt, such numerical solutions are

approximate solutions, but, in many cases,

approximate solutions to the required accuracy

are quite sufficient.

The development of numerical methods for

approximating solutions of initial value

problems (IVPs) in Ordinary Differential

Equations (ODEs) has attracted considerable

attention in recent decades and many

individuals have shown interest in

constructing efficient methods with good

stability properties for the numerical

integration of ordinary differential equations.

Although, a very wide variety of numerical

methods have been proposed, the number of

methods with high order and good stability

properties remains relatively small. Solutions

to Ordinary Differential Equations were

derived using analytic or even exact methods.

Most of their solutions are very useful such

that it provides brilliant insight into the

behavior of some system. These include those

that can be approximated with linear model

and those that have simple geometry and low

dimensionality. Conversely, many differential

equations cannot be solved analytically,

because most real life problems are non-linear

and involve complex shapes and processes.

A broad variety of classical techniques have

since been developed for finding approximate

solutions to these problems. Many researchers

have developed different methods, for solving

first order ordinary differential equations of

(1.0) among them are Aladeselu (2007),

Erwin (1999), Awoyemi (2001), Multi-Step

Collocation methods of Yusuph and

Onumanyi (2002), Badmus and Mshelia

(2011,2012), Mshelia et al (2016) just to

mention a few. In this research work, we

considered the Numerical computational

method for first order ordinary differential

equation which is of the form:

mailto:[email protected]

mailto:[email protected]


Academy Journal of Science and Engineering 10 (1), 2016 Page 73

y′ = f(x, y), y(x0) = y0 , for a ≤ x ≤b (1.1) at various step length k =2 and k =3

respectively

1.2 Aim of the Research:

The aim of this research work is to develop

some K-step implicit hybrid block Linear

Multistep Methods and also reformulated them

into S- stages Runge-Kutta type methods for

solution of equation (1.0).

1.3 Objectives of the Research:

The objectives of this work are to:

(i) To develop an implicit hybrid block

Linear Multistep Method (BILM) by

incorporating two off-grid points

through interpolation and collocation

procedures.

(ii) Obtain continuous formulation of each

of the two different D-Matrices and

use Maple17 Mathematical software to

obtain their discrete schemes and

equally reformulated them into

equivalent Runge-Kutta methods for

solution of equation (1.0).

(iii)To analyze basic properties of each of

the methods developed which include

local errors, and convergence analysis.

(iv) To test the performance of developed

methods using some problems of non-

linear Ordinary Differential Equations.

1.4 Definition of terms

i) Linear Multistep Method (LMM).

If a computational method for determining a

sequence [𝑦𝑛] takes the form of a linear

relationship between 𝑦𝑛+𝑖, 𝑓𝑛+𝑖 , 𝑖 =0,1,2, … , 𝑘, then we call it a LMM of step

number k or a linear k-step method.

A linear k-step is mathematically express as

αkyn−k + ⋯ + α1yn+1α0

= h(βk

fn+k + ⋯ + β1

fn+1

+ β0

fn) (1.2)

and can be written in general as

∑ 𝛂𝐢

𝐤

𝐢=𝟎

𝐲𝐧+𝐢 = 𝐡 ∑ 𝛃𝐢𝐟𝐧+𝐢

𝐤

𝐢=𝟎

(𝟏. 𝟑)

For a linear k-step method, it is required that

𝛼𝑘 ≠ 0 and either 𝛼 0 ≠ 0 𝑜𝑟 𝛽0 ≠ 0. Usually, the coefficients are normalized such

that 𝛼𝑘 = 1, 𝐼𝑓 𝛽𝑘 ≠ 0, then it is implicit

scheme and it is explicit if otherwise.

ii) Zero stable:

A linear multi-step method is said to be Zero-

stable if the roots 𝑹𝒋, 𝒋 = 𝟏(𝟏)𝒌 of the first

characteristics polynomials

𝝆(𝑹) = 𝒅𝒆𝒕 [∑ 𝑨𝒊𝑹𝒌−𝒊

𝒌

𝒊=𝟎

] = 𝟎, 𝑨𝟎

= −𝟏, 𝒔𝒂𝒕𝒊𝒔𝒇𝒊𝒆𝒔 |𝑹𝒋| ≤ 𝟏

If one of the roots is +𝟏, we call this the

principal root of 𝝆(𝑹). Fatunla (1991)

iii) Order and Error constants

A linear multi-step method

𝒚(𝒙) = ∑ 𝜶𝒋𝒌𝒋=𝟎 (𝒙)𝒚𝒏+𝒋 =

𝒉 ∑ 𝜷𝒋𝒌𝒋=𝟎 (x)𝒇𝒏+𝒋 𝟏. 𝟒)

we associate the linear differential operator

𝑳[𝒚(𝒙); 𝒉] = ∑_(𝒋 = 𝟎)^𝒌▒[𝜶_𝒋 ; 𝒚(𝒙 +𝒋𝐡) − 𝒉𝜷_𝒋 𝒚′(𝒙; 𝒋𝐡)] ( 𝟏. 𝟓)

where 𝑦(𝑥) is an arbitrary function,

continuously differentiable on [a, b].

Expanding the test function 𝑦(𝑥 + 𝑗ℎ) and it’s

derivative 𝑦′(𝑥 + 𝑗ℎ) as Taylor series about

𝑥 and collecting terms in (1.5) gives



𝐿 [𝑦(𝑥); ℎ] = 𝐶0𝑦(𝑥) + 𝐶1ℎ𝑦 ′(𝑥) + ⋯ +𝐶𝑞ℎ𝑞𝑦𝑞(𝑥) (1.6)

where 𝐶𝑞 are constants.

A simple calculation yields the following

formulae for the constants 𝐶𝑞 in term of the

coefficients 𝛼𝑗 , 𝛽𝑗 .

𝐶0 = ∝0 + ∝1+∝2 + ∝3+ … 𝑘 ∝𝑘

𝐶1 = ∝1+ 2 ∝2 + 3 ∝3+ … +𝑘 ∝𝑘−(𝛽0 + 𝛽1 + 𝛽2 + ⋯ + 𝛽𝑘)

𝐶𝑞 = 1

𝑞!( ∝1+ 2𝑞 ∝2 + 3𝑞 ∝3+

… +𝑘𝑞 ∝𝑘) − 1

(𝑞−1)!(𝛽1 + 2𝑞−1𝛽2 + … +

𝑘𝑞−1𝛽𝑘), 𝑞 = 2,3, … (1.7)

Following Henrici, P (1962) we say that the

method has order P if

𝐶0 = 𝐶1 = 𝐶2 = ⋯ 𝐶𝑝 = 0 , 𝐶𝑝+1 ≠ 0

Hence, we say that the method has order P

𝑖𝑓 𝑪𝟎 = 𝑪𝟏 = 𝑪𝟐 = 𝑪𝒑 = 𝟎, But 𝑪𝒑+𝟏 ≠ 𝟎

then 𝑪𝒑+𝟏 is then the error constant and

𝑪𝒑+𝟏𝒉𝒑+𝟏𝒚𝒑+𝟏(𝒙𝒏) the principal local

truncated error at the point 𝒙𝒏. (see Lambert

1973)

iv) Runge-Kutta methods

Runge-Kutta Method comprises of implicit

and explicit iterative methods used in temporal

discretization for the approximate solution of

Ordinary Differential Equations. Runge-Kutta

Methods are among the most popular ODE

solvers. They were first studied by Carle

Runge and Martin Kutta around 1900. Modern

developments are mostly due to John Butcher

in the 1960s.

Second order Runge-Kutta Methods

We consider the general first-order ODEs of

equation (1.1)

Using Taylor expansion

𝒚(𝒙 + 𝒉) = 𝒚(𝒙) + 𝒉𝒚^′ (𝒙)+ 𝒉^𝟐/𝟐 𝒚^′′ (𝒙) + 𝑶(𝒉^𝟑)

The first derivatives can be replaced by the

RHS of the differential equation (i) and the

second derivatives is obtained as

𝒚′′(𝒙) = 𝒇𝒙(𝒙, 𝒚) + 𝒇𝒚(𝒙, 𝒚)𝒇(𝒙, 𝒚)

with Jocobian 𝑓𝑦. We will from now on

neglect the dependence of y on x when it

appears as an argument to f . Therefore, the

Taylor expansion becomes

𝑦(𝑥 + ℎ) = 𝑦(𝑥) + ℎ/2 𝑓(𝑥, 𝑦)+ ℎ/2 {𝑓(𝑥, 𝑦) + ℎ𝑓_𝑥 (𝑥, 𝑦)+ ℎ𝑓_𝑦 (𝑥, 𝑦)𝑓(𝑥, 𝑦) }+ 𝑂(ℎ^3 )

Recalling the multivariate Taylor expansion

𝑓(𝑥 + ℎ, 𝑦 + 𝑘)= 𝑓(𝑥, 𝑦) + ℎ𝑓𝑥(𝑥, 𝑦)+ 𝑓𝑦(𝑥, 𝑦)𝑘 + ⋯

we see that the expression in brackets can be

interpreted as

𝑓(𝑥 + ℎ, 𝑦 + ℎ𝑓(𝑥, 𝑦))

= 𝑓(𝑥, 𝑦) + ℎ𝑓𝑥(𝑥, 𝑦)+ ℎ𝑓𝑦(𝑥, 𝑦)𝑓(𝑥, 𝑦) + 𝑂(ℎ2),

Therefore , we get

𝑦(𝑥 + ℎ) = 𝑦(𝑥) +ℎ

2𝑓(𝑥, 𝑦)

+ ℎ

2𝑓(𝑥 + ℎ, 𝑦 + ℎ𝑓(𝑥, 𝑦))

+ 𝑂(ℎ3)

Hence, RK method of order 2 is of the form

𝑦𝑛+1 = 𝑦𝑛 + ℎ(1

2𝑘1 +

1

2𝑘2)

where

𝑘1 = 𝑓(𝑥𝑛,𝑦𝑛),

𝑘2 = 𝑓(𝑥𝑛 + ℎ, 𝑦𝑛 + ℎ𝑘1)



Its Butcher tableaux is of the form

0 0 0 0 0 1

2

1

2

0 0 0

1

2

0 1

2

0 0

1 0 0 0 0

1 1

6

1

3

1

3

1

6

The Runge-Kutta Method for first order ODEs is given by

𝑦𝑛+1 = 𝑦𝑛 + ℎ (∑ 𝑏𝑗𝑘𝑗

𝑠

𝑗=1

)

where

𝑘𝑖 = 𝑓(𝑥 + 𝑐𝑖, 𝑦 + ∑ 𝑎𝑖𝑗𝑘𝑗𝑠𝑗=1 ) 𝑖 = 1,2, … 𝑠 (1.8)

This method can be expressed in a Table (Butcher’s Table) as follows.

𝑪 A

𝑪𝟏 𝒂𝟏𝟏 𝒂𝟏𝟐 . 𝒂𝟏𝒔

𝑪𝟐 𝒂𝟐𝟏 𝒂𝟐𝟐 . 𝒂𝟐𝒔

. . . . . = 𝑪 𝑨

. . . . . 𝒃𝑻

. . . . .

𝑪𝒔 𝒂𝒔𝟏 𝒂𝒔𝟐 . 𝒂𝒔𝒔

𝑏𝑇 𝑏1 𝑏2 . 𝑏𝑠

𝐴 = (𝑎𝑖𝑗) 𝑖, 𝑗 = 1,2, … . 𝑠 is an 𝑆 x 𝑆 Matrix. 𝑏𝑇 = (𝑏1, 𝑏2, 𝑏3, … . . 𝑏𝑠) (1.9)

𝐶𝑖 = ∑ 𝑎𝑖𝑗

𝑠−1

𝑗=1

In this research work we developed some

implicit block linear multistep method for

values of k=2 and k=3. Also the same implicit

block method were reformulated into S-stage

Runge-Kutta type methods which when

implemented on both linear and non linear

problems of ordinary differential equations,

we obtained better accuracy than the existing

methods.



2.0 Methodology:

Consider a first Order Differential Equation

(1.1) of the form

𝑦′ = 𝑓(𝑥, 𝑦), 𝑦(𝑥𝑜) = 𝑦𝑜

We assume power series solution of the form

𝑦(𝑥) = ∑ 𝑎𝑖

𝑝+𝑐−1

𝑖=0

𝑥𝑖 (2.1)

and the first derivative is

𝑦 ′ (2.2)

where p is the interpolation and c is the

collocation points and 𝐚𝐢’s are the parameters

to be determined.

Our interest is to derive each different method

at k =2, k =3 and reformulated them into its

equivalent Runge-Kutta type Method

2.1 Derivation of LMM of K=2 (Method I)

In Method I.

In equation (2.1) and (2.2) 𝑝 = 3, 𝑐 = 3, then

𝑡 = 𝑝 + 𝑐 − 1 = (3 + 3 − 1) = 5 (t, is the

degree of the polynomials.

Now, we interpolate (2.1) and collocate (2.2)

at the same point of 𝑥 = {𝑥𝑛+1/2, 𝑥𝑛+1𝑥𝑛+3/2}

which gives the system of non-linear

equations of the form

𝑦(𝑥) = ∑ 𝑎𝑖

5

𝑖=0

𝑥𝑖

Also its first derivative is

𝑦 ′(𝑥) = ∑ 𝑖𝑎𝑖𝑥𝑖−15

𝑖=1 (2.3)

Specifically equation (2.3) gives

a0 + a1xn+1/2 + a2xn+1/22 + a3xn+1/2

3 + a4xn+1/24 + a5xn+1/2

5 = yn+1/2

a0 + a1xn+1 + a2xn+12 + a3xn+1

3 + a4xn+14 + a5xn+1

5 = yn+1

a0 + a1xn+3/2 + a2xn+3/22 + a3xn+3/2

3 + a4xn+3/24 + a5xn+3/2

5 = yn+3/2

a1xn+1/2 + 2a2xn+1/2 + 3a3xn+1/22 + 4a4xn+1/2

3 + 5a5xn+1/24 = fn+1/2

a1xn+1 + 2a2xn+1 + 3a3xn+12 + 4a4xn+1

3 + 5a5xn+14 = fn+1

a1xn+3/2 + 2a2xn+3/2 + 3a3xn+3/22 + 4a4xn+3/2

3 + 5a5xn+3/24 = fn+3/2

(𝟐 . 𝟒) Re-arranging (2.4) in matrix form:

= (2.5)



Using Maple Mathematical software to

determine the values of 𝑎𝑛′𝑠 in equation (2.5)

we obtain the following result as our

continuous formula as

y(x) = αn+

1

2

yn+

1

2

+ αn+1yn+1 + αn+

3

2

yn+

3

2

+

h [βn+

1

2

fn+

1

2

+ βn+1

fn+1 +

βn+

3

2

fn+

3

2

] (2.6)

Specifically our continuous formula is

y(x) = {10 −66

h1(x − xn) +

166

h2 (x − xn)2 −198

h3 (x − xn)3 +112

h4 (x − xn)4 −24

h5(x − xn)5 }y

n+3

2

+{−18 +114

h1(x − xn) −

254

h2 (x − xn)2 +

262

h3 (x − xn)3 −

128

h4 (x − xn)4

+24

h5(x − xn)5 }y

n+12

+ {−9h

2+ 24(x − xn) −

97

2h (x − xn)2 +

47

h2 (x − xn)3 −

22

h3 (x − xn)4

+4

h4(x − xn)5 }f

n+1

2

+{−9h

2+ 24(x − xn) −

97

2h (x − xn)2 +

47

h2 (x − xn)3 −

22

h3 (x − xn)4

+4

h4(x − xn)5 }f

n+1

2

+{−9h + 57(x − xn) −136

h (x − xn)2 +

162

h2 (x − xn)3 −80

h3 (x − xn)4

+16

h4(x − xn)5 }fn+1 +{−

3h

2+ 10(x − xn) −

51

2h (x − xn)2 +

31

h2 (x − xn)3 −18

h3 (x − xn)4

+4

h4(x − xn)5 }f

n+32

(2.7)

Evaluate (2.7) at 𝑥 = 𝑥𝑛+𝑗 , 𝑗 = 0, 2 also the first derivatives of (2.7) is evaluated at 𝑥 =

𝑥𝑛+𝑗 , 𝑗 = 0, 2 to obtain our discrete block method called Method I.

yn+2 + 18yn+

3

2

− 9yn+1 − 10yn+

1

2

=3h

2fn+

1

2

+ 9hfn+1 +9h

2fn+

3

2

−10yn+

3

2

− 9yn+1 + 18yn+

1

2

+ yn = −9hfn+1 −9

2hf

n+1

2

−3

2hf

n+3

2

114yn+

3

2

− 48yn+1 − 66yn+

1

2

= 10hfn+

1

2

+ 57hfn+1 + 24hfn+

3

2

− hfn+2

66yn+

32

+ 48yn+1 − 114yn+

12

= −hfn + 24hfn+

12

+ 57hfn+1 + 10hfn+

32

(2.8)

2.2 Derivation of LMM At K =3 ( Method II)

In Method II

With respect to the equation (2.1) and (2.2)

choosing 𝑝 = 4, 𝑐 = 3, 𝑡 = 𝑝 + 𝑐 − 1 (t is the

degree of the polynomials.

(i.e. 4 + 3 − 1 = 6).

Now, we interpolate (1.1) at point of 𝑥 ={𝑥𝑛, 𝑥

𝑛+1

2

𝑥𝑛+1, 𝑥𝑛+

3

2

} and collocate at point

𝑥 = {𝑥𝑛+

1

2

, 𝑥𝑛+1, 𝑥𝑛+

3

2

} which gives the

system of non-linear equations of the form


Academy Journal of Science and Engineering 10 (1), 2016 8

a0 + a1xn + a2xn2 + a3xn

3 + a4xn4 + a5xn

5 + a6xn6 = yn

a0 + a1xn+

12

+ a2xn+

12

2 + a3xn+

12

3 + a4xn+

12

4 + a5xn+

12

5 + a6xn+

12

6 = yn+

12

a0 + a1xn+1 + a2xn+12 + a3xn+1

3 + a4xn+14 + a5xn+1

5 + a6xn+16 = yn+1

a0 + a1xn+

32

+ a2xn+

32

2 + a3xn+

32

3 + a4xn+

32

4 + a5xn+

32

5 + a6xn+

32

6 = yn+

32

a1xn+

12

+ 2a2xn+

12

+ 3a3xn+

12

2 + 4a4xn+

12

3 + 5a5xn+

12

4 + 6a6xn+

12

5 = fn+

12

a1xn+1 + 2a2xn+1 + 3a3xn+12 + 4a4xn+1

3 + 5a5xn+14 + 6a6xn+1

5 = fn+1

a1xn+

32

+ 2a2xn+

32

+ 3a3xn+

32

2 + 4a4xn+

32

3 + 5a5xn+

32

4 + 56xn+

32

5 = fn+

32

(2.9)

Re-arranging (2.9) in matrix form as:

=

(3.0)

The same Maple 17 Mathematical software will be use to determine the values of 𝑎𝑛′𝑠 in equation

(3.0) to obtain the following result as our continuous formula which is of the form

𝐲(𝐱) = 𝛂𝐧 𝐲𝐧 + 𝛂𝐧+

𝟏

𝟐

𝐲𝐧+

𝟏

𝟐

+ 𝛂𝐧+𝟏𝐲𝐧+𝟏 + 𝛂𝐧+

𝟑

𝟐

𝐲𝐧+

𝟑

𝟐

+ 𝐡[𝛃𝐧+

𝟏

𝟐

𝐟𝐧+

𝟏

𝟐

+ 𝛃𝐧+𝟏𝐟𝐧+𝟏 +𝛃𝐧+

𝟑

𝟐

𝐟𝐧+

𝟑

𝟐

]

(3.1)



Specifically, (3.1) is of the form

y(x) = { 66

9h(x − xn) −

436

9h2 (x − xn)2 +1098

9h3 (x − xn)3 −1312

9h4 (x − xn)4 +248

3h5 (x − xn)5 −160

9h6 (x −

xn)6} yn+

3

2

+ { 18

h(x − xn) −

175

h2 (x − xn)2 +224

h3 (x − xn)3 −216

h4 (x − xn)4

+96

h5 (x − xn)5 − 16

h6 (x − xn)6} yn+1 +{− 18

h(x − xn) +

132

h2 (x − xn)2 −314

h3 (x − xn)3 +336

h4 (x −

xn)4 −168

h5(x − xn)5 +

32

h6 (x − xn)6} y

n+1

2

+ {1 − 66

9h(x − xn) +

193

9h2 (x − xn)2

−288

9h3 (x − xn)3 +

232

9h4 (x − xn)4 −

32

3h5(x − xn)5 +

16

9h6 (x − xn)6} yn

+{ −9(x − xn) +48

h (x − xn)2 −

97

h2 (x − xn)3 +94

h3 (x − xn)4 −44

h4 (x − xn)5 +8

h5 (x −

xn)6} fn+

1

2

+ {−9(x − xn) +57

h (x − xn)2 −

126

h2 (x − xn)3 +152

h3 (x − xn)4

−80

h4 (x − xn)5 +16

h5 (x − xn)6 } fn+1 +{−1(x − xn) +20

3h (x − xn)2 −

51

3h2 (x − xn)3

+62

3h3 (x − xn)4 −

12

h4(x − xn)5 +

8

3h5 (x − xn)6} f

n+32

(3.2)

Evaluate (3.2) at 𝑥 = 𝑥𝑛+𝑗 , 𝑗 = 2, 3 , also the first derivative of (3.2) is evaluated at 𝑥 =

𝑥𝑛+𝑗 , 𝑗 = 0, 2 𝑎𝑛𝑑 3 to obtain the following discrete scheme called Method II.

yn+2 + 28yn+

32

− 28yn+

12

− yn = 6hfn+

12

+ 18hfn+1 + 6hfn+

32

yn+3 + 1800yn+

3

2

+ 675yn+1 − 2376yn+

1

2

− 100yn = 540hfn+

1

2

+ 1350hfn+1 + 300hfn+

3

2

562yn+

32

+ 54yn+1 − 594yn+

12

− 22yn = 129hfn+

12

+ 369hfn+1 + 105hfn+

32

− 3hfn+2

15520yn+

3

2

+ 7020yn+1 − 21600yn+

1

2

− 940yn = 4968hfn+

1

2

+ 12015hfn+1 + 2520hfn+

3

2

−

3hfn+3 22𝑦

𝑛+3

2

+ 54𝑦𝑛+1 − 54𝑦𝑛+

1

2

− 22𝑦𝑛 = 3ℎ𝑓𝑛 + 27ℎ𝑓𝑛+

1

2

+ 27ℎ𝑓𝑛+1 + 3ℎ𝑓𝑛+

3

2

(3.3)

3.3 Conversion of our proposed LMM into Runge–Kutta type methods:

The method I at k=2 are of order [5,5,5,5]𝑇 with an Error constants of [1

1280,

1

1920,

1

1280, −

1

1920]𝑇.

Also when Method 1 is arranged in Matrix equation form we have

+

+ (3.4)



We obtain the inverse of the matrix

as (3.5)

Multiply equation (3.4) by (3.5) to obtain

= + +

(3.6)

Arranging (3.6) in Butcher array table we have

C A

1

2

251

1440

323

720

−11

60

53

720

−19

1440

1 29

180

31

45

2

15

1

45

−1

180

3

2

27

160

51

80

9

20

21

80

−3

160

1 7

45

32

45

4

15

32

45

7

45

𝐛 = 𝟐 𝟕

𝟒𝟓

𝟑𝟐

𝟒𝟓

𝟒

𝟏𝟓

𝟑𝟐

𝟒𝟓

𝟕

𝟒𝟓

(3.7)



The above table (3.7) does not satisfy Butcher array condition. Hence by restructuring (3.7) to

satisfy Butcher array condition as C A

1

4

251

2880

323

1440

−11

120

53

1440

−19

2880

1

2 29

360

31

90

2

30

1

90

−1

360

3

4

27

320

51

160

9

40

21

160

−3

320

1 7

90

32

90

4

30

32

90

7

90

b = 1 7

90

32

90

4

30

32

90

7

90

(3.8)

Note:

Butcher array conditions for 𝑦′ = 𝑓(𝑥, 𝑦) are

(𝑖) ∑ 𝑎𝑖𝑗 = 𝐶𝑖

𝑠

𝑗=1

(𝑖𝑖) ∑ 𝑏𝑗 = 1

𝑠

𝑗=1

The method (3.4) is formally given as Runge-Kutta type method as

𝑦𝑛+1 = 𝑦𝑛 + ℎ{7

90𝑘1 +

32

90𝑘2 +

4

30𝑘3 +

32

90𝑘4 +

7

90𝑘5}

𝑘1 =f(𝑥𝑛, 𝑦𝑛)

𝑘2 =f(𝑥𝑛 +1

4ℎ, 𝑦𝑛 + ℎ(

251

2880𝑘1 +

323

1440𝑘2 −

11

120𝑘3 +

53

1440𝑘4 −

19

2880𝑘5))

𝑘3 =f(𝑥𝑛 +1

2ℎ, 𝑦𝑛 + ℎ(

29

360𝑘1 +

31

90𝑘2 −

2

30𝑘3 +

1

90𝑘4 −

1

360𝑘5))

𝑘4=f(𝑥𝑛 +3

4ℎ, 𝑦𝑛 + ℎ(

27

320𝑘1 +

51

160𝑘2 +

9

40𝑘3 +

21

160𝑘4 −

3

320𝑘5))

𝑘5 =f(𝑥𝑛 + 1ℎ, 𝑦𝑛 + ℎ(7

90𝑘1 +

32

90𝑘2 +

2

30𝑘3 +

32

90𝑘4 +

7

90𝑘5)) (3.9)

Also the method II at k=3 are of order [6,6,6,6,6]𝑇 with an Error constants of

[1

4480,

47

8960,

15

448,

39

112, −

3

8960]𝑇.



Also when Method 11 is arranged in Matrix equation form we have

=

+

+

(3.10)

We obtain a normalized equation as

=

+

+

(3.11)

Arranging (3.11) in Butcher array table ,we have



C A

1

6

959

17280

35

216 −

487

5760

49

1080 −

211

17280

1

1920

1

3

169

3240

32

135

11

360

8

405 −

7

1080

1

3240

1

2

103

1920

9

40

81

640

13

120 −

9

640

1

1920

2

3

7

135

32

135

4

45

32

135

7

135

0

1 11

120

0 27

40 −

8

15

27

40

11

120

𝑏 = 1 11

120

0 27

40 −

8

15

27

40

11

120

(3.12)

The method 11 is formally given as Runge-Kutta type Method as

𝑦𝑛+1 = 𝑦𝑛 + ℎ{11

120𝑘1 + (0)𝑘2 +

27

40𝑘3 +

8

15𝑘4 +

27

40𝑘5 +

11

120𝑘6}

𝑘1 =f(𝑥𝑛, 𝑦𝑛)

𝑘2=f(𝑥𝑛 +1

6ℎ, 𝑦𝑛 + ℎ(

959

17280𝑘1 +

35

216𝑘2 −

487

5760𝑘3 +

49

1080𝑘4 −

211

17280𝑘5 +

1

1920𝑘6))

𝑘3=f(𝑥𝑛 +1

3ℎ, 𝑦𝑛 + ℎ(

169

3240𝑘1 +

32

135𝑘2 +

11

360𝑘3 +

8

405𝑘4 −

7

1080𝑘5 +

1

3240𝑘6))

𝑘4 =f(𝑥𝑛 +1

2ℎ, 𝑦𝑛 + ℎ(

103

1920𝑘1 +

9

40𝑘2 +

81

640𝑘3 +

13

120𝑘4 −

9

640𝑘5 +

1

1920𝑘6))

𝑘5 =f(𝑥𝑛 +2

3ℎ, 𝑦𝑛 + ℎ(

7

135𝑘1 +

32

135𝑘2 +

4

45𝑘3 +

32

135𝑘4 +

7

135𝑘5 + (0)𝑘6))

𝑘6 =f(𝑥𝑛 + (1)ℎ, 𝑦𝑛 + ℎ(11

120𝑘1 + (0)𝑘2 +

27

40𝑘3 −

8

15𝑘4 +

27

40𝑘5 +

11

120𝑘6))

(3.13)

4.0 Numerical Experiments:

We shall implements all the two methods

derived in this paper with numerical

experiments to ascertain their efficiency and

accuracy

i) Example 1

𝑦′ = 1 − 𝑦2 𝑦(0) = 0, ℎ = 0.1, Exact solution:𝑦(𝑥) = tanh (𝑥)

ii) Example 2 (Logistic Population

Model)

A logistic law has useful applications to

human populations and animal populations.

The mathematical model is defined as

y(t) = Ay(t) – B y2(t), y(to) = go A, B are positive constant. If B = 0, it gives

the Maltus population model. If 0<y(0)< 𝐴

𝐵 ,

the population is monotonic increasing to a

large population, to limit 𝐴

𝐵, if, y(0) >

𝐴

𝐵 , it

decrease to limit 𝐴

𝐵.

Now we consider a logistic model of the form

𝑦′ − 𝐴𝑦 = − 𝐵𝑦2, 𝑦(𝑡0) = 𝑦0, A, B

positive constant, with A = 5, B = 3, t0 =0, y0

= 2.

Thus the model reduces to

𝑦′ = 5y – 3y2, y(o) = 2

Exact solution:𝑦(𝑥) =−10

𝑒5𝑥−6



Table I: Approximate results of Problem 1 on Method 1 at k=2

𝑿 Theoretical solution LMM at K =2 Runge- Kutta type at K =2

0.1 0.099667994624956 0.099667998229271 0.099667994611037

0.2 0.197375320224904 0.197375318495844 0.197375320197890

0.3 0.291312612451591 0.291312619120627 0.291312612413497

0.4 0.379948962255225 0.379948959311176 0.379948962209550

0.5 0.462117157260010 0.462117160571878 0.462117157211492

0.6 0.537049566998036 0.537049564003004 0.537049566951809

0.7 0.604367777117163 0.604367775846507 0.604367777077680

0.8 0.664036770267849 0.664036768330667 0.664036770238037

0.9 0.716297870199025 0.716297867121717 0.716297870179974

1.0 0.761594155955765 0.761594155378706 0.761594155955765

Table II :-Absolute error of problem 1 on Method I at k =2

𝑿 LMM at K =2 Runge- Kutta type at K =2

0.1 3.604315 𝑋 10−9 1.3919 𝑋 10−11

0.2 1.72906 𝑋 10−9 2.7014 𝑋 10−11

0.3 6.669036 X 10−9 3.8094 𝑋 10−11

0.4 2.944049 𝑋 10−9 4.5675 𝑋 10−11

0.5 3.311868 𝑋 10−9 4.8518 𝑋 10−11

0.6 2.995032 𝑋 10−9 4.6227 𝑋 10−11

0.7 1.270656 𝑋 10−9 3.9483 𝑋 10−11

0.8 1.937182 𝑋 10−9 2.9812 𝑋 10−11

0.9 3.077308 𝑋 10−9 1.9051 𝑋 10−11

1.0 5.77059 𝑋 10−10 8.834 𝑋 10−12

Figure 1: Error graph of Problem1 on LMM and RKTM at k=2

0

1E-09

2E-09

3E-09

4E-09

5E-09

6E-09

7E-09

8E-09

1 2 3 4 5 6 7 8 9 10

LMM at K =2

R-KTM at K =2



Table III :- Approximate results of Problem 1on Method 1I at k=3


0.1 0.099667994624956 0.099667992657626

0.099667994600974

0.2 0.197375320224904 0.197375318789478 0.197375320184291

0.3 0.291312612451591 0.291312575574453 0.291312612406103

0.4 0.379948962255225 0.379948927677656 0.379948962216385

0.5 0.462117157260010 0.462117133102505 0.462117157235355

0.6 0.537049566998036 0.537049544740135 0.537049566989716

0.7 0.604367777117163 0.604367786002063 0.604367777122827

0.8 0.664367702678490 0.664036777800830 0.664036770282710

0.9 0.716297870199025 0.716297897139883 0.716297870217936

1.0 0.761594155955765 0.761594179114741 0.761594155974577

Table IV:- Absolute error of problem 1 on Method 1I at k =3

𝑋 LMM at K =3 Runge- Kutta type at K =3

0.1 1.96733 𝑋 10−9 2.3982 𝑋 10−11

0.2 1.435426 𝑋 10−9 4.0613 𝑋 10−11

0.3 3.6877138X 10−9 4.5488 𝑋 10−11

0.4 3.4577569 𝑋 10−8 3.884 𝑋 10−11

0.5 2.4157505 𝑋 10−8 2.4655 𝑋 10−11

0.6 2.2257701 𝑋 10−8 8.32 𝑋 10−12

0.7 8.8849 𝑋 10−9 5.664𝑋 10−12

0.8 7.532981𝑋 10−9 1.4861 𝑋 10−11

0.9 2.6940858 𝑋 10−8 1.8911 𝑋 10−11

1.0 2.3158976 𝑋 10−8 1.8812 𝑋 10−11

Figure 2: Error graph of Problem1 on LMM and RKTM at k=3

0

1E-08

2E-08

3E-08

4E-08

5E-08

6E-08

7E-08

8E-08

9E-08

0.0000001

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

LMM at K =3

R-KTM at K =3



Table V :- Approximate results of Problem 2 on Method I at k=2


0.1 1.854094158893870 1.854088073248640 1.854094299304560

0.2 1.775530174744680 1.775536691936170 1.775530282469000

0.3 1.731041251962320 1.717028151383760 1.731041322529170

0.4 1.705127314681140 1.698044370133650 1.705127359188870

0.5 1.689784323984830 1.668657502326590 1.689784351719930

0.6 1.680612125128940 1.707788617573350 1.680612142333990

0.7 1.675097259033450 1.691368500032100 1.675097269685340

0.8 1.671769922367190 1.681562363330350 1.671769928955270

0.9 1.669758223053270 1.675669684573270 1.669758227125390

1.0 1.668540422822470 1.672115726053510 1.668540425338330

Table VI :-Absolute error of problem 2 on Method I at k=2

𝑿 LMM at K =2 Runge- Kutta type at K =2

0.1 6.08564 𝑋 10−6 1.40411 𝑋 10−7

0.2 6.51719 𝑋 10−6 1.07724 𝑋 10−7

0.3 1.40131 𝑋 10−2 7.05669 𝑋 10−8

0.4 7.08294 𝑋 10−3 4.45077 𝑋 10−8

0.5 2.11268 𝑋 10−2 2.77351 𝑋 10−8

0.6 2.71765 𝑋 10−2 1.72050 𝑋 10−8

0.7 1.62712 𝑋 10−2 1.06519 𝑋 10−8

0.8 9.79244 𝑋 10−3 6.58808 𝑋 10−9

0.9 5.91146 𝑋 10−3 4.07212 𝑋 10−9

1.0 3.57530 𝑋 10−3 2.51586 𝑋 10−9

Figure 3: Error graph of Problem 2 on LMM and RKTM at k=2

0

0.005

0.01

0.015

0.02

0.025

0.03

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

LMM at K =2

R-KTM at K =2



Table V11 :- Approximate results of Problem 2 on Method II at k=3


0.1 1.854094158893870 1.854090747704210 1.854093965245180

0.2 1.775530174744680 1.775529842733920 1.775530033710300

0.3 1.731041251962320 1.731032257932580 1.731041162771980

0.4 1.705127314681140 1.705144394868270 1.705127259867930

0.5 1.689784323984830 1.689794472704050 1.689784290526310

0.6 1.680612125128940 1.680618226597400 1.689784290526310

0.7 1.675097259033450 1.675099868186440 1.675097246602610

0.8 1.671769922367190 1.671761367728220 1.67176994788700

0.9 1.669758223053270 1.669760196768810 1.669758218432030

1.0 1.668540422822470 1.668548169179810 1.668540420004080

Table VI11 :-Absolute error of problem 2 on Method II at k=3

𝑋 LMM at K =3 Runge- Kutta type at K =3

0.1 3.4111896 𝑋 10−6 1.9364869 𝑋 10−7

0.2 3.3201076 𝑋 10−7 1.4103438 𝑋 10−7

0.3 8.9940297 𝑋 10−6 8.919034 𝑋 10−8

0.4 1.7080187 𝑋 10−5 5.481321 𝑋 10−8

0.5 1.0148719 𝑋 10−5 3.345852 𝑋 10−8

0.6 6.1014684 𝑋 10−6 2.03946 𝑋 10−8

0.7 2.6091529 𝑋 10−6 1.243084 𝑋 10−8

0.8 8.554638 𝑋 10−6 7.57849 𝑋 10−9

0.9 1.97377155 𝑋 10−6 4.62124𝑋 10−9

1.0 7.774635 𝑋 10−6 2.5158 𝑋 10−9

Figure 4: Error graph of Problem 2 on LMM and RKTM at k=3

0

0.000005

0.00001

0.000015

0.00002

0.000025

0.00003

0.000035

0.00004

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

LMM at K =3

R-KTM at K =3



5.0 SUMMARY, DISCUSSION OF

RESULTS AND CONCLUSION

In this research work, we derived hybrid block

Linear Multi-step methods from step length

𝑘 = 2, 3 respectively. The two block discrete

methods were later reformulated to its

equivalent Runge-Kutta type methods.

(RKTM)

In the first problem solved, RKT methods at

𝑘 = 2, 3 converges well than its equivalent

LMM at 𝑘 = 2, 𝑎𝑛𝑑 3 respectively. ( see

tables 11 and1V, figures 1 and 2)

Similarly in the second problem solved, RKT

methods at 𝑘 = 2, 3 performed excellently

well than its equivalent LMM at 𝑘 = 2, 𝑎𝑛𝑑 3

respectively. ( see tables V1 andV111, figures

3 and 4)

We want to conclude that Runge-Kutta type

Methods are more efficient than its equivalent

linear multi-step methods though it consumed

human efforts and time wasting during the

computational processes

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construction of some implicit linear multi-step …

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