Derivation of Three - Step Sixth Stage Runge-Kutta Method For The Solution Of First Order...

International journal of scientific and technical research in engineering (IJSTRE)

www.ijstre.com Volume 1 Issue 1 ǁ April 2016.

Manuscript id. 371428197 www.ijstre.com 8

Derivation of Three - Step Sixth Stage Runge-Kutta Method For

The Solution Of First Order Differential Equations

Mshelia DWa, Yakubu DG

b , Badmus AM

c and Manjak NH

b

a) Department of Mathematics, Umar Ibn Ibrahim Elkanami. College of Education, Science and technology, Bama

Borno State

Email [email protected]

b) Department of Mathematical Sciences, Abubakar Tafawa Balewa University Bauchi. Nigeria Email

[email protected]

c) Department of Mathematics, Nigerian Defence Academy,Kaduna. Nigeria

Email [email protected]

Abstract In this research paper, we extended the idea of Hybrid Block method at 𝑘 = 3 through interpolation

and collocation approaches to an effectively Sixth Stage Implicit Runge-Kutta method for the solution of initial

value problem of first order differential equations. The new approach displays a uniform order 6 schemes and

zero stable. The new method demonstrates superiority over its equivalent linear multi-step method (LMM) with

some numerical experiments tested.

Keyword: (Linear Multi-step method, Sixth stage, Runge-Kutta, Uniform order and Zero stable)

I. Introduction 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. Among the methods for the numerical solutions of ODEs, the Discrete Value Methods (DVMs), which

systematically generate the approximate solutions {yn} along the step-point, {xn}, xn = a + nh, are the most

popular. Here h denotes the positive step size and yn is assumed to approximate y(xn), many of the existing

methods such as Euler method, Runge-Kutta methods and linear multistep methods fall into this category of

DVMs.

Runge-Kutta methods are among the most popular (ODEs) solvers, it was first studied by Carle Runge and

Martins Kutta around 1900. Runge-Kutta (RK) methods are single-step methods, they do not require successive

high derivatives of a function and therefore are general purpose initial value problem solvers. The most popular Runge-Kutta method is the Classical Runge-Kutta method of fourth order. It’s symmetrical in form and has

simple coefficients. The method is well suited for Computers because it needs no special starting procedure,

makes light demand on storage and repeatedly uses the same straight forward computational procedure. Its

numerically stable, unfortunately the early Runge-Kutta methods are explicit. Explicit RK methods are generally

unstable for the solution of Stiff equations because their region of absolute stable is small, in particular it’s

bounded. The instability of explicit Runge-Kutta methods motivates the development of implicit methods. We

consider implicit linear multistep methods and reformulated them into implicit RK methods with order 𝒑 =𝒔(𝒔𝒕𝒂𝒈𝒆𝒔)

1.1 Statement of the Problem

We want to develop highly efficient-Implicit Runge-Kutta integration formula at 𝒌 = 𝟑 for solution of first

order initial value problem of ordinary differential equations of the form

𝑦 ′ 𝑥 = 𝑓 𝑥, 𝑦 , 𝑦 0 = 𝑦0 (1.0)

1.2 Aim and Objectives

The aim of this research work is to Reformulation Linear multistep methods to its equivalent Runge-Kutta type

methods at 𝒌 = 𝟑 for the integration of Stiff and non stiff ordinary differential equations for first ODEs.

mailto:[email protected]



Derivation of Three - Step Sixth Stage Runge-Kutta Method For The Solution Of First Order

Differential Equations

Manuscript id. 371428197 www.ijstre.com Page 19

Objectives:

i) To derive self starting high order discrete and hybrid Schemes at 𝒌 = 𝟑 in Block form with their

Continuous formulations. The single Continuous formulation is evaluated at some selected points to obtain some discrete schemes which form the Block methods for solving (1.0)

ii) To Reformulate the block discrete methods into its equivalent Runge-Kutta methods of s stages

iii) To compare and contrast the efficiency of the methods and also implementation costs.

iv) To analysis the Block stability of the methods.

Theorem 1.0 Let I denote the identity matrix of dimension (𝑚 + 𝑡) x (𝑚 + 𝑡) and consider the matrices C and

D. Then

𝑖. 𝐷𝐶 = 𝐼

𝑖𝑖 𝑦 𝑥 = 𝐶𝑖+1,𝑗+1𝑦𝑛+𝑗

𝑡−1

𝑗 =0

+ 𝐶𝑖+1,𝑗+1𝑓𝑛+𝑗

𝑚−1

𝑗 =0

𝑥𝑖

𝑡+𝑚−1

𝑖=0

i.e. 𝐶 = 𝐷−1

for proof (see Onumanyi et al,1994)

1.3 Definition of Basic terms

Definition 1:-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)

Definition 2: Order and Error constants

A linear multi-step method

𝒚 𝑥 = 𝛼𝑗

𝑘

𝑗 =0

𝑥 𝑦𝑛+𝑗 = ℎ 𝛽𝑗

𝑘

𝑗 =0

(x)𝑓𝑛+𝑗 (𝟏. 𝟏)

we associate the linear differential operator

𝐿 𝑦 𝑥 ; ℎ = 𝛼𝑗 ;𝑦 𝑥 + 𝑗h − ℎ𝛽𝑗 𝑦′(𝑥; 𝑗h)

𝑘

𝑗 =0

( 1.2)

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 (1.1) gives

𝐿 𝑦 𝑥 ; ℎ = 𝐶0𝑦 𝑥 + 𝐶1ℎ𝑦 ′ 𝑥 + ⋯ + 𝐶𝑞ℎ𝑞𝑦𝑞(𝑥) (1.3)

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, … (1.4)

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)

II. Derivation of Sixth stage Runge-Kutta Methods at .𝒌 = 𝟑

Now we assume a power series solution of the form

𝑦 𝑥 = 𝛼𝑗 𝑥𝑗

𝑚+𝑡−1

𝑗 =0

(2.1)

as a basis solution for (1.0). Also the second derivative of (2.1) gives

𝑦′ 𝑥 = 𝑗𝛼𝑗 𝑥𝑗−1

𝑚+𝑡−1

𝑗 =0

(2.2)

Interpolating (2.1) at 𝑥 = 𝑥𝑛+𝑗 , 𝑗 = 0,1

2 ,1,

3

2 and collocating (2.2) at 𝑥 = 𝑥𝑛+𝑗 , 𝑗 = ,1,2 ,3. Specifically for this

method 𝑡 = 4 and 𝑚 = 3, the degree of the polynomial is 𝑚 + 𝑡 − 1 and we have the following system of non

linear equations

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

𝑎1 + 2𝑎2𝑥𝑛+1 + 3𝑎3𝑥𝑛+12 + 4𝑎4𝑥𝑛+1

3 + 5𝑎5𝑥𝑛+14 + 6𝑎6𝑥𝑛+1

5 = 𝑓𝑛 +1

𝑎1 + 2𝑎2𝑥𝑛+2 + 3𝑎3𝑥𝑛+22 + 4𝑎4𝑥𝑛+2

3 + 5𝑎5𝑥𝑛+24 + 6𝑎6𝑥𝑛+2

5 = 𝑓𝑛 +2

𝑎1 + 2𝑎2𝑥𝑛+3 + 3𝑎3𝑥𝑛+32 + 4𝑎4𝑥𝑛+3

3 + 5𝑎5𝑥𝑛+34 + 6𝑎6𝑥𝑛+3

5 = 𝑓𝑛 +3

(2.3)

By arranging (2.3) in Matrix equation form we have

1 𝑥𝑛 𝑥𝑛2 𝑥𝑛

3 𝑥𝑛4 𝑥𝑛

5 𝑥𝑛6

1 𝑥𝑛+

12

𝑥𝑛+

12

2 𝑥𝑛+

12

3 𝑥𝑛+

12

4 𝑥𝑛+12

5 𝑥𝑛+

12

6

1 𝑥𝑛+1 𝑥𝑛+12 𝑥𝑛+1

3 𝑥𝑛+14 𝑥𝑛+1

5 𝑥𝑛+16

1 𝑥𝑛+

32

𝑥𝑛+

32

2 𝑥𝑛+

32

3 𝑥𝑛+

32

4 𝑥𝑛+32

5 𝑥𝑛+

32

6

0 1 2𝑥𝑛+1 3𝑥𝑛+12 4𝑥𝑛+1

3 5𝑥𝑛+14 6𝑥𝑛+1

5

0 1 2𝑥𝑛+2 3𝑥𝑛+22 4𝑥𝑛+2

3 5𝑥𝑛+24 6𝑥𝑛+2

5

0 1 2𝑥𝑛+3 3𝑥𝑛+32 4𝑥𝑛+3

3 5𝑥𝑛+34 6𝑥𝑛+3

5

𝑎0

𝑎1

𝑎2

𝑎3

𝑎4

𝑎5

𝑎6

=

𝑦𝑛

𝑦𝑛+

12

𝑦𝑛+1

𝑦𝑛+

32

𝑓𝑛 +1

𝑓𝑛 +2

𝑓𝑛 +3

(2.4)

The continuous formula for (2.4) will be of the form

𝑦 𝑥 = 𝛼0𝑦𝑛 + 𝛼12𝑦

𝑛+12

+ 𝛼1𝑦𝑛+1 + 𝛼32𝑦

𝑛+3 2

+ ℎ 𝛽1𝑓𝑛 +1 + 𝛽2𝑓𝑛+2 + 𝛽3𝑓𝑛 +3 (2.5)

Using Maple 11 Mathematical software to evaluate the values of 𝑎𝑗 , ( 𝑗 = 0,1

2 , 1,

3

2 ,1 2,3) in (2.4) and

substituted in (2.5) to obtain our Continuous formula for this method as




.]

4095

)(31

1365

)(322

16380

)(1801

8190

)(999

16380

)(1045

8190

)(102[]

455

)(48

455

)(894

455

)(2334

455

)(1111

455

)(610

455

)(123[]

35

)(47

35

)(2022

140

)(50971

70

)(3653

140

)(4705

70

)(534[

]1755

)(2848

585

)(48716

1755

)(431216

1755

)(566116

1755

)(334016

1755

)(70816[]

65

)(12

65

)(24

65

)(287

65

)(2844

65

)(6552

65

)(1024[]

455

)(736

455

)(13948

455

)(142416

455

)(229716

455

)(176016

455

)(51616[]

12285

)(2204

4095

)(8778

12285

)(779811

12285

)(358924

12285

)(693252

12285

)(1662641[)(

35

6

4

5

3

4

2

32

25

6

4

5

3

4

2

32

15

6

4

5

3

4

2

32

6

6

5

5

4

4

3

3

2

2

16

6

5

5

4

4

3

3

2

2

6

6

5

5

4

4

3

3

2

2

6

6

5

5

4

4

3

3

2

2

2

3

2

1

nnnn

nnnn

nn

nnnnn

n

nnnnn

n

nnnnn

nn

nnnn

nn

n

nnn

nnnn

nn

nnnn

fh

xx

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xx

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xxxxf

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xxxxf

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xx

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xx

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xxy

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xx

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h

xxxy

(2.6)

Evaluating (2,6) at 𝑥 = 𝑥𝑛+𝑗 , 𝑗 = 2,3 and the first derivative of (2.6) is evaluated at 𝑥 = 𝑥𝑛+𝑗 , 𝑗 =1

2,

3

2 to obtain

the following discrete schemes as our Block method at 𝑘 = 3

𝑦𝑛+2 +72

65𝑦𝑛+1 +

256

455𝑦

𝑛+12−

61

1365𝑦𝑛 =

27

35ℎ𝑓𝑛+1 +

102

455ℎ𝑓𝑛 +2 −

1

455ℎ𝑓𝑛+3

𝑦𝑛+3 +320

39𝑦

𝑛+32−

45

13𝑦𝑛+1 −

576

91𝑦

𝑛+12

+160

273𝑦𝑛 =

45

7ℎ𝑓𝑛 +1 +

180

91ℎ𝑓𝑛+2 +

25

91ℎ𝑓𝑛+3

26432𝑦𝑛+

32

+ 25704𝑦𝑛+1 − 74304𝑦𝑛+

12

+ 22168𝑦𝑛 = −4095ℎ𝑓𝑛 + 31239ℎ𝑓𝑛 +1

+1107ℎ𝑓𝑛+2 − 51ℎ𝑓𝑛+3

2032𝑦𝑛+

32

+ 5724𝑦𝑛+1 − 7344𝑦𝑛+

12− 412𝑦𝑛 =

1

1872ℎ𝑓

𝑛+12

+ 3159ℎ𝑓𝑛+1 + 72ℎ𝑓𝑛 +2

−3ℎ𝑓𝑛 +3

263312𝑦𝑛+

32− 212436𝑦𝑛+1 − 54864𝑦

𝑛+12

+ 3988𝑦𝑛 = 94419ℎ𝑓𝑛+1 − 4968ℎ𝑓𝑛+2

+65520ℎ𝑓𝑛+

32

+ 129ℎ𝑓𝑛+3

(2.7)

The method (2.7) is of Order 6,6,6,6,6]𝑇 with Error constants of 11

127400, −

19

10192, −

2487

560, −

249

1120, − 7041

1120 𝑇

By arranging (2.7) in Matrix equation form, we have




256

455

72

65

−512

195 1 0

−576

91

−45

13

320

39 0 1

74304 −25704 −26432 0 07344 −5724 −2032 0 0−54864 −212436 263312 0 0

𝑦

𝑛+12

𝑦𝑛+1

𝑦𝑛+

32

𝑦𝑛+2

𝑦𝑛+3

=

0 0 0 0

61

1365

0 0 0 0 −160

2730 0 0 0 −22168

0 0 0 0 4120 0 0 0 −3988

𝑦

𝑛−52

𝑦𝑛−2

𝑦𝑛−

32

𝑦𝑛−1

𝑦𝑛

+

0

−27

35 0

102

455

−1

455

0 45

7 0

180

91

25

910 −31239 0 −1107 51

−1872 −3159 0 −72 3 0 94419 65520 −4968 129

𝑓

𝑛+12

𝑓𝑛 +1

𝑓𝑛+

32

𝑓𝑛 +2

𝑓𝑛 +3

+

0 0 0 0 00 0 0 0 00 0 0 0 40950 0 0 0 00 0 0 0 0

𝑓

𝑛−52

𝑓𝑛−2

𝑓𝑛−

32

𝑓𝑛−1

𝑓𝑛

(2.8)

Let 𝐴(0) =

256

455

72

65

−512

195 1 0

−576

91

−45

13

320

39 0 1

74304 −25704 −26432 0 07344 −5724 −2032 0 0−54864 −212436 263312 0 0

By multiplying (2.8) by the inverse of 𝐴(0) and rearrange it in Butcher Table as

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

1200

1 11

120

0 27

40

−8

15

27

40

11

1200

(2.9)

The Table (2.9) satisfies Runge-Kutta conditions for solution of first order ODEs since

𝑖 𝑎𝑖𝑗 = 𝑐𝑖

𝑠

𝑗 =1

𝑖𝑖 𝑏𝑗 = 1

𝑠

𝑗 =1

see Butcher 1996

The method (2.7) is formally given as Runge-Kutta type method as

)120

11

40

27

15

8

40

270

120

11( 6543211 kkkkkkhyy nn

where

nn yxfk ,1




)

1920

1

17280

211

1080

49

5760

487

216

35

17280

959(,

6

15543212 kkkkkkhyhxfk nn

)

3240

1

1080

7

405

8

360

11

135

32

3240

169(,

3


)

1920

1

640

9

120

13

640

81

40

9

1920

103(,

3


)*0

135

7

135

32

45

4

135

32

135

7(,

3


)

120

11

40

27

15

8

40

27*0

120

11(, 6543216 kkkkkkhyhxfk nn

(2.10)

III. Consistency and Stability of the block scheme The sixth stage RK method

256

455

72

65

−512

195 1 0

−576

91

−45

13

320

39 0 1

74304 −25704 −26432 0 07344 −5724 −2032 0 0−54864 −212436 263312 0 0

𝑦

𝑛+12

𝑦𝑛+1

𝑦𝑛+

32

𝑦𝑛+2

𝑦𝑛+3

=

0 0 0 0

61

1365

0 0 0 0 −160

2730 0 0 0 −22168

0 0 0 0 4120 0 0 0 −3988

𝑦

𝑛−52

𝑦𝑛−2

𝑦𝑛−

32

𝑦𝑛−1

𝑦𝑛

+

0

−27

35 0

102

455

−1

455

0 45

7 0

180

91

25

910 −31239 0 −1107 51

−1872 −3159 0 −72 3 0 94419 65520 −4968 129

𝑓

𝑛+12

𝑓𝑛 +1

𝑓𝑛+

32

𝑓𝑛 +2

𝑓𝑛 +3

+

0 0 0 0 00 0 0 0 00 0 0 0 40950 0 0 0 00 0 0 0 0

𝑓

𝑛−52

𝑓𝑛−2

𝑓𝑛−

32

𝑓𝑛−1

𝑓𝑛

Let 𝐴(0) =

256

455

72

65

−512

195 1 0

−576

91

−45

13

320

39 0 1

74304 −25704 −26432 0 07344 −5724 −2032 0 0−54864 −212436 263312 0 0

When the first characteristic equation is multiplied by the inverse of 𝐴(0), and applied the condition of zero

stability we obtained the following

𝜌 𝑧 = 𝜆

1 0 0 0 00 1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 1

−

0 0 0 0 −10 0 0 0 −10 0 0 0 −10 0 0 0 −10 0 0 0 −1

= 0




𝜆4 𝜆 − 1 = 0

𝜆1 = 0, 𝜆2 = 0, 𝜆3 = 0, 𝜆4 = 0, 𝜆5 = 1

From definition 1, the newly hybrid block method (2.7) is zero stable and also consistent since the order of all

the integrators in (2.7) are [6,6,6,6,6]𝑇 > 1

IV. Numerical Experiments

The following examples are used to confirm the efficiency of our method

Example 4.1

𝑦 ′ − 2𝑦 = 𝑒−𝑥 , 𝑦 0 =3

4 ℎ = 0.1 0 ≤ 𝑥 ≤ 1.0

Exact Solution: 𝑦 𝑥 = −1

3𝑒−3𝑥 +

13

12 𝑒2𝑥

Example 4.2

𝑦 ′ = 20𝑥2 − 20𝑦 + 2𝑥, 𝑦 0 =1

3 ℎ = 0.05 0 ≤ 𝑥 ≤ 1.0

Exact Solution: 𝑦 𝑥 = 𝑥2 +1

3𝑒−20𝑥

Example 4.3

𝑦 ′ = −𝑦, 𝑦 0 = 1 ℎ = 0.05 0 ≤ 𝑥 ≤ 1.0

Exact Solution: 𝑦 𝑥 = 𝑒−𝑥

Table 1: Approximate solution of Example 4.1 at k=3

Mesh value Exact solution Present Method at

k=3

Error of Present

Method at k=3

0.1 1.021573848000000 1.021573848000000 ---------------

0.2 1.343233171000000 1.343233171000000 -----------------

0.3 1.72702262600000 1.72702262600000 ------------------

0.4 2.18756265600000 2.18756265600000 ------------------

0.5 2.742628426000000 2.742628426000000 ----------------

0.6 3.413856121000000 3.413856119000000 2.0 E (-09)

0.7 4.227604866000000 4.227604860000000 3.0 E (-09)

0.8 5.216008802000000 5.216008801000000 1.0 E (-09)

0.9 6.418261528000000 6.418261529000000 1.0 E (-09)

1.0 7.882184294000000 7.882184288000000 6.0 E (-09)


Mesh value Yakubu et al (2004) Kwami (2011) Present Method k=3

0.05 4.6657 (E-04) 1.70624598 (E-07)

0.10 1.0629 (E-02) 4.2107 (E-05) 1.25538478 (E-07)

0.15 5.6175 (E-04) 6.927449 (E-08)

0.20 5.3890 (E-03) 6.4073 (E-04) 3.3979523 (E-08)

0.25 1.7049 (E-05) 1.562545 (E-08)

0.30 1.2320 (E-02) 5.3864 (E-04) 6.897933 (E-09)

0.35 1.9855 (E-04) 2.96054 (E-09)

0.40 1.3008 (E-03) 7.2861 (E-05) 1.24471 (E-09)

0.45 2.7302 (E-05) 5.1514 (E-10)

0.50 4.1148 (E-04) 1.0418 (E-05) 2.10566 (E-10)

0.55 3.6440 (E-06) 8.5209 (E-11)

0.60 3.9430 (E-4) 1.7999 (E-06) 3.4196 (E-11)

0.65 6.6250 (E-07) 1.363 (E-11)

0.70 4.0724 (-05) 2.4356 (E-07) 5.400 (E-12)

0.75 9.0031 (E-08) 2.131 (E-12)




0.80 1.3629 (E-05) 3.3443 (E-8) 8.360 (E-13)

0.85 1.2140 (E-8) 3.260 (E-13)

0.90 1.3672 (E-05) 4.8625 (E-09) 1.280 (E-13)

0.95 3.5210 (E-07) 4.70 (E-14)

1.00 1.4145 (E-06) 1.2680 (E-07) 1.0 (E-14)

Figure 1: Error graph of Example 4.2


Mesh values Yakubu et al

(2004) k = 3

Kwami (2011) k= 3 Present method k = 3

0.05 2.0291 x 10-11 1.0 x 10-15

0.10 2.1541 x 10-7 1.0482 x 10-11 2.0 x 10-15

0.15 3.0476 x 10-11 3.0 x 10-15

0.20 6.9544 x 10-8 4.9204 x 10-11 4.0 x 10-15

0.25 3.7974 x 10-11 5.0 x 10-15

0.30 2.8062 x 10-7 5.6549 x 10-11 6.0 x 10-15

0.35 6.8823 x 10-11 6.0 x 10-15

0.40 4.1350 x 10-7 5.8933 x 10-11 7.0 x 10-15

0.45 7,1250 x 10-11 7.0 x 10-15

0.50 2.8127 x 10-7 8.2750 x 10-11 7.0 x 10-15

0.55 7.2173 x 10-11 8.0 x 10-15

0.60 4.1578 x 10-7 8.3785x 10-11 7.0 x 10-15

0.65 9.0835 x 10-11 8.0 x 10-15

0.70 4.9444 x 10-7

8.1565 x 10-11

9.0 x 10-15

0.75 8.8840 x 10-11

9.0 x 10-15

0.80 3.7858 x 10-7 9.5601 x 10-11 9.0 x 10-15

0.85 8.6093 x 10-11 9.0 x 10-15

0.90 4.6203 x 10-7 9.3104 x 10-11 9.0 x 10-15

0.95 9.6813 x 10-11

9.0 x 10-15

1.0 5.0564 x 10-7 8.8506 x 10-11 9.0 x 10-15

0

0.002

0.004

0.006

0.008

0.01

0.012

1 2 3 4 5 6 7 8 9 10

Yakubu e tal 2004

Kwami 2011

Present Method




Figure 2: Error graph of Example 4.3 𝑎𝑡 𝑘 = 3

V. Discussion of Results In the three problems tested with our method, the numerical solution and absolute errors of example 4.1

is display in Table 1. The comparison of Absolute error of problem 2 with Yakubu et al (2004), Kwami (2011)

and our present method is displayed in Table 2 and figure 1, while the errors of problems 4.3 with different

methods were displayed in Table 3 and figure 2 respectively.

VI. Conclusion The implicit Block linear multi-step method 𝑎𝑡 𝑘 = 3 is reformulated to Sixth stage implicit Runge-

Kutta method. The advantage of this method is that in all the problems tested, the result obtained converges

better than the existing methods. (see Tables 1,2, 3 and error graphs 1 and 2)

References [1] Butcher J C (1996) A history of Runge-Kutta methods. Journal of Applied Numerical Mathematics.

Volume 20 pp247-260

[2] Fatunla S.O (1991)“Block Method for Second Order Differential Equations” International Journal of

Computer mathematics. 41: 55-63. [3] Henrici P (1962) Discrete variable methods in ordinary differential equations. John Willey and Sons

[4] Kwami AM (2011) A class of Continuous General Linear method for ordinary differential equations,

Ph.D Thesis (unpublished), Abubakar Tafawa Balawa University, Bauchi. Nigeria

[5] Lambert J.D (1973): Computational Methods in Ordinary Differential Equations .John Wiley and

Sons, New York. 278

[6] Mshelia DW, Yakubu DG , Badmus AM and Manjak NH (2015) A fifth stage Runge- Kutta method for

the Solution of ordinary differential equations submitted to Journal of Scientific Research and Report

[7] Yakubu, D.G, Onumanyi P and Chollom J P. (2004), A new family of general linear methods based on

the block Adams-Moulton multistep methods J. of pure and Applied Sciences, 7 (1): 98 – 106.

[8] Onumanyi P and Awoyemi D.O, Jator S.N. and Siriseria U.W.(1994) “ New Linear Multistep Methods

with continuous coefficients for first order ivps” Journal of Nigeria Mathematics society 13: 37- 51

0

0.000000

0.000000

0.000000

0.000000

0.000000

0.000000

1 2 3 4 5 6 7 8 9 10

Yakubu etal 2004

kwami 2011

Present method

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