A Fifth-Order Hybrid Block Integrator for Third-Order Initial Value Problems

(1)

Çankaya University

Journal of Science and Engineering

https://dergipark.org.tr/cankujse

A Fifth-Order Hybrid Block Integrator for Third-Order Initial Value Problems

Rotimi Oluwasegun Folaranmi^1* , Abayomi Ayotunde Ayoade² , Tolulope Latunde³

1* Department of Mathematics and Computing, Thomas Adewumi University, Kwara State, Nigeria

2 Department of Mathematics, University of Lagos, Lagos State, Nigeria

3 Department of Mathematics, Federal University Oye-Ekiti, Ekiti State, Nigeria

Keywords Abstract

Block Method, Chebyshev Polynomials, Linear Multistep Method.

The formulation of hybrids block method as integrator of third-order Initial Value Problems in Ordinary Differential Equations is our focus in this paper. Chebyshev polynomials were used as trial function to develop a hybrid One-step Method (HBOSM3) adopting collocation and interpolation technique. The basic properties of HBOSM3 were integrated and findings revealed that the method was accurate and convergent. One of desirable features of these methods is the production of exact solutions at the grid points.

1. Introduction

This work is concerned with the class of the Problems )

, '' , ' , , ( )

( ⁽ ¹⁾

)

(^m x = f x y y y y ^m−

y 

) ( ) (

₀ ₀

)

( x y s

y ^s

=

s=0,1,2, ,m−1 (1)

for the case m=3.

The analytical solution of many of such problems does not exist. Thus, the need for formulation of numerical schemes to integrate (1) becomes necessary.

Researchers have reduced higher order of (1) to first order ODEs but with a set back see [1,2,3,4] and the inability of the method to utilize additional information associated with a specific ODE such as the oscillatory nature of the solution [5] occasioned by the increase dimension of the problem and low order of accuracy of the methods employed to solve the system of first-order IVPs of ODEs.

It has commonly been reported by scholars that implementation of linear multistep methods in predictor- corrector mode is very expensive.

To circumvent the setbacks encounter in predictor-corrector approach, the block methods have been introduced to solve IVPs in ODEs. [6,7] first proposed block methods as a means of obtaining stating values for predictor- corrector algorithms. The block methods however provide the advantage of being self starting, possess uniform order and they are obtained from a single continuous formula. Of recent, [8,9,10,11,12,13,14,15] developed different numerical methods and considered different trial functions.

Problems arising from ODEs can either be formulated as an IVP or a BVP. However, our concern shall be with IVP. Several researchers such as [16,17,18] attempt solving (1) directly using derived LMMs without reduction to system of first order ODEs. [6,18] developed new block methods which are self-starting using power series

(2)

function, [19] employed a newly derived orthogonal polynomials as trial function to develop one step hybrid block method for solution of general second order Initial value problem. Despite elegant properties of Chebyshev polynomials, it is rare seeing scholars considering it as basis function.

Thus, in this paper,we propose the development of one-step hybrid block linear multistep method for the solution of Initial Value Problems of Third Order Differential Equations with the use of Chebyshev polynomial as basis function.

2. Methodology

The approximation of analytical solution of problem (1) employing Chebyshev polynomial of the form



⁺ ⁻

=

¹

0

) ( )

(

s r

n n

nT x

a x

y , (2)

as basis function on the partition a=x₀ x₁ ...x_k x_k₊₁...x_k =b is considered here.

The function y(x) is integrated in the interval [a, b], with a constant step size h, given byh= x_k₊₁−x_k; 1

,...

1 ,

0 −

= K

k .

and



=

− 

= ⁿ

j

n n j

n x Cos nCos x C x

T

0 ) ( 1 )

( )

( , (3)

which is the nth degree Chebyshev polynomial which is valid in the range of definition of (2).

The Chebyshev polynomials T_n(x)satisfied the recurrence relation )

( ) ( 2 )

( ₁

1 x xT x T x

T_n₊ = _n − _n₋ , n1(T₀(x)=1) (4)

for interval [-1, 1].

Thus t

h h x x x

x

x x X x

x

T ^k

k k

k

k − − =

− =

−

= −

=

+

+ 2 2

) 2 (

1 1

1 , x_k xx_k₊₁

(5) wheret =t(x),a function of

x

, is given by (5).

The first, second and third derivative of (2) is given by



⁺ ⁻

=

¹

0

'

' ( ) )

( '

s r

n n

nT x

a x

y (6)



⁺ ⁻

=

¹

0

) ( '' )

( ''

s r

n n

nT x

a x

y (7)



⁺ ⁻

=

¹

0

) ( ' ' ' )

( ' ' '

s r

n n

nT x

a x

y (8)

wherex ( ba, ), the a_n's are constant, r and s are points of collocation and interpolation respectively.

Conventionally, we need to interpolate at least three points to be able to approximate the solution of (2) and, for this purpose, we proceed by arbitrarily selecting two off-step point,

v

xk₊ , v(0,1)in(x_k,x_k₊₁) ensuring that the zero-stability of the main method is guaranteed.

Thus, equation (2) is interpolated atx_k₊_i, i=0, v and 1 and its third derivative is collocated at x_k₊_i i=0, v, and 1 so as to obtain a system of equations.

In what follows, we shall develop one step methods with three off step points

1 3 1 ,

and

v = .

(3)

2.1. Development of the Proposed Method

In this section, the derivation of continuous one-step method with three off-step point is considered. Using (2) with 𝑟 = 5 and 𝑠 = 3, we have a polynomial of degree seven as follow;



=

⁷

0

) ( )

(

n n nT x a x

y (9)

with third derivative given by



=

⁷

0

) ( '' ' )

( '' '

n n

nT x

a x

y (10)

Collocating (10) at x=x_k₊_i,

, 1 4 , 3 2 , 1 4 , 1

= 0

i and interpolating (9) at x= x_k₊_i,

4 , 1

= 0

i and

2

1

lead to a

system of equations written in matrix form AX =Bas:

) 11 (

0 1

2 1 1

2 1 2

1 1 2

1 2

1 1

56448 21504

6720 1536

192 0

0 0

4032 768

960 768

192 0

0 0

2688 0

960 0

192 0

0 0

4032 768

960 768

192 0

0 0

56448 21504

6720 1536

192 0

0 0

2 1 4 1 1 3

4 3 3

2 1 3

4 1 3

3

7 6 5 4 3 2 1 0

















=





























−

+ + + + + +

k k k

k k k k k

y y y

f h

a a a a a a a a

The values of a_j,j =0,1, 7below are obtained using Maple software to solve (11)

) 12 (

161280 1 40320

1 26880

1 40320

1 161280

1

46080 1 23040

1 23040

1 46080

1

46080 1 5760

1 2560

1 5760

1 46080

1

46080 1 11520

7 11520

7 46080

1

46080 1 5760

7 2560

7 5760

7 46080

1

3 161280 4

13 80640

163 26880

377 80640

793 161280

13

2 23040 2

1 4608

5 2560

21 23040

143 11520

1

1 3

4 3 3

2 1 3

4 1 3 3

7

1 3

4 3 3

4 1 3 3

6

1 3

4 3 3

2 1 3

4 1 3 3

5

1 3

4 3 3

4 1 3 3

4

1 3

4 3 3

2 1 3

4 1 3 3

3

2 1 4

1 1

3

4 3 3

2 1 3

4 1 3 3

1

2 1 4

1 1

3

4 3 3

2 1 3

4 1 3 3

0











+

− +

−

=

+

− +

−

=

+ +

− +

=

+ +

−

=

+ +

=

+

− + +

+ +

+

=

+

− + +

+ +

+

=

+ + +

+

+ + +

+

+ + +

+

+ + +

+ +

+

+ + +

+ +

+

k k k

k k

k k k k

k k k

k k

k k k k

k k k

k k

k k k

k k

k k k

k k

f h f

h f

h a

f h f

h f

h a

f h f

h f

h a

f h f

h f

h a

f h f

h f

h a

y y

y f h f

h f

h a

y y

y f h f

h f

h a

Substituting the values of a_i(0 i7)into (2), we obtain a continuous scheme in the form

(4)

) 13 ( ) ) ( )

( )

( ( )

( )

( ) (

1

0 4

3 4 3 2 1 2 1 4 1 4 1 3

2 1 2 1 4 1 4 1

0

  

+ +

=

^x ^y^k ^x ^y_k₊ ^x ^y_k₊ ^h



^j ^x ^f^k⁺^j ^x ^f_k₊ ^x ^f_k₊ ^x ^f_k₊

x

y

      

Simplifying (13), we have













 



 + − − + +

=



 



− − + + − −

=



 



 + + − +

=



 



 + − + + −

=



 



 − + − − +

=

+ +

=

−

= +

=

7 6

5 4

2 3

1

7 6

5 4

2 3

4 3

7 5

3 2

3

2 1

7 6

5 4

2 3

4 1

7 6

5 4

2 3

0

2

2 1

2

4 1

2 0

2520 1 1440

1 2880

1 1152

1 23040

7 161280

) 13 (

630 1 720

1 180

1 144

1 2304

5 80640 ) 47

(

420 1 96

1 48

1 1280

21 26880

) 97 (

630 1 720

1 180

1 144

1 11520

193 80640

) 583 (

2520 1 1440

1 2880

1 1152

1 23040

1 161280

) 13 (

2 3 1 ) (

4 4 ) (

2 )

(

t t

h t

t t

h t

t t

t h

t

t t

h t

t t

h t

t t t





(14)

where

h h x t x− ^k −

=2 2

Evaluating (13) with the expressions in (14) at

4 +3

=

k

x

x and x= x_k₊₁the following main methods are obtained as

 



 

 + + − +

+ +

−

=

₊

+ +

+

+ 1

4 3 2

1 4

1 3

2 1 4

1 4

3

116 126 4

15360 3

3

_k

k k

k k k

k k

k h f f f f f

y y

y

y (15)

 



 

 + + + +

+ +

−

=

₊

+ +

+ + 1

4 3 2

1 4

1 3

2 1 4

1

86 126 26

6 3840 8

3

_k

k k

k k k

k k

k h f f f f f

y y

y

y (16)

Differentiating (13), we obtain

(5)

(17)

The second derivatives of continuous functions are given as















 



 − − + +

=



 



− + + − −

=



 



 + − +

=



 



 − + + −

=



 



− + − − +

=

−

=

5 4

3 2

1

5 4

3 2

4 3

5 3

2 1

5 4

3 2

4 1

5 4

3 2

0

2 2

1

2 4

1 0 2

15 1 12

1 36

1 24

1 2880 ) 7

( ' '

15 4 6

1 9

4 3

1 288 ) 5

( ' '

5 2 6

5 2 1 160 ) 21

( ' '

15 4 6

1 9

4 3

1 1440 ) 193

( ' '

15 1 12

1 36

1 24

1 2880 ) 1

( ' '

) 16 ( ' '

) 32 ( ' '

) 16 ( ' '

t t

h t

t t

h t

t t

t h

t

t t

h t

t t

h t t h t h t h





(18) The additional methods to be coupled with the main methods are obtained by evaluating the first and second derivatives of (13) at x_k

4 +1 k

x ,

2 +1 k

x

4 +3 k

x andx_k₊₁. Thus, we have the following discrete schemes:

) 19 40320 (

79 4480

19 40320

793 80640

2 307 8

6 ₁

4 3 2

1 4

1 3

2 1 4

1

' 







 + − + −

= +

−

+ ₊

+ +

+ k

k k

k k k

k k

k y y y h f f f f f

hy

( )













 



 + − − + +

=



 



− − + + − −

=



 



 + − + + −

=



 



 − + − − +

= +

=

+

−

= +

=

6 5

4 3

2 1

6 5 4 3 2

4 3

6 5 4

3 2

4 1

6 5

4 3

2 0

2 1 4 1 0

180 1 120

1 288

1 144

1 5760

7 80640 ) 13

( '

45 1 60

1 18

1 576

5 40320 ) 47

( '

45 1 60

1 18

1 2880

193 40320 ) 583

( '

180 1 120

1 288

1 144

1 5760

1 80640 ) 13

( '

8 1 6 ) ( '

16 1 8 ) ( '

8 1 2 ) ( '

t t

t h

t

t t t t t h

t

t t t

t t h

t

t t

t h

t h t t

h t t





(6)

) 20 32256 (

1 8064

1 8960

3 40320

383 161280

2 79

2 ₁

4 3 2

1 4

1 3

2 1 '

4

1 





− − − − +

=

−

+ ₊

+ +

+ k

k k

k k k k

k y y h f f f f f

hy

) 21 80640 (

13 40320

47 13440

97 40320

583 80640

6 13 8

2

₁

4 3 2

1 4

1 3

2 1 4

1 '

2

1







 

 + + − +

=

− +

−

₊

+ +

+

+ k

k k

k k k

k k

k y y y h f f f f f

hy

) 22 32256 (

1 40320

163 1792

117 40320

1801 161280

10 89 16

6 ₁

4 3 2

1 4

1 3

2 1 4

1 '

4

3 





 + + + +

=

− +

− ₊

+ +

+

+ k

k k

k k k

k k

k y y y h f f f f f

hy

) 23 80640 (

391 8064

503 4480

569 40320

3061 16128

14 11 24

10 ₁

4 3 2

1 4

1 3

2 1 4

1 '

1 





 + + + +

=

− +

− ₊

+ +

+ + k

k k

k k k

k k

k y y y h f f f f f

hy

) 24 192 (

1 1440

41 480

31 480

101 2880

16 233 32

16

₁

4 3 2

1 4

1 3

2 1 4

1 ''

2







 

 − − + − −

=

− +

−

₊

+ +

+ k

k k

k k k

k k

k y y y h f f f f f

y h

) 25 720 (

1 120

1 480

13 72

1 160

16 1 32

16

₁

4 3 2

1 4

1 3

2 1 4

1 ''

4 1

2







 

 + − + −

=

− +

−

₊

+ +

+

+ k

k k

k k k

k k

k y y y h f f f f f

y h

) 26 2880 (

7 288

5 160

21 1440

193 2880

16 1 32

16 ₁

4 3 2

1 4

1 3

2 1 4

1 ''

2 1

2 





− + + − +

=

− +

− ₊

+ +

+

+ k

k k

k k k

f f

f h

y y

h

) 27 240 (

1 360

37 480

139 120

13 288

16 1 32

16

₁

4 3 2

1 4

1 3

2 1 4

1 ''

4 3

2

 



 

 + + + −

=

− +

−

₊

+ +

+

+ k

k k

k k k

f f

f h

y y

h

) 28 2880 (

239 480

157 96

19 1440

209 320

16 1 32

16

₁

4 3 2

1 4

1 3

2 1 4

1 ''

4 1

2







 

 − + + + +

=

− +

−

₊

+ +

+

+ k

k k

k k k

k k

k y y y h f f f f f

y h

Equations

( ) 19

,

( ) ( ) 20  28

are solved simultaneously to obtain the block method as shown below.

(7)











+ +

+

=

− +

+ +

+

=

− +

+ +

+

=

− +

+

=

+ +

+

=

− +

+ +

=

− +

+ +

=

− +

+ +

=

− +

+ +

+

=

− +

+ +

+

=

− +

+ +

+

=

− +

+ +

+

=

+ + +

+ +

+ + +

+ +

+ + +

+ +

+ + +

+ +

+ + +

+ +

+ + +

+ +

+ + +

+ +

+ + +

+ +

+ + +

+ +

+ + +

+ +

+ + +

+ +

1 3 4 3 3 2 1 3 4 1 3 3

'' 2 ''

1

1 3 4

3 3 2

1 3 4 1 3 3

'' 2 ''

4 3

1 3 4

3 3 2 1 3 4 1 3 3

'' 2 ''

2 1

1 3 4

3 3 2

1 3 4

1 3 3

'' 2 ''

4 1

4 3 3 2 1 3 4 1 3 3

'' 2 ' '

1

1 3 4

3 3 2

1 3 4

1 3 3

'' 2 ' '

4 3

1 3 4

3 3 2 1 3 4 1 3 3

'' 2 ' '

2 1

1 3 4

3 3 2

1 3 4

1 3 3

'' 2 ' '

4 1

1 3 4

3 3 2

1 3 4

1 3 3

'' 2 ' 1

1 3 4

3 3 2

1 3 4

1 3 3

'' 2 '

4 3

1 3 4

3 3 2

1 3 4

1 3 3

'' 2 ' 2

1

1 3 4

3 3 2

1 3 4

1 3 3

'' 2 '

4 1

90 7 45

16 15

2 45

16 90

7

320 3 160

21 40

9 160

51 320

27

360 1 90

1 15

1 90

31 360

29

2880 19 1440

53 120

11 1440

323 2880

251

45 4 15

1 15

4 90

7

2560 9 128

3 1280

27 640

117 2560

147 4

3

480 1 90

1 48

1 10

1 1440

53 2

1

7680 7 5760

29 3840

47 128

3 9103

145 4

1

504 1 105

2 210

1 315

34 840

31 2

1

71680 81 7168

45 35840

243 35840

1863 71680

1431 32

9 4

3

40320 19 5040

13 168

1 5040

83 40320

331 8

1 2 1

645120 47 107520

43 107520

103 64512

107 71680

113 32

1 4

1

k k k

k

k k k

k k k k

k k k

k k k k

k k k

k k k k

k k

k k k

k k

k k k

k k

k k k

k k

k k k

k k

k k k

f h f

h f

h f h y h y

f h f

h f

h y

f h f

h f

h f h y

h y

f h f

h f

h y

f h f

h f

h f h y h hy y

f h f

h f

h y

h hy y

f h f

h f

h f h y

h hy y

f h f

h f

h y

h hy y

f h f

h f

h y

h hy y y

f h f

h f

h y

h hy y y

f h f

h f

h k

f h y

h hy y y

f h f

h f

h y

h hy y y

(29)

3. Basic Properties of the Method

3.1. Order and Error Constant

Here under, the basic properties of the derived schemes are discussed.

The implicit schemes (15) and (16) belong to the class of LMM of the form



= +

= + = ^k

j

j n j k

j

j n

jy h f

0 3

0





(30)

Following [21], the approach adopted in [9,20,3], we define the local truncation error associated with equation (30) by the difference operator

 



=

+

− +

= ^k

j

n j n

jy x jh h f x jh

h x y L

0

3 ( )

) (

] : ) (

[

 

(31)

Where y(x) is an arbitrary function, continuously differentiable on [a, b].

Expanding (31) in Taylor series about the point

x

, we obtain the expression

) ( )

( '' )

( ' )

( ]

);

(

[

y x h C₀y x C₁hy x C₂h²y x C ₃h ³y ³ x

L

= + + +  +

_p₊ ^p⁺ ^p⁺

Where the C₀ , C₁ , C₂ C_p C_p₊₂ are obtained as



=

= ^k

j

C j 0

0



,



=

= ^k

j

j j

C

1



,



=

= ^k

j

j j

C

1 2 2 2!

1



 

 



 − − −

=  

=

−

=

k q j k

j q

q j q q q j

C q

( 1 )( 2 )

³