A Class of One-Step Hybrid Third Derivative Block ... - Boffin Access

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Advances in Computer Sciences

ISSN 2517-5718

A Class of One-Step Hybrid Third Derivative Block Method for the Direct Solution of Initial Value Problems of Second-Order Ordinary Differential Equations Skwame Y1 Raymond D2*

1 2

Department of Mathematical Science, Adamawa State University, Mubi-Nigeria Department of Mathematics and Statistics, Federal University, Wukari-Nigeria

Article Information

Abstract In this paper, we consider the development of a class of one-step hybrid third derivative block method with three off-grid points for the direct solution of initial value problems of second order Ordinary Differential Equations. We adopted method of interpolation and collocation of power series approximate solution to generate the continuous hybrid linear multistep method, which was evaluated at grid points to give a continuous block method. The discrete block method was recovered when the continuous block method was evaluated at selected grid points. The basic properties of the method was investigated and was found to be zero-stable, consistent and convergent. The efficiency of the method was tested on some stiff equations and was found to give better approximation than the existing method, which we compared our result with.

Keywords

One-Step; Hybrid Block Method; Third Derivative; Stiff Odes; Collocation and Interpolation Method.

DOI:

10.31021/acs.20181109

Article Type:

Research Article

Journal Type:

Open Access

Volume:

1 Issue: 2

Manuscript ID: ACS-1-109 Publisher:

Boffin Access Limited

Received Date:

13 February 2018

Accepted Date: 16 April 2018 Published Date: 08 May 2018

Introduction

This paper solves second order initial value problems in the form

= y '' f ( x, y ( x ) , y ' ( x )= y '0 ) , y ( x0 ) y= 0 , y ' ( x0 )

(1)

where f is continuous within the interval of integration. Solving higher order derivatives method by reducing them to a system of first-order approach involves more functions to evaluate which then leads to a computational burden as in [1-3]. The setbacks of this approach had been reported by scholars, among them are Bun and Vasil’yer and Awoyemi et al [4,5].

The method of collocation and interpolation of the power series approximation to generate continuous linear multistep method has been adopted by many scholars; among them are Fatunla, Awoyemi, Olabode, Vigor Aquilar and Ramos, Adeniran et al, Abdelrahim et al, Mohammad et al to mention a few [5-11]. Block method generates independent solution at selected grid points without overlapping. It is less expensive in terms of number of function evaluation compared to predictor corrector method, moreover it possess the properties of Runge Kutta method for being self-starting and does not require starting values. Some of the authors that proposed block method are: [12-17]. In this paper, we developed a on e-step hybrid third derivative method with three offgrid, which is implemented in block method. The method is self-starting and does not require starting values or predictors. The implementation of the method is cheaper than the predictor-corrector method. This method harnesses the properties of hybrid and third derivative, this makes it e¢cient for sti¤ problems.

The paper is organized as follows: In section 2, we discuss the methods and the materials for the development of the method. Section 3 considers analysis of the basis properties of the method which include convergence and stability region, numerical experiments where the e¢ciency of the derived method is tested on some numerical examples and discussion of results. Lastly, we concluded in section 4.

*Corresponding author: Raymond D Department of Mathematics and Statistics Federal University, Wukari-Nigeria Tel. No: 0233207482596 E-mail: [email protected]

Citation:

Skwame Y, Raymond D. A Class of One-Step Hybrid Third Derivative Block Method for the Direct Solution of Initial Value Problems of Second-Order Ordinary Differential Equations. Adv Comput Sci. 2018;1(2):109

Copyright:

© 2018 Skwame Y, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 international License, which permits unrestricted use, distribution and reproduction in any medium, provided the original author and source are credited.

Derivation of the Method

We consider a power series approximate solution of the form

y ( x) =

2 r + s −1

∑ j =0

 x − xn  ai    h 

j

(2)

where r=2 and s=5 are the numbers of interpolation and collocation points respectively, is considered t o be a solution to (1). The second and third derivative of (2) gives

Adv Comput Sci

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aj j!

 x − xn    2 h ( j − 2)  h 

2 r + s −1

∑

y '' ( x ) =

j =2

j −2

= f ( x, y , y ' ) ,

 x − xn  ∑   3 h  j =3 h ( j − 3 ) 

(3)

= g ( x, y , y ' ) ,

Substituting (3) into (1) gives

j −2 2 r + s −1 a j j !  x − xn  j −3  x − xn  = f ( x, y, y '') ∑ 2   + ∑ 3   h ( j − 2)  h  h  =j 2= j 3 h ( j − 3)  (4)

aj j!

2 r + s −1

1 Collocating (4) at all points xn + r , r = 0  4 1 and Interpolating   1

Equation (2) at xn + s , s = 0, , gives a system of non linear equation 4 of the form

AX = U

4031 ( x - xn )5 85862 ( x - xn )6 54 405 h3 h4

+

208100 ( x - xn )7 75920 ( x - xn )8 66080 ( x - xn )9 + 567 189 243 h5 h6 h7

j −3

aj j!

2 r + s −1

y ''' ( x ) =

+



(5)

Where

A = [ a0 , a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 , a9 , a10 , a11 ] T ,

-

b1 = 4

7168 ( x - xn )5 270592 ( x - xn )6 45 405 h3 h4

+

841216 ( x - xn )7 567 h5

-

-

753664 ( x - xn )10 32768 ( x - xn )11 + 1215 h8 297 h9

2

xn2

x2

n+

xn3 1 4

x3

n+

xn4 1 4

2 2

6 xn 6x 1

2

6x

2

6x

2

6 xn +1

0 0

n+

n+

x4

x5

n+

12 x 12 x

n+

4 1 2

1 4 2 n 2

xn5

12 x 2

n+

1 4 3 n 3

xn6 x6

n+

1 4 1 2

20 x 20 x

n+

20 x3

n+

1 4 4 n 4

xn7 x7

n+

1 4 1 2

30 x 30 x

n+

30 x 4

n+

1 4 5 n 5

xn8 x8

n+

1 4 1 2

42 x 42 x

n+

42 x5

n+

1 4 6 n 6

xn9 x9

n+

1 4 1 2

56 x 56 x

n+

56 x 6

n+

1 4 7 n 7

x10 n x10 1

n+

1 4 1 2

72 x 72 x

n+

72 x 7

n+

n+

4 8 n 8

90 x 1 4 1 2

90 x

n+

90 x8

n+

1 4 1 2

12 x 2

20 x3

30 x 4

42 x5

56 x 6

72 x 7

90 x8

12 x

20 x

30 x

6 6

24 xn 24 x 1

60 602

120 120

42 x 210 210

56 x 336 336

72 x 504 504

90 xn8+1 7207n 7207 1

0

6

24 x

60

120

210

336

504

7207

0

6

24 x

60

0

6

24 xn +1

60

n+

3 4

3 4 2 n +1

n+

n+

n+

3 4 3 n +1 2 n

n+

4 1 2

3 n+ 4

1 n+ 4 2 1 n+ 2 2 3 n+ 4 2 n +1

3 4 4 n +1 3 n 3 1 n+ 4 3 1 n+ 2 3 3 n+ 4 3 n +1 n+

120

120

3 4 5 n +1 4 n 4 1 n+ 4 4 1 n+ 2 4 3 n+ 4 4 n +1

n+

210

210

3 4 6 n +1 5 n 5 1 n+ 4 5 1 n+ 2 5 3 n+ 4 5 n +1 n+

336

336

3 4 7 n +1 6 n 6 1 n+ 4 6 1 n+ 2 6 3 n+ 4 6 n +1 n+

504

504

n+

n+

n+

7207

3 4

4 1 2

3 n+ 4 7 n +1

720

  yn   y  x11 1   n+ 1  n+ 4   a0   4  9 110 xn     f n   a1   110 x 9 1     f 1  a2 n+ n+ 4 4     110 x 9 1   a3   f 1  n+  n+ 2  2a  4 110 x 9 3   a   f 3   5   n+  n+ 4   =  4 110 xn9+1   a6   f n +1  9908n   a7   g n      9908 1   a8   g 1  n+   a9   n + 4  4 9908 1   a10   g 1     n+  n+ 2  a   2  9908 3   11   g 3  n+ n+  4  4  g n +1  9908n +1  x11 n

Solving (5) for a i ' s using Gaussian elimination method, gives a continuous hybrid linear multistep method of the form  + h ∑ β f 

   1 1 3 , ,  + h  ∑ γ j g n + j + γ k g n + k  , k= 4 2 4   

1 1 2 3 j n+ j j n+ j k n+k 1 = j 0=j 0 j = 0, 4

∑α

y

+β f

Differentiating (6) once yields

    1 1 1 α j yn + j + h  ∑ β j f n + j + ∑ β j f n + j  + h 2  ∑ γ j g n + j + ∑ γ j g n + j  ∑  113   113  h = 1 j 0= j 0 =j 0, = =  j 4 , 2 , 4   j 4 , 2 , 4  4 p ' ( x )=

Where

(6)

(7)

4( x − xn ) α0 = 1 − h

α1 = 4

364288 ( x - xn )8 358912 ( x - xn )9 + 189 h6 243 h7

b1 = -

and

y ( x=)

148231 128 ( x - xn ) 4 ( x - xn )h 11975040 9 h2

+

  U =  yn , y 1 , f n , f 1 , f 1 , f 3 , f n +1 , g n , g 1 , g 1 , g 3 , g n +1  T , n+ n+ n+ n+ n+ n+ n+   4 4 2 4 4 2 4

1 xn  1 xn + 1 4  0 0  0 0  0 0  0 0   0 0 0 0  0 0  0 0   0 0  0 0 

5120 ( x - xn )11 126464 ( x - xn )10 + 297 h9 1215 h8

Adv Comput Sci

3264 ( x - xn )8 2048 ( x - xn )9 2048 ( x - xn )10 + 7 9 45 h6 h7 h8

+

b3 = 4

-

x)

243193 128 ( x - xn ) 4 ( x - xn )h + 11975040 9 h2

17408 ( x - xn )5 213248 ( x - xn )6 + 135 405 h3 h4

-

685568 ( x - xn )7 313088 ( x - xn )8 + 567 189 h5 h6

326144 ( x - xn )9 720896 ( x - xn )10 + 243 h7 1215 h8

32768 ( x - xn )11 297 h9

-

Equation (6) is evaluated at the non-interpolating points and (7) at all points, produces the following general equations in block form

2602339 1 485 ( x ( x - xn )h + ( x - xn ) 38320128 2 36

456 ( x - xn )5 4424 ( x - xn )6 10496 ( x - xn )7 5 h3 15 h4 21 h5

b1 = -

4( x − xn ) h

=-

-

12( x - xn ) 4 1807 ( x - xn )h 80640 h2

382169 53 ( x - xn ) 4 ( x - xn )h + 191600640 36 h2

1241 ( x - xn )5 23758 ( x - xn )6 + 90 405 h3 h4

-

80356 ( x - xn )7 38992 ( x - xn )8 + 567 189 h5 h6

-

43552 ( x - xn )9 103936 ( x - xn )10 + 243 h7 1215 h8

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5120 ( x - xn )11 297 h9

-

Equation (6) is evaluated at the non-interpolating points



28343 1 ( x - xn )h 2 + ( x - xn )3 g0 = 18247680 6



-

4546 ( x - xn )7 1600 ( x - xn )8 1360 ( x - xn )9 + 189 h4 63 h5 81 h6

4

Where,

551 16 ( x - xn ) 4 ( x - xn )h 2 49896 3 h

20768 ( x - xn )8 19328 ( x - xn )9 38912 ( x - xn )10 + 63 81 405 h5 h6 h7 8192 ( x - xn )11 + 495 h8

-

g1 = 2

6( x - xn ) 4 32027 ( x - xn )h 2 3548160 h

264 ( x - xn )5 3124 ( x - xn )6 3224 ( x - xn )7 + h2 h3 h4 5 15 7 608( x - xn )8 4288 ( x - xn )9 1024 ( x - xn )10 + 9 5 h5 h6 h7 2048 ( x - xn )11 + 55 h8 +

g3 = 4

3959 16 ( x - xn ) 4 ( x - xn )h 2 1596672 9 h

+

736 ( x - xn )5 9184 ( x - xn )6 30208 ( x - xn )7 + 45 135 189 h2 h3 h4

-

14176 ( x - xn ) 15232 ( x - xn ) 34816 ( x - xn ) + 63 81 405 h5 h6 h7

+

8

9

10

8192 ( x - xn )11 495 h8

14339 1 ( x - xn ) 4 ( x - xn )h 2 127733760 12 h 5 6 47 ( x - xn ) 452 ( x - xn ) 1538 ( x - xn )7 + + 60 135 189 h2 h3 h4 752 ( x - xn )8 848 ( x - xn )9 2048 ( x - xn )10 + 63 h5 81 h6 405 h7 512 ( x - xn )11 + 495 h8 g1 =

Adv Comput Sci



AYL = B R1 + C R2 + D R3 + E R4 + G R5

608 ( x - xn )5 18848 ( x - xn )6 7424 ( x - xn )7 + 15 135 27 h2 h3 h4

+

4

produces the following general equations in block form

512 ( x - xn )11 512 ( x - xn )10 + 495 h8 81 h7

g1 =

2

and (7) at all points

   xn , xn + 1 , xn + 1 , xn + 3 , xn +1  4 2 4  

25 ( x - xn ) 4 209 ( x - xn )5 398 ( x - xn )6 + 18 36 27 h h2 h3 +



 xn + 1 , xn + 3 , xn +1 

 −2  −3   −4  − 4  h  4 A =  − h  4 −  h  4 − h  − 4  h

1 0 0 0 0 0 0 0 1 0 0 0 0 0  0 0 1 0 0 0 0  0 0 0 0 0 0 0   0 0 0 1 0 0= 0 B   0 0 0 0 1 0 0   0 0 0 0 0 1 0  0 0 0 0 0 0 1  

y 1  n+ 4  y   n + 12     yn + 3   4  yn +1  C  Y = = y '  n+ 1  4  y' 1   n+ 2     y 'n + 3  4   y 'n +1   16463h 2  435456   6149h 2  72576  2  9925h  72576   −148231h  D =  11975040  83485h  1197504  26921h   147840  1156801h   5987520  2735353h  11975040

Volume: 1.2

445h 2 32256 767 h 2 10752 767 h 2 5376 −1807 h 80640 1489h 80640 767 h 5376 21521h 80640 24817h 80640

 1 0  2 0    3 0         =  R                     

 37111h 2   6967296     12679h 2   1161216    2  19709h   1161216     − 260233h   38320128  =  , R2  1961683h   95800320   1403419h     63866880   2182739h     95800320   5021969h   191600640 

2225h 2 435456 1403h 2 72576 5179h 2 72576 −243193h 11975040 89279h 5987520 103853h 3991680 165731h 1197504 2640391h 11975040

(8)

  

[ fn ]

3217 h 2  6967296  1381h 2  1161216   8411h 2  f 1 1161216   n+ 4   −382169h  f  1 191600640  , R =  n + 2    3  137161h   f n+ 3   4 95800320   f n +1  5303h   2365440  358217 h   95800320  3530299h  38320128 

3/6

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 −17 h3  9072   −3053h3  967680  3  −29h  7560  2  551h  49896 G= 2  −23851h  2280960  2  −95h  16632  2  −72269h  15966720  2  −61h  249480

−83h3 30720 −83h3 15360 −83h3 15360 32027 h 2 3548160 −12053h 2 1774080 −56677 h 2 3548160 −12053h 2 1774080 32027 h 2 3548160

−871h3 1451520 −347 h3 193536 −269h3 241920 3959h 2 1596672 −5755h 2 3193344 −7951h 2 2661120 −123463h 2 15966720 15701h 2 1140480

−599h3  23224320  −127 h3  1935360   −1117 h3  g 1  3870720    n+ 4  14339h 2  g   n + 12  127733760  , = R  5   2 −2567 h  E  g n + 3 =  4 31933440     g n +1  −5311h 2  42577920   −6439h 2  31933440   −312317 h 2  127733760 

Multiplying equation (8) by the inverse of (A) gives the hybrid block formula of the form

 4253h3   23224320     23h3   60480    3  467 h   774144    2  −28343h   18247680  =  , R4 2  551h   798336   2   3629h   4730880   2   3239h   3991680   2   25651h   25546752 

[ gn ]

I YL = B R1 + C R2 + D R3 + E R4 + G R5

1 0 0 0 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

0 0 0 0 0

0 0 0 0 0

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

148231h 2  47900160  196 h 2   4455 2  18531h  197120  13952 h 2  + 93555  89371 h   1088640  1654 h  8505  921h  4480  2048h   8505

1807 h

2

y 1   n+ 4    y n + 1  1 2  0   0   y 3  1 n+ 4  0   y 0   n +1  1 = y' 0  n+ 1   4  1 0  y' 1  0 0 n+ 2   1   y '  0 0 3  n + 4  0 y '   n +1 

243193h

2

322560 2 h

47900160 2 68 h

40 2 3159 h

4455 2 6813h

35840 2 52 h

197120 2 8578 h

315 103h

93555 38341h

2520 52 h

1088640 394 h

315 81h

8505 711h

280 104 h

4480 2048 h

315

8505

Adv Comput Sci

 2602339 h 2   153280512   35h 2    h  891   2 4  39015 h   h   630784   2  6353h 2   3h   yn  +  74844  [ f ] n 4   y 'n   1539551h  h    1   17418240   24463h  1   272160  1   6501h  1    71680   1601h     17010 

 28343h3   −551h3  382169 h  72990720   199584   1277 h3  766402560  −41h3 2    13h    13305603   5544 3 8910  2    9747 h   −4509 h 8469 h    6307840   394240 3153920    269 h 2   −464 h3 2  f 1 3457 h  n + 4   124740      374220   f 1  + 26051h 2 [ g n ] +  31185 2   n+   59681h  −31207 h   2   11612160  f  1451520 17418240  n + 3  421h 2   −19 h 2 4 1153h        1134 181440 272160   fn +1    2  −279 h 2 411h   339 h     143360  71680  179202 1601h   2   −32 h 29 h    17010   2835  11340  2

Volume: 1.2

−32027 h

3

−3959 h

3

14192640 3 −5 h

6386688 3 −17 h

693 3 −19197 h

9240 3 −9 h

1576960 3 −10 h

2464 3 −16 h

693 2 −81h

4455 2 −1243h

1520 2 −h

290304 2 −31h

40 2 −81h

5670 2 −183h

1520

17920 2 32 h

0

2835

2

  510935040 2  −109 h  1330560  2 −27 h  g 1  180224  n+ 2  4 −5 h g 1   n+ 12474   2 2 −2237 h   g  3 n+   4 11612160  2   g n +1  −43h  181440 2  −9 h  28672  2 −29 h   11340  −14339 h

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Analysis of Basic Properties of the Method Order of the Block According to [7] the order of the new method in Equation (8) is obtained by using the Taylor series and it is found that the developed method has a uniformly order eleven, with an error constants vector of:

6.1829 ×10−14 ,1.752 ×10−13 ,3.0454 ×10−13 , 4.8542 ×10−13 ,  C11 =  −13 −13 −13 −13   4.1791×10 , 4.8542 ×10 ,5.5292 ×10 ,9.7083 ×10 

T

Consistency

The hybrid block method [4] is said to be consistent if it has an order more than or equal to one. Therefore, our method is consistent.

Zero Stability of Our Method

 k ( i ) k −i  = ρ ( z ) det = ∑ A z  0  j =0 

−h

1 0 0 0 0 0 0 0 1 0 0 0 0 0  0 0 1 0 0 0 0  0 0 0 1 0 0 0 ρ ( z) = z 0 0 0 0 1 0 0  0 0 0 0 0 1 0 0 0 0 0 0 0 1  0 0 0 0 0 0 0

 0 0  0  0  0   0 0 − 0  0   0  0 0  0   1  0  0

0 0 1 0 0 0

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

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

h 4  h 2 3h  2 4 = z 6 ( z − 1) h  1 1  1  1

ρ ( z )= z 6 ( z − 1) = 0, 2

Definition: A block method is said to be zero-stable if as h → 0 , the root zi , i = 1(1) k of the first characteristic polynomial ρ ( z ) = 0 that is

7219   4 = h12   w   15107327262720000 

Satisfies zi ≤ 1 and for those roots with zi = 1 , multiplicity must not exceed two. The block method for k=1, with three off grid collocation point expressed in the form

Hence, our method is zero-stable.

Regions of Absolute Stability (RAS)

Using Mat Lab package, we were able to plot the stability region of the block methods (see Figure 1 below). The stability polynomial for K=1 with three offstep point using Scientific Workplace software package we have in the following one:

79 1171 3423139  3  4    3  11   w +   w   w  − h    30519853056000    156959244288000     139591703907532800 

−

4666483 585349 73634299 2899283  3   4  9   3   3   w +   w  + h   w −   w   23930006384148480    1495625399009280     3625758543052800    2913555972096000 

10  

12604072069 775601  4  6897035147  3  7    4  707100649  3    w +   w  − h   w +   w  1495625399 00928000 8157956721 868800 9442079539 200       109258348953600     6   1953881  4  26693444639  3  5   2850443  4  357859  3  + h   w +   w  w −   w  − h    1077753600    145677798604800     36883123200    256927334400  −h

−h

8 

2587  4  12221035  3  3   25  4  269  3  2   16399  4  1009733  3  4 3    w  + w − w w +   w  − h   w +   w  + h   w +  4257792 1034643456 1782 114740 228096 4790016               

4 

x-values 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.00

Exact Solution 0.90483741803595957316 0.81873075307798185867 0.74081822068171786607 0.67032004603563930074 0.60653065971263342360 0.54881163609402643263 0.49658530379140951470 0.44932896411722159143 0.40656965974059911188 0.36787944117144232160

x-values .0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.00

Exact Solution -0.10517091807564762480 -0.22140275816016983390 -0.34985880757600310400 -0.49182469764127031780 -0.64872127070012814680 -0.82211880039050897490 -1.01375270747047652160 -1.22554092849246760460 -1.45960311115694966380 -1.71828182845904523540

Table 1: Comparison of the proposed method

Table 2: Comparison of the proposed new method

Adv Comput Sci

Computed Solution 0.90483741803595957245 0.81873075307798185795 0.74081822068171786534 0.67032004603563930001 0.60653065971263342285 0.54881163609402643185 0.49658530379140951390 0.44932896411722159058 0.40656965974059911099 0.36787944117144232065 Computed Solution -0.10517091807564762481 -0.22140275816016983391 -0.34985880757600310397 -0.49182469764127031779 -0.64872127070012814679 -0.82211880039050897478 -1.01375270747047652150 -1.22554092849246760440 -1.45960311115694966350 -1.71828182845904523500

Volume: 1.2

Error in our method 7.100E(-19) 7.200E(-19) 7.300E(-19) 7.300E(-19) 7.500E(-19) 7.800E(-19) 8.000E(-19) 8.500E(-19) 8.900E(-19) 9.500E(-19) Error in our method 1.000E(-20) 1.000E(-20) -3.000E(-20) -1.000E(-20) -1.000E(-20) -1.000E(-19) -1.000E(-19) -2.000E(-19) -3.000E(-19) -4.000E(-19)

Error in [13] 1.054712E(-14) 1.776357E(-14) 2.342571E(-14) 2.797762E(-14) 3.130829E(-14) 3.397282E(-14) 3.563816E(-14) 3.674838E(-14) 3.730349E(-14) 3.741452E(-14) Error in [16] 8.326679(-17) 2.775557(-16) 5.551115(-16) 9.436896(-16) 2.109424(-15) 3.219647(-15) 4.440892(-15) 5.995204(-15) 7.771561(-15) 1.065814(-14)

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2. SJ kayoed, O Adeyeye. A 3-step hybrid method for direct solution of second order initial value problems. Aust J Basic Appl Sci. 2011;5(12):2121-2126. 3. A James, A Adesanya, S Joshua. Continuous block method for the solution of second order initial value problems of ordinary differential equation. Int J Pure Appl Math. 2013;83(3):405-416. 4. RA Bun, TD Vasil’s. A numerical method for solving Differential equations of any Order Comp. Math Phys. 1992;32(3):317-330.

5. DO Awoyemi, AO Adesanya, SN Ogunyebi. Construction of Self Starting Numeral Method for the Solution of initial value Problem of General Second Ordinary Differential Equation. Journ Num math. 2008;4(2):267-278. 6. SO Fatunla. Block methods for second order. Int J Comput Maths. 2007;41(1-2):55-63.

7. BT Olabode. An accurate scheme by block method for the third order ordinary differential equation. Pacific journal of science and technology. 2009;10(1).

Figure 1: Absolute Stability Region

Using Mat Lab software, the absolute stability region of the new method is plotted and shown in Figure 1.

Numerical Example

Problem I: We consider a highly stiff problem

y ''+ 1001 y '+ 1000 y , y ( 0 ) = 1, y ' ( 0 ) = −1

Exact Solution:

y ( x ) = exp ( − x ) h =

1 10

Exact Solution:

Conclusion

)

9. AO Adeniran, BS Ogundare. An efficient hybrid numerical scheme for solving general second order Initial Value Problems (IVPs). International Journal of Applied Mathematical Research. 2015;4(2):411-419.

10. R Abdelrahim, O Zurni. Direct Solution of Second-Order Ordinary Differential Equation Using a Single-Step Hybrid Block Method of Order Five. Mathematical and Computational Applications. 2016;21(2):12.

12. SN Jator. A six order linear multi step method for the direct solution of y ' ' = f x, y, y 1 . Intern J Pure and Applied of Maths. 2007;40(4):457-472.

(

1 y (= 0 ) 1, y ' ( = 0) , 0 ≤ x ≤ 1. 2

)

13. Jator SN, Li J. A self-starting linear multistep method for direct solution general second order initial value problems. Intern J Comput Math. 2009;86(5):827-836.

1 y ( x) = 1 − e x with h = 100

It is evident from the above tables that our proposed methods are indeed accurate, and can handle stiff equations. Also in terms of stability analysis, the method is A-stable. Comparing the new method with the existing method, the result presented in the Tables 1 and 2 shows that the new method performs better than the existing method, and even the order of new method is higher than the order of the existing method [11,18]. In this article, a one-step block method with three off-step points is derived via the interpolation and collocation approach. The developed method is consistent, A-stable, convergent, with a region of absolute stability and order Ten.

References

1. Z Omar, M Suleiman. Parallel r-point implicit block method for solving higher order ordinary differential equations directly. J ICT , 2004;3(1):53-66.

Adv Comput Sci

(

11. A Mohammad, O Zurni. Implicit One-Step Block Hybrid ThirdDerivative Method for the Direct Solution of Initial Value Problems of Second-Order Ordinary Differential Equations. Hindawi Journal of Applied Mathematics. 2017.

Problem II:

f ( x , y , y= ') y ' ,

8. J Vigo-Aguilar, H Ramos. Variable stepsize implementation of multistep methods for y ' ' = f x, y, y 1 . J Comput Appl Math. 2006;192(1):114–131.

14. ZA Majid, MB Sulaiman, Z Omar. 3 points implicit block method for solving ordinary differential equations, Bull. Malaysia Mathematical Science Soc. 2006;29(1) 23-31. 15. DO Awoyemi, EA Adebile, AO Adesanya, TA Anake. Modified block for the direct solution for the second order ordinary differential equations. Intern J Applied Math and Comp. 2011;3(3):181-188.

16. Anake TA, Awoyeni DO, Adesanya AO. A one step method for the solution of general second order ordinary differential equations. Intern J Sci and Engin. 2012;2(4):159-163.

17. Adesanya AO, Odekule MR, Alkali MA. Three steps block predictor-block Corrector method for solution of general second order of ordinary differential equations. Intern J Engin Research and Application. 2012;2(4):2297-2301.

18. M Alkasassbeh, Z Omar. Generalized two-hybrid One-Step Implicit Third derivatives block method for the Direct Solution of Second Order Ordinary Differential Equations. International Journal of Pure and Applied Mathematics. 2017;112(3):497-517.

Volume: 1.2

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