Collocation Techniques for Block Methods for the Direct Solution of Higher Order Initial Value Problems of Ordinary Differential Equations

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http://www.sciencepublishinggroup.com/j/ijdst doi: 10.11648/j.ijdst.20170304.11

ISSN: 2472-2200 (Print); ISSN: 2472-2235 (Online)

Collocation Techniques for Block Methods for the Direct

Solution of Higher Order Initial Value Problems of Ordinary

Differential Equations

Kamoh Nathaniel Mahwash

1, *

, Awari Yohanna Sani

1

, Chun Pamson Bentse

2

1

Department of Mathematics/Statistics Bingham University, Karu, Nigeria

2_{Department of Mathematics/Statistics Plateau State University Bokkos, Bokkos, Nigeria}

Email address:

[email protected] (K. N. Mahwash), [email protected] (A. Y. Sani), [email protected] (C. P. Bentse)

To cite this article:

Kamoh Nathaniel Mahwash, Awari Yohanna Sani, Chun Pamson Bentse. Collocation Techniques for Block Methods for the Direct Solution of Higher Order Initial Value Problems of Ordinary Differential Equations. International Journal on Data Science and Technology.

Vol. 3, No. 4, 2017, pp. 39-44. doi: 10.11648/j.ijdst.20170304.11

Received: June 1, 2017; Accepted: August 21, 2017; Published: September 26, 2017

Abstract:

This paper presents the derivation techniques of block method for solving higher order initial value problems of ordinary differential equations directly. The method was developed via interpolation and collocation of the shifted Legendre polynomials as basis function. The method is capable of providing the numerical solution at several points simultaneously.

Keywords:

Collocation, Interpolation, Shifted Legendre Polynomials, Block Method, Higher Order, Direct Solution, Initial Value Problems

1. Introduction

This work considers solving an ordinary differential equation (ODE) of the th order ( ≥ 2). There are currently two well-known techniques for solving higher order ODEs. The first is to reduce it to a system of first order ordinary differential equations and then solve using predictor corrector or Runge-Kutta method. The second technique is a block method; it is formulated in terms of linear multistep methods. It preserves the traditional advantage of one step methods, of being self-starting and permitting easy change of step length (Lambert, 1973). Their advantage over Runge-Kutta methods lies in the fact that they are less expensive in terms of the number of functions evaluation for a given order. The method generates simultaneous solutions at all grid points as suggested by many researchers such as Anake (2011), Onumanyi and Okunuga (1985), Onumanyi and Yusuph (2002), Onumanyi et al., (1993, 1994), Awoyemi (2001, 2005 and 1991), Areo et al. (2008), Fatunla (1991, 1995), Lambert (1991), Awoyemi and Kayode (2005), Awari et al. (2014), Okunuga and Ehigie (2009), Adesanya et al., (2009, 2008), Serisina et al., (2004), Owolabi (2015), Yahaya and Badmus (2009) and much recently by Warren and Zill (2013) and Omar and Kuboye (2015). These methods solve higher order initial value problems of ordinary differential equations

without going through the process of reduction.

This present method is aimed at developing a general block method for the direct solution of higher order ( ≥ 2) initial value problems of ordinary differential equations, with the block approach, the non-self-starting nature associated with the predictor corrector method has been eliminated. Unlike the approach in predictor corrector method where additional equations were supplied from a different formulation, all our additional equations are obtained from the same continuous formulation.

2. Derivation of the k-Step Block Method

Consider solving an ordinary differential equation (ODE) of the th order ( ≥ 2)

1

0 1 2 1

( , , , ,..., )

( ) , ( ) , ( ) ,..., ( ) , [ , ]

y f x y y y y

y a y y a y y a y y a y x a b

ω ω

ω

ω −

−

′ ′′ =

′ ′′

= = = = ∈

(1)

(2)

Consider an approximate solution of (1) given by the shifted Legendre polynomials of the form;

0

( ) ( )

m

i i i

y x C P t

=

∑

(2)

where ∈ ℝ, ( , ). The derivative of (2) gives

0

( ) m

i i i

yω C Pω t

=

∑

(3)

Substituting (3) into (1) gives

' '' '" 1

0

( ) ( , ( ), ( ), ( ), ( ),... ( ))

m

i i i

yω C Pω t f x y x y x y x y x yω− x

=

∑

= (4)

Evaluating (4) at , = 0(1) and (2) at , =

0(1) − 1 respectively; gives a system of nonlinear equations of the form

AX =B (5)

where

0 1 2

(0) (0) (0) . . . (0)

(( 1) ) (( 1) ) (( 1) ) . . . (( 1) )

(0) (0) (0) . . . (0)

( ) ( ) ( ) . . . ( )

(2 ) (2 ) (2 ) . . . (2 )

. . . .

. . .

( ) ( ) ( ) (

m

P P P P

P k h P k h P k h P k h

P P P P

P h P h P h P h

P kh P kh P kh P

ω ω ω ω

− − − − 0

1 1

2

3 1

4 2

. .

)

n

k

n

m n k

C y

C f

kh

−

+ +

+

 _{   } _

 _{   } _

=

 _{   } _

     

 

Solving for ′",# = 0(1)2 − 1 in (5) using inverse of a matrix method which are then substituted into (2) to produce a continuous linear multistep method;

1

1 1

( ) ( ) ( )

k k

j n j j n j

j j

y x α x y hω β x f

−

+ +

= =

=

∑

+

∑

(6)

where $_%( ) and &_%( ) are coefficients to be determined. Evaluating $_%( ) and &_%( ) at _' and its first derivatives

evaluated at the points , = 0(1) and substituting in (6)

gives the discrete block method of the form

(())_*

+ = ∑ -'.) ())/++ ℎ ∑ 2'.) ())3+ (7)

Where (()), -()) and 2()) are square matrices and

*+= [ 5, 6, … , ', ℎ 8 5, ℎ 8 6, … , ℎ 8 ']:

/+ = [ , ℎ 8 ]:

3+= [; , ; 5, … , ; 6, … , ; ']:

which can be modified to obtain explicitly values for 5, 6, … , ', 8 5, 8 6, 8 <, … and 8 _' needed for the implementation of an th order initial value problem of ordinary differential equation.

3. Numerical Example

Consider a specific case with = 5, = 2 and

approximating the solution of (1) by the shifted Legendre polynomials of the form (2).

Evaluating (4) at , = (1) and (2) at and _>

respectively; gives a system of nonlinear equations of the form

AX=B(8)

where

0 1 2 3 4 5 6

'' '' '' '' '' '' ''

0 1 2 3 4 5 6

'' '' '' '' '' '' ''

0 1 0 3 4 5 6

'' ''

0 1

(0) (0) (0) (0) (0) (0) (0)

(4 ) (4 ) (4 ) (4 ) (4 ) (4 ) (4 )

( ) ( ) ( ) ( ) ( ) ( ) ( )

(2 ) (2 ) ( ) (2 ) (2 ) (2 ) (2 ) (3 ) (3 )

p p p p p p p p h p h p h p h p h p h p h

p h p h p h p h p h p h p h p h p h p h p h p h p h p h A

p h p h p

=

'' '' '' '' ''

0 3 4 5 6

'' '' '' '' '' '' ''

0 1 0 3 4 5 6

'' '' '' '' '' '' ''

0 1 0 3 4 5 6

( ) (3 ) (3 ) (3 ) (3 ) (4 ) (4 ) ( ) (4 ) (4 ) (4 ) (4 ) (5 ) (5 ) (5 ) (5 ) (5 ) (5 ) (5 )

h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h

 

 

 

0

1 4

2 1

3 2

4 3

5 4

6 5

,

n

c

y

c

y

c

f

c

f

X

B

c

f

c

f

c

f

+ + + + + +

 





 





 





 





 





=

 

=





 





 





 





 





 





Solving for ′",# = 0(1)6 in (8) using inverse of a matrix method and then substituting into (6) to give a continuous linear multistep method;

( )

4

( )

2 5

( )

0 0

j n j j n j

j j

y x

α

x y

₊

h

β

x f

₊

= =

=

∑

+

∑

(9)

(3)

( )

0 4

2 3 4 5 6 7

0 2 3 4 5

3 4 5 6 7

1 2 3 4 5

6 4 3 7 5

2 4 2 5 3

3 1 1 4 1 1 4

1 137 5 17 1 1 94

2 360 32 480 240 5040 315

5 77 71 7 1 356

6 144 450 360 1008 315

13 107 5 1 44 59

6 315

360 144 504 315

5

x t

x t t t t t t ht

h h h h h

x t t t t t ht

h h h h h

x t t t t ht t

h

h h h h

x α α β β β β = − = − = − + − + − − = − + − + − = + − − − − =

( )

3 4 5 6 7

2 3 4 5

4 3 6 7 5

4 ₂ ₄ ₅ ₃

7 4 3 6 5

5 ₅ ₂ ₄ ₃

13 49 1 1 152

9 24 240 360 504 315

61 4 5 11 1 41

63 24

288 3720 1008 480

1 5 4 1 1 7

315 30

5040 144 3360 480

t t t t t ht

h h h h h

x t ht t t t t

h

h h h h

x t t ht t t t

h

h h h h

β β                           ₋ ₊ ₋ ₊ ₋     = + − + − −      ₌ ₋ ₋ ₊ ₊ ₊                                   (10)

Evaluating (10) at @ = −1,−2, −3 and 5 with its first derivative evaluated at @ = 0, −1, −2, −3, −4 and 5 with @ = ( −

) and the results substituted in (9), to give the following discrete schemes needed for the implementation of the main scheme

2 2 2 2 2 2

1 1 2 3 4 5 4

2 2 2 2 2

2 1 2 3 4 4

2 2 2

3 1 2 3

3 337 53 71 1 1 1 13

4 480 120 240 240 480 4 240

1 8 13 8 1 1 1

2 15 15 15 30 2 30

1 127 29 161 1

4 480 60 240 15

n n n n n n n n n

n n n n n n n n

n n n n n

y y h f h f h f h f h f y h f

y y h f h f h f h f y h f

y y h f h f h f

+ + + + + + +

+ + + + + +

+ + + +

= − − − − − + −

= − − − − + −

= − − − − 2 2 2

4 5 4

2 2 2 2 2 , 2

4 1 2 3 4 5

2 2 2 2 2 2

5 1 2 3 4 5 4

1 3 1

480 4 60

1424 176 608 16 16 376

4

315 315 315 63 315 315

1 23 13 11 15 7 5 1

4 96 24 16 16 96 4 48

n n n n

n n n n n n n n n

n

h f h f y h f

y y h f h f h f h f h f hy h f

y y h f h f h f h f h f y h f

y + + + + + + + + + + + + + + + + + + − = + + + − + + + = − + + + + + + + ,

4 1 2 3 4 5

1

,

4 1 2 3 4 5

2

,

4 3

1 1 317 1403 3497 149 571 61

4 4 10080 10080 5040 1008 10080 10080

1 1 4 191 1 103 1 1

4 4 315 630 63 315 315 630

1 1

4 4

n n n n n n n n

n

n n

n

y y hf hf hf hf hf hf h h

y y y hf hf hf hf hf hf h h

y y y h h + + + + + + + + + + + + + + + + = − + + − − − − + = − + + + + − − −

= − + 1 2 3 4 5

,

4 1 2 3 4 5

4

,

5 4

41 481 2887 1159 683 61

2016 2016 5040 5040 10080 10080

1 1 4 92 124 296 118 4

4 4 315 315 315 315 315 315

1 1 317 1733

4 4 10080

n n n n n n

n n n n n n n n

n

n n n n

hf hf hf hf hf hf

y y y hf hf hf hf hf hf h h

y y y hf h h + + + + + + + + + + + + + + + + + + − + = − + + + + + + −

= − + + + 1 2 3 4 5

3671 1943 2753 3197

10080hfn+ +5040hfn+ +5040hfn+ +2016hfn+ +10080hfn+

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4. Analysis of the Block Method

4.1. Order and Error Constants

Expanding the block solution of (11) in Taylor’s series and collecting like terms in powers of ℎ, the following result is obtained;

C₎_{= C}₅_{= ⋯ = C}_E_{= (0,0,0,0,0,0,0,0,0,0)}:_,

C_F_{= G0,0,0,0,0,0,0, −} 29

2240 , −945 , −8 12096K275

:

C_L

= G−_{24192 , −}199 _{945 , −}19 _{4480 , −}141 _{189 , −}8 _{24192 , −}1375 _{60480 , −}863 _3780K37 :.

Hence the block method has varying order of NO = 5 and 6 with varying error constants of C_F= P−_66>)6Q , −_Q>RL , −_56)QE6FRS: and C_L= P−_6>5Q65QQ , −_Q>R5Q, −_>>L)5>5, −_5LQL , −_6>5Q65<FR, −_E)>L)LE< , −_<FL)<F S:.

Consistency

Following Lambert (1973, 1991), the block method is consistent since it has orders NO = 5,6 > 1

Zero stability

The block solution of the block method (11) is said to be zero stable if the roots U ; = 1, … , W of the first characteristic polynomial X(U), defined by

X(U) = YZ@|U\ − ]|

satisfies |U | ≤ 1 and every root with |U | = 1 has

multiplicity not exceeding two in the limit as ℎ → 0 From the block method (11), we have

Q=

1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1

                                andT =

0 0 0 0 0 1 0 0 0 0

0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0

                               

Substituting, we get

U5)_{− U}L_{= 0}

U = (0,0,0,0,0,0,0,0, −1,1)

This shows that the block method is zero stable, since all roots with modulus one do not have multiplicity exceeding the order of the differential equation in the limit as ℎ → 0.

Convergence

According to Lambert (1991), the block method is convergent since it is both consistent and zero stable.

Region of absolute stability

Reformulating (11) as a General Linear Multistep Method (GLMM) containing a partition of matrices A, B, C and DI where

1

1 0 0 0

4

1

0 1 0 0

2

3

0 0 1 0

4

0 0 0 1 0

5

0 0 0 1

4 A   −          ₋        =                  ₋    3

0 0 0 0

4

1

0 0 0 0

2

1

0 0 0 0

4

0 0 0 0 1

1

0 0 0 0

4 B                   =                 ₋   

1 4 0 3 3 4 9 7 1 4 9 5 7 1 6 1 1 0 0 8 0 5 0 4 0 1 0 0 8 1 0 0 8 0 1 0 0 8 0

1 9 1 1 1 0 3 1 1

6 3 0 6 3 3 1 5 3 1 5 6 3 0

4 8 1 2 8 8 7 1 1 5 9 6 8 3 6 1 2 0 1 6 5 0 4 0 5 0 4 0 1 0 0 8 0 1 0 0 8 0

9 2 1 2 4 2 9 6 1 1 8 4

3 1 5 3 1 5 3 1 5 3 1 5 3 1 5

1 7 3 3 3 6 7 1 1 9 4 3 2 7 5 3 3 1 9 7 1 0 0 8 0 5 0 4 0 5 0 4 0 2 0 1 6 1 0 0 8 0

C   −      ₋ ₋ ₋     = _ −    −                          

3 3 7 5 3 7 1 1 1

4 8 0 1 2 0 2 4 0 2 4 0 4 8 0

8 1 3 8 1

0

1 5 1 5 1 5 3 0

1 2 7 2 9 1 6 1 1 1

4 8 0 6 0 2 4 0 1 5 4 8 0

1 4 2 4 1 7 6 6 0 8 1 6 1 6

3 1 5 3 1 5 3 1 5 6 3 3 1 5

2 3 1 3 1 1 1 5 7

9 6 2 4 1 6 1 6 9 6

D I   − − − − −          ₋ ₋ ₋ ₋          = −_ − − − _       −              

Substituting these matrices into the stability

polynomial (( − U − 2`U6) − - , the stability matrix

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3 3 2 3 3 3 4 3 5 3 6 5 4

3 5 9 5 1 0 5 7 5 8 5 5 2 5 3

1 2 3 3 3 9 8 3 6 3 9 7 2 8 3 2 2 6 5 6 7 9 0 2 1 1 9 1 1 1 9 9 2 6 7

2 7 0 4 5 3 6 0 0 1 6 2 0 0 4 3 2 0 0 3 6 2 8 8 0 0 6 0 4 8 0 0

1 7 1 7 3 0 3 1 7 5 9 9 3 7 3 1 2 6 6 4 1 4 2 7

4 5 0 1 2 6 1 2 6 0 0 1 1 3 4 0 0 0 2 1 0 4 7 2 5 1 1 3 4 0 0

9 1 2 7 9

r z r z r z r z r z r z r r

r r z r z r z r z r z r z r z

+ + + + + + −

+ − − − − − + −

+ 5 4 5 5 5 6 4 7 4 8 4

4 2 4 3 4 4 4 5 4 6

5 9 0 6 5 7 1 0 9 1 2 3 9 2 8 3 7 1 9 7 1 9 7 6 4 5 9 2 6 5 1 3 4 0 2 0 1 1 3 4 0 0 0 1 0 2 0 6 0 0 0 8 1 6 4 8 0 0 2 7 2 1 6 0 0 1 0 0 8

4 8 0 9 7 3 4 6 5 3 5 3 5 3 3 4 7 0 4 7 0 3 2 9 7 0 5 4 6 1 8 9 0 2 0 5 7 1 9 3 7 8 0 0 5 1 0 3 0 0 0 2 7 2 1 6 0 0 3 6 2 8 8 0 0 1 6 3 2 9 6 0 0 0

r z r z r z r z r z r z

r z r z r z r z r z

+ + − − −

− − − − −

and

3 3 3 2 3 3 3 4 3 5 5

4 5 8 5 9 5 6 5 7 5 5 5 2

5 3

1 2 3 3 3 9 8 3 6 3 9 7 2 8 3 2 2 6 5 6 7 9 0 2 1 1 9 1 1 1 9 9 2 6 7

7 0 2 2 6 8 0 0 5 4 0 0 1 0 8 0 0 7 2 5 7 6 0 1 0 0 8 0 0

1 7 5 7 3 0 3 1 7 5 9 9 3 7 3 2 5 3 2 8 1 4 2 7

2

5 0 6 3 1 8 0 0 1 4 1 7 5 0 2 1 0 4 7 2 5 3 7 8 0 0

9 1 2 7 9 5 9 0 6 5 7 8 5 0 5

r r z r z r z r z r z r

r r z r z r z r z r r z r z

r z

+ + + + + +

− − − − − − + −

+ + 5 4 5 5 4 6 4 7 4

4 4 2 4 3 4 4 4 5

1 0 9 1 2 3 9 2 8 3 7 1 9 7 1 9 7 6 4 5 9 2 6 5 1 2 2 6 8 0 0 1 7 0 1 0 0 0 1 1 6 6 4 0 0 3 4 0 2 0 0 1 0 0 8

4 8 0 9 7 3 4 6 5 3 5 3 5 3 3 4 7 0 4 7 0 3 2 9 7 0 5 4 6 1 8 9 0 2 0 5 7 1 9 1 8 9 0 0 1 7 0 1 0 0 0 6 8 0 4 0 0 7 2 5 7 6 0 2 7 2 1 6 0 0 0

r z r z r z r z r

r z r z r z r z r z

+ − − −

− − − − −

The region of absolute stability is shown in Figure 1

Figure 1. The Region of Absolute Stability of the Block Method (11).

5. Conclusion

In this paper, it has been shown that continuous collocation methods for solving ordinary differential equations can equally be derived through the approach in this study. In this study, a new block method approach which is capable of solving higher order initial value problems of ordinary differential equations is presented, with the block approach; the non self starting nature associated with the predictor

corrector method and the Runge Kutta method has been eliminated. Unlike the approach in predictor corrector method where additional equations were supplied from a different formulation, all the required additional equations are obtained from the same continuous formulation. The basic property of the method was investigated and was found to be zero stable, consistent and convergent. The absolute stability region of the block method was also investigated and revelations showed that the newly constructed method is not A-stable as revealed by Figure 1. This method is very simple and effective for a wide-range of ordinary differential equations.

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