The Numerical Computation of Three Step Hybrid Block Method for

Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 1 (2017), pp. 89–103 © Research India Publications http://www.ripublication.com/gjpam.htm

The Numerical Computation of Three Step Hybrid Block Method for Directly Solving Third Order Ordinary Differential Equations M. Hijazi Department of Mathematics, College of Art and Sciences, Al Jouf University. R. Abdelrahim Department of Mathematics, College of Art and Sciences, Al Jouf University.

Abstract This paper consider a three step block method with two hybrid points chosen within grid interval [xn , xn+1 ]. The new hybrid block method is adopted for directly solving third order ordinary differential equations (ODEs). The numerical properties of the method were investigated with the method exhibiting convergence following from satisfying the properties of consistency and zero-stability. Tow certain numerical examples are tested to demonstrate the accuracy of the method. The new results show improved accuracy in terms of error comparison when compared with existing method in this field. AMS subject classification: 65L05, 65L06, 65L20. Keywords: Block method, Hybrid method.

1.

Introduction

In this work we study the solution to general third order initial value problem of the form

y = f (x, y, y , y ), y(a) = δ 0 , y (a) = δ 1 , y (a) = δ 2 . x ∈ [a, b].

(1)

90

M. Hijazi and R. Abdelrahim

These equations have been found to adequately model real life application ranging from, fluid flows, mechanics,electric circuits model, vibrations. In literature, scholars proposed several numerical methods for solving equation (1). Famous authors as [1], [2], [11] have discussed the direct solution of general third ordinary differential equation to eliminate the burden inherent in the method of reduction. sequentially, researchers have developed direct block methods generally to numerically approximate these equations include [3], [10], [8], [9] amongst others. This research developed a hybrid block method for the numerical solution of (1) above with the aim of outperforming existing methods in terms of error.

2.

Methodology

In this part, derivation of a three-step hybrid block method with two off-step points; xn+ 1 3 and xn+ 3 for solving (1) is shown. 5 Let the power series polynomial of the form: y(x) =

q+d−1

ai

i=0

x − xn h

i (2)

.

as an approximate solution of (1), where i x ∈ [xn , xn+1 ] for n = 0, 1, 2, . . . , N − 1, ii q is the number of interpolation points which is same as the order of the differential equation; which is 3 in this case, iii d is the number of collocation points, iv h = xn − xn−1 is the step size of partitioning the interval [a, b] given by a = x0 < x1 < · · · < xN−1 < xN = b. The next step in the derivation of the new hybrid block method includes differentiating (1) thrice to give

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

q+d−1 i=2

i(i − 1)(i − 2) ai h3

x − xn h

i−3 .

(3)

The approximate solution (2) is interpolated at xn+k , k = 0, 1, 2 while differential system (3) is collocated at all points in the selected interval to produce nine equations which can

The Numerical Computation of Three Step Hybrid Block Method be written as a system in matrix form AX = B, where ⎛ 1 0 0 0 0 0 0 0 ⎜ 1 1 1 1 1 1 1 1 ⎜ ⎜ 1 2 4 8 16 32 64 128 ⎜ ⎜ ⎜ 0 0 0 6 0 0 0 0 ⎜ h3 ⎜ 70 40 8 20 6 ⎜ ⎜ 0 0 0 3 3 3 3 ⎜ h h (3h ) (9h ) (27h3 ) ⎜ 6 24 60 120 210 A=⎜ ⎜ 0 0 0 ⎜ h3 h3 h3 h3 h3 ⎜ ⎜ 108 648 3402 72 ⎜ 0 0 0 6 ⎜ 3 3 3 3 h (5h ) (5h ) (25h ) (125h3 ) ⎜ ⎜ 960 3360 48 240 ⎜ 0 0 0 6 ⎜ h3 h3 h3 h3 h3 ⎜ ⎝ 72 540 3240 17010 6 0 0 0 3 3 3 3 h h h h h3 ⎛ ⎞ ⎞ ⎛ yn a0 ⎜ yn+1 ⎟ ⎜ a1 ⎟ ⎜ ⎟ ⎟ ⎜ ⎜ yn+2 ⎟ ⎜ a2 ⎟ ⎜ ⎟ ⎟ ⎜ ⎜ fn ⎟ ⎜ a3 ⎟ ⎜ ⎟ ⎟ ⎜ ⎜ f 1 ⎟ ⎟ a and B = X=⎜ ⎜ n+ 3 ⎟ ⎜ 4 ⎟ ⎜ ⎟ ⎜ a5 ⎟ ⎜ fn+ 3 ⎟ ⎟ ⎜ 5 ⎟ ⎜ ⎜ a6 ⎟ ⎜ fn+1 ⎟ ⎟ ⎜ ⎜ ⎟ ⎝ a7 ⎠ ⎝ fn+2 ⎠ a8 fn+3

91

⎞

0 1 256

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ 0 ⎟ ⎟ 112 ⎟ ⎟ 3 (81h ) ⎟ ⎟ ⎟, 336 ⎟ ⎟ h3 ⎟ 81648 ⎟ ⎟ (3125h3 ) ⎟ ⎟ 10752 ⎟ ⎟ ⎟ h3 ⎟ 81648 ⎠ h3

(4)

The unknown values of ai s, i = 0(1)8 in (4) can be obtained by matrix inverse approach, where X = A−1 B. The values obtained are then substituted back into equation (2) to give a continuous implicit scheme of the form y(x) =

2

α i (x)yn+i +

i=0

3

β i (x)fn+i + β 1 (x)fn+ 1 + β 3 (x)fn+ 3 3

3

5

5

(5)

i=0

It is necessary to also obtain the corresponding first and second derivative schemes of equation (5), and these are obtained as given below

y (x) =

2

α i (x)yn+i +

i=0

y (x) =

2 i=0

3

β i (x)fn+i + β s (x)fn+s + β r (x)fn+r

(6)

i=0

α i (x)yn+i +

3 i=0

β i (x)fn+i + β s (x)fn+s + β r (x)fn+r

(7)

92


where (2x − 2xn ) (x − xn )2 − h h2 2 (x − xn ) (x − xn ) − α2 = (2h2 ) (2h) 2 (3x − 3xn ) (x − xn ) α0 = − +1 2 (2h ) (2h (13(x − xn )4 ) (x − xn )3 (131(x − xn )5 ) (7(x − xn )6 ) + + − + β0 = − (48h) 6 (540h2 ) (60h3 ) (26(x − xn )7 ) (5(x − xn )8 ) (451h(x − xn )2 ) (31h2 (x − xn )) − − − (945h4 ) 10080 15120 (2016h5 ) α1 =

(3(x − xn )4 ) (33(x − xn )5 ) (163(x − xn )6 ) (89(x − xn )7 ) − + − (32h) (160h2 ) (960h3 ) (1680h4 ) (5(x − xn )8 ) (2651h(x − xn )2 ) (1259h2 (x − xn )) + + − 13440 6720 (896h5 ) β1 =

(729(x − xn )4 ) (5103(x − xn )5 ) (5913(x − xn )6 ) − + β1 = 3 (1280h) (6400h2 ) (12800h3 ) (2673(x − xn )7 ) (81(x − xn )8 ) (9477h(x − xn )2 ) (12393h2 (x − xn )) − + + − (22400h4 ) 35840 89600) (7168h5 ) (x − xn )4 (37(x − xn )5 ) (5(x − xn )6 ) (59(x − xn )7 ) − + − (2304h (34560h2 ) (4608h3 ) (120960h4 ) (5(x − xn )8 ) (29h(x − xn )2 ) (149h2 (x − xn )) + − + 107520 483840 (64512h5 )

β3 =

(18125(x − xn )5 ) (3125(x − xn )4 ) (8125(x − xn )6 ) − − 5 (24192h2 ) (8064h) (16128h3 ) (11875(x − xn )7 ) (3125(x − xn )8 ) (3125h(x − xn )2 ) (625h2 (x − xn )) + + − + 225792 338688 (84672h4 ) (225792h5 )

β3 =

(3(x − xn )4 ) (9(x − xn )5 ) (13(x − xn )6 ) (37(x − xn )7 ) + − + (560h) (700h2 ) (1050h3 ) (7350h4 ) (x − xn )8 (181h(x − xn )2 ) (481h2 (x − xn )) − + − 23520 58800 (1568h5 Evaluating equation (5) at the non-interpolating points xn+ 1 , xn+ 3 and xn+3 and evalu3 5 ating equation (7) at all points give the discrete schemes and its derivative. The discrete scheme and its derivatives are combined in a matrix form as below β2 = −

AYM = BR1 + h3 [CR2 + DR3 ]

(8)

The Numerical Computation of Three Step Hybrid Block Method

93 ⎤

⎡

1 −5 0 0 0 0 0 0 0 0 0 ⎢ 9 9 ⎢ ⎢ 3 −21 ⎢ 0 1 0 0 0 0 0 0 0 0 ⎢ 25 25 ⎢ ⎢ 0 3 0 −3 1 0 0 0 0 0 0 0 ⎢ ⎢ ⎢ 1 ⎢ 0 −2 0 0 0 0 0 0 0 0 0 ⎢ h (2h) ⎢ ⎢ 1 ⎢ 0 −4 0 0 1 0 0 0 0 0 0 ⎢ (3h) (6h) ⎢ ⎢ −1 ⎢ 0 0 1 0 0 0 0 0 0 0 ⎢ 0 ⎢ (2h) ⎢ −1 −4 ⎢ 0 0 0 0 1 0 0 0 0 ⎢ 0 ⎢ (5h) (10h) ⎢ 2 −3 ⎢ 0 0 0 0 1 0 0 0 0 0 A=⎢ ⎢ (2h) h ⎢ ⎢ 4 −5 ⎢ 0 0 0 0 0 0 0 1 0 0 ⎢ h (2h) ⎢ ⎢ −1 2 ⎢ 0 0 0 0 0 0 0 0 0 0 ⎢ h2 h2 ⎢ ⎢ −1 2 ⎢ 0 0 0 0 0 0 0 1 0 0 ⎢ 2 h h2 ⎢ ⎢ 2 −1 ⎢ 0 0 0 0 0 0 0 0 0 1 ⎢ h2 h2 ⎢ ⎢ 2 −1 ⎢ 0 0 0 0 0 0 0 0 0 0 ⎢ 2 h h2 ⎢ ⎢ −1 2 ⎢ 0 0 0 0 0 0 0 0 0 ⎢ 0 2 ⎢ h h2 ⎣ −1 2 0 0 0 0 0 0 0 0 0 0 h2 h2 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 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

94

M. Hijazi and R. Abdelrahim ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ B=⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

5 9 7 25 1

⎤ 0

0

0

0

0

0

−3 −1 0 (2h) −7 0 0 (6h) −1 0 0 (2h) −9 0 0 (10h) 1 0 0 (2h) 3 0 0 (2h) 1 0 −1 h2 1 0 0 h2 1 0 0 h2 1 0 0 h2 1 0 0 h2 1 0 0 h2

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎡ ⎥ yn+ 1 ⎥ 3 ⎥ ⎢ ⎥ ⎢ yn+ 35 ⎥ ⎢ ⎥ ⎢ yn+1 ⎥ ⎢ ⎥ ⎢ yn+2 ⎥ ⎢ ⎥ ⎢ yn+3 ⎥ ⎢ ⎥ ⎢ y 1 ⎥ ⎢ n+ 3 ⎥ ⎢ ⎥ ⎢ y 3 ⎥ ⎢ n+ 5 ⎥ ⎢ ⎥ , YM = ⎢ yn+1 ⎥ ⎢ ⎥ ⎢ y ⎥ ⎢ n+2 ⎥ ⎢ y ⎥ ⎢ n+3 ⎥ ⎢ ⎥ ⎢ y 1 ⎥ ⎢ n+ ⎥ ⎢ 3 ⎥ ⎢ yn+ 3 ⎥ ⎢ 5 ⎥ ⎢ y ⎥ ⎢ n+1 ⎥ ⎢ ⎥ ⎣ yn+2 ⎥ ⎥ yn+3 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

The Numerical Computation of Three Step Hybrid Block Method ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ ⎤ ⎢ yn ⎢ R1 = ⎣ yn ⎦ , C = ⎢ ⎢ ⎢ yn ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

311) − 157464 1932014216144431) − 844424930131968000 17) − 360 31) − 15120h 22579) − 5511240h 41) 3780h 206756055532931) 141863388262170624h 817) − 15120h 1033) 7560h 451) − 5040h2 87413) 3674160h2 439) 15120h2 30892227186823) 1688849860263936h2 2809) − 15120h2 3581) 5040h2

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ , R2 = fn , ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

95

96

M. Hijazi and R. Abdelrahim ⎡

2183 ⎢ 103680 ⎢ 9003879 ⎢ ⎢ ⎢ 500000000 ⎢ 2187 ⎢ ⎢ ⎢ 6400 ⎢ ⎢ 12393 ⎢ ⎢ 89600h ⎢ ⎢ 9059 ⎢ ⎢ 1209600h ⎢ ⎢ −81 ⎢ ⎢ 1120h ⎢ ⎢ −17158473 ⎢ ⎢ 700000000h ⎢ ⎢ 29079 ⎢ D=⎢ ⎢ 89600h ⎢ −729 ⎢ ⎢ ⎢ 1280h ⎢ ⎢ −9477 ⎢ 2 ⎢ 17920h ⎢ ⎢ −32906574933547761 ⎢ ⎢ 157625986957967360h2 ⎢ ⎢ −14499 ⎢ ⎢ 89600h2 ⎢ ⎢ −3039249564193869 ⎢ ⎢ ⎢ 39406496739491840h2 ⎢ 97767 ⎢ ⎢ ⎢ 89600h2 ⎣ −322947 89600h2

2873 69984 6198968448095049 140737488355328000 127 160 1259 6720h 59813834750055 985162418487296h −41 240h −38252993519933 985162418487296h 3221 6720h 1513 3360h −2651 6720h2 −1450607258448441 3940649673949184h2 −1433 6720h2 −186579933893557 492581209243648h2 1889 1344h2 −12857 6720h2

⎡ ⎢ ⎢ R3 = ⎢ ⎢ ⎣

−30406247484175 171597954441921588 −5731 1008000 −625 1152 35184372088832 19066439382435729h −156148243330236416 13899434309795646441h 3125 42336h −147557439391042 6355479794145243h −9042383626829824 19066439382435729h 115625 169344h 3125 112896h2 −451521047015981056 4633144769931882147h2 6931321301499904h2 19066439382435729 211915473091035136 3972174871340776875h2 −30997431810260992 19066439382435729h2 554375 112896h2

fn+s fn+1 fn+r fn+2 fn+3

1129 612360 983095266258333 492581209243648000 89 200 481 58800h 384830477418243 137922738588221440h −5 588h −897478534702293 551690954352885760h 3503 58800h 6289 5880h −181 11760h2 −330515220668921 19703248369745920h2 −1073 58800h2 −1127787290386461 68961369294110720h2 18919 58800h2 94351 58800h2

⎤ −355 ⎥ 5038848 ⎥ −4128960186368303 ⎥ ⎥ 54043195528445952000 ⎥ ⎥ 107 ⎥ ⎥ ⎥ 11520 ⎥ ⎥ −149 ⎥ ⎥ 483840h ⎥ ⎥ −3793 ⎥ ⎥ 35271936h ⎥ ⎥ 1 ⎥ ⎥ 3024h ⎥ 1112477978993159 ⎥ ⎥ ⎥ 18158513697557839872h ⎥ ⎥ −25 ⎥ ⎥ ⎥ 13824h ⎥ 15383 ⎥ ⎥ ⎥ 241920h ⎥ ⎥ 29 ⎥ 2 ⎥ 53760h ⎥ ⎥ 76261 ⎥ ⎥ 2 117573120h ⎥ ⎥ 359 ⎥ ⎥ 2 483840h ⎥ 1402029110795591 ⎥ ⎥ ⎥ 2269814212194729984h2 ⎥ ⎥ −3323 ⎥ ⎥ 2 ⎥ 483840h ⎦ 3139 10752h2

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

Multiplying Equation (8) by the inverse of A gives

¯ 2 + DR ¯ 1 + h2 CR ¯ 3 I Ym = BR

(9)


97

where I is 8 × 8 identity matrix and ⎡

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ B¯ = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

1

h 3

1

h

3h) 5 1 2h 1

1

3h

0

1

0

1

0

1

0 0 0 0 0 0 0

1 1 0 0 0 0 0

h2 18 h2 2 9h2 50 2h2 9h2 ) 2 h 3 h 3h 5 2h 3h 1 1 1 1 1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ¯ C=⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

81119 22044960 283 6048 53373748954636193 3546584706554265600 173 945 81 224 15311 551124h 43 420h 608574480094322017 10639754119662796800h 8 63h 57 140h 3716 32805h2 16 135h2 19111835666679647 177329235327713280h2 13h − 135h2 4 5h2

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

98


⎡

62863 14515200 ⎢ ⎢ ⎢ ⎢ 22599 ⎢ ⎢ 179200 ⎢ ⎢ ⎢ 422967987 ⎢ ⎢ ⎢ 14000000000 ⎢ ⎢ ⎢ 2187 ⎢ ⎢ 2800 ⎢ ⎢ ⎢ ⎢ 59049 ⎢ ⎢ 25600 ⎢ ⎢ ⎢ 27493 ⎢ ⎢ 604800h ⎢ ⎢ ⎢ ⎢ 891 ⎢ ⎢ 2800h ⎢ ⎢ ⎢ 54069201 ⎢ D¯ = ⎢ ⎢ 350000000h ⎢ ⎢ ⎢ 3483 ⎢ ⎢ 2800h ⎢ ⎢ ⎢ ⎢ 19683 ⎢ ⎢ 22400h ⎢ ⎢ ⎢ 1441542818226213 ⎢ ⎢ ⎢ 4503599627370496h2 ⎢ ⎢ ⎢ 2349 ⎢ ⎢ 6400h2 ⎢ ⎢ ⎢ 3560178765734487 ⎢ ⎢ ⎢ 7881299347898368h2 ⎢ ⎢ ⎢ 81 ⎢ ⎢ 50h2 ⎢ ⎢ ⎣ −19683 6400h2

5077 9797760

−2276905104986785373009615724674029 977166249626370586284007196728472064

−49 1749600

19 1920

−7502996979199245341 478338831226547569152

−19 39200

2605295792686847 985162418487296000

−704066407075204174733 59792353903318446144000

−489459088801973 3448068464705536000

29 70

−784627135079659637 13287189756292988032

53 3675

8991 4480

−321746164578602927273 478338831226547569152

19197 39200

30788911142989 6333186975989760h

−3889200551806103211911 174354503982076588955904h

−1611820699970681 6206523236469964800h

61 1680h

31777398970726214801 717508246839821353728h

−19 14700h

777006043710241 73887181386547200h

−9966409193986778027 239169415613273784576h

−4736453742504059 8275364315293286400h

227 210h

−63554797941452438557 119584707806636892288h

302 3675h

81 56h

428994886104803902301 717508246839821353728h

5427 4900h

1559333725146793 59109745109237760h2

−7272610836661481000839 58118167994025529651968h2

−572448285968749 413768215764664320h2

29 160h2

8925133661531129739 26574379512585976064h2

−1 350h2

116106316830821 7388718138654720h2

3837120925715191659827 149480884758296115360000h2

−199176197120087 206884107882332160h2

9 5h2

−395452076080148511773 239169415613273784576h2

59 175h2

−243 160h2

625h 128h2

81 50h2

1573 705438720

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 1827605764783219 ⎥ ⎥ 162129586585337856000 ⎥ ⎥ ⎥ ⎥ −1 ⎥ ⎥ 2160 ⎥ ⎥ ⎥ ⎥ 279 ⎥ ⎥ 35840 ⎥ ⎥ ⎥ 1817 ⎥ ⎥ 88179840h ⎥ ⎥ ⎥ ⎥ 1 ⎥ ⎥ 10080h ⎥ ⎥ ⎥ 62043750194477069 ⎥ ⎥ 1361888527316837990400h ⎥ ⎥ ⎥ ⎥ −13 ⎥ ⎥ 5040 ⎥ ⎥ ⎥ ⎥ 279 ⎥ ⎥ 4480h ⎥ ⎥ ⎥ 917 ⎥ ⎥ 2 ⎥ 8398080h ⎥ ⎥ ⎥ 7 ⎥ ⎥ 34560h2 ⎥ ⎥ ⎥ 888064810521187 ⎥ ⎥ 11349071060973649920h2 ⎥ ⎥ ⎥ ⎥ −1 ⎥ ⎥ 2 135h ⎥ ⎥ ⎦ 373 2 1280h 37 967680

3. Analysis of the Method 3.1.

Order of the Methhod

The linear difference operator L associated with (9) is defined as ¯ 1 − h3 CR ¯ 3 ¯ 2 + DR L[y(x); h] = I YM − BR

(10)

where y(x) is an arbitrary test function continuously differentiable on [a,b]. YM and R3 component’s are expanded in Taylor’s series respectively and its terms are collected in powers of h to give

L[y(x), h] = C¯0 y(x) + C¯1 hy (x) + C¯ 2 hy (x) + · · ·

⎤

(11)

Definition 3.1. Hybrid block method (9) and associated linear operator (10) are said to be of order p, if C¯0 = C¯1 = 2¯ = · · · = C¯ p+2 = 0 and C¯ p+3 = 0 with error vector constants C¯ p+2 .


99

Expanding (9) in Taylor series about xn gives ⎡

∞ 1 j j (3) h h h2 81119h3 j − yn − yn − yn − y − y ⎢ ⎢ j! 3 18 n 22044960 n ⎢ j =0 ⎢ ∞ ⎢ 62863 (s)j hj +3 j +3 ⎢ yn − ⎢ 14515200 j! ⎢ j =0 ⎢ ⎢ ∞ 5077 (1)j hj +3 j +3 ⎢ ⎢ yn − ⎢ 9797760 j! ⎢ j =0 ⎢ ∞ 3 ⎢ 2276905104986785373009615724674029 ( 5 )j hj +3 j +2 ⎢ yn + ⎢ ⎢ 977166249626370586284007196728472064 j! ⎢ j =0 ⎢ ∞ (2)j hj +3 j +3 ⎢ 49 ⎢ yn + ⎢ 1749600 j! ⎢ j =0 ⎢ ⎢ ∞ (3)j hj +3 j +3 ⎢ 1573 ⎢ yn − ⎢ 705438720 j! ⎢ j =0 ⎢ ⎢ ⎢ ⎢ ∞ ⎢ (1)j hj h2 283h3 j ⎢ − yn − yn − (h)yn − y − y ⎢ j! 2 n 6048 n ⎢ j =0 ⎢ ⎢ ∞ ∞ 1 ⎢ 19 (1)j hj +3 j +3 22599 ( 3 )j hj +3 j +3 ⎢ yn yn − − ⎢ j! 1920 179200 j! ⎢ j =0 j =0 ⎢ ⎢ ∞ 3 ∞ 7502996979199245341 ( 5 )j hj +3 j +2 19 (2)j hj +3 j +3 ⎢ ⎢ + yn + yn ⎢ 478338831226547569152 j ! 39200 j! ⎢ j =0 j =0 ⎢ ∞ ⎢ 37 (3)j hj +3 j +3 ⎢ yn − ⎢ ⎢ 967680 j! ⎢ j =0 ⎢ ⎢ ⎢ ⎢ ∞ ( 3 )j hj ⎢ 3h h2 53373748954636193h3 j 5 ⎢ yn − y − yn − yn − ( )yn − ⎢ 5 2 3546584706554265600 n j ! ⎢ j =0 ⎢ ⎢ ∞ 1 ∞ 422967987 ( 3 )j hj +3 j +3 2605295792686847 (1)j hj +3 j +3 ⎢ ⎢ − yn − yn ⎢ 14000000000 j ! 985162418487296000 j! ⎢ j =0 j =0 ⎢ ∞ 3 ⎢ 704066407075204174733 ( 5 )j hj +3 j +2 ⎢ yn + ⎢ ⎢ 59792353903318446144000 j! ⎢ j =0 ⎢ ∞ ⎢ 489459088801973 (2)j hj +3 j +3 ⎢ yn + ⎢ 3448068464705536000 j! ⎢ j =0 ⎢ ⎢ ∞ j j +3 (3) h ⎢ 1827605764783219 j +3 ⎢ yn − ⎢ 162129586585337856000 j! ⎢ j =0 ⎢ ⎢ ⎢ ⎢ ∞ ⎢ (2)j hj 173h3 j ⎢ − yn − yn − (2h)yn − 2h2 yn − y ⎢ j! 945 n ⎢ j =0 ⎢ ⎢ ∞ 1 ∞ ⎢ 2187 ( 3 )j hj +3 j +3 29 (1)j hj +3 j +3 ⎢ − yn − yn ⎢ 2800 j! 70 j! ⎢ j =0 j =0 ⎢ ⎢ ∞ ∞ 3 ⎢ 784627135079659637 ( 5 )j hj +3 j +2 53 (2)j hj +3 j +3 ⎢ yn − + yn ⎢ 13287189756292988032 j! 3675 j! ⎢ j =0 j =0 ⎢ ∞ ⎢ 1 (3)j hj +3 j +3 ⎢ yn + ⎢ ⎢ 2160 j! ⎢ j =0 ⎢ ⎢ ⎢ ⎢ ∞ ⎢ 9h2 81h3 (3)j hj j ⎢ yn − y − yn − yn − (3h)yn − ⎢ 2 224 n j ! ⎢ j =0 ⎢ ⎢ ∞ 1 ∞ 59049 ( 3 )j hj +3 j +3 8991 (1)j hj +3 j +3 ⎢ ⎢ yn − yn − ⎢ 4480 j! 25600 j ! ⎢ j =0 j =0 ⎢ ∞ 3 j j +3 ∞ ⎢ 19197 (2)j hj +3 j +3 j +2 ⎢ + 321746164578602927273 ( 5 ) h yn yn − ⎢ ⎢ 478338831226547569152 39200 j! j! ⎢ j =0 j =0 ⎢ ∞ ⎢ 279 (3)j hj +3 j +3 ⎣ yn − 35840 j! j =0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

100


Comparing the coefficients to power of h yields, C¯1 = C¯2 = · · · = C¯10 = 0, and C¯11 = 0. This yields, the main method Bˆ = [yn+ 1 , yn+ 3 , yn+1 , yn+2 , yn+3 ]T 5

3

having order [8, 8, 8, 8, 8]T 3.2.

Zero Stability

The hybrid block method Bˆ is said to be zero stable if the first characteristic polynomial π (x) having roots such that |xz | = 1, and if |xz | = 1, then, the multiplicity of xz must not exceed three. This is clear below, ˆ (x) = |x I − B| ⎛ 1 0 ⎜ 0 1 ⎜ = x ⎜ ⎜ 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

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

1 1 1 1 1

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

= x 4 (x − 1) which implies x = 0, 0, 0, 0, 1. Hence, our method is zero stable for all s, r ∈ (0, 3) 3.3.

Consistency

The hybrid block method Bˆ is said to be consistent if its order greater than or equal one i.e. P ≥ 1. This proves that our method is consistent for all s, r ∈ (0, 3). 3.4.

Convergence

Theorem 3.2. thm1 Consistency and zero stability are sufficient conditions for a linear multistep method to be convergent. Since the method is consistent and zero stable, it implies the method is convergent for all s, r ∈ (0, 3). 3.5.

Numerical Results

In finding the accuracy of our methods, the following third order ODEs are examined. The new block methods solved the same problems the existing methods solved in order to compare results in terms of error.

The Numerical Computation of Three Step Hybrid Block Method Problem 1: 3,

y = 2y +3y −10y+34xe−2x −16e−2x −10x 2 +6x+34, y(0) =

Exact solution:

101

y (0) = 0, y (0) = 0 1 2+x y(x) = 1 + ln 2 2−x

Problem 2: Exact solution:

y + y = 0, y(0) = 0, x y(x) = 1 − e

Problem 3: Exact solution:

y − 3 sin x = 0, y(x) = 1 − ex

y (0) = 1, y (0) = 2.

y(0) = 1,

y (0) = 0, y (0) = −2.

Table 1: Comparison of the new method with some existing methods for solving problem 1 h Method Error at x=1. New method Adams(2014) Awoyemi(2005)

7.56e−12 3.56e−8 7.60e−6

New method 0.00625 Adams(2014) Awoyemi(2005)

1.19e−13 3.23e−9 9.54e−7

0.0125

h 0.01

New method Adams(2014) Awoyemi(2003)

Error at x=4. 5.80e−10 2.91e−6 1.16e−3

102


Table 2: Comparison of the new method with some existing methods for solving problem 2 h Method Error at x=1.

0.01

New method Awoyemi(2006) Adesanya(2012)

h New method 0.025 Awoyemi(2003) Adams(2014)

1.62e−10 1.07e−6 5.14e−5 Error at x=5. 1.41e−12 3.53e−6 9.72e−8

Table 3: Comparison of the new method with some existing methods for solving problem 2 h Method Error at x = 1. New method 0.01 Adams(2014) Adesanya(2012)

4.

1.11e−15 6.40e−10 1.75e−14

Conclusion

We have developed a new hybrid block method for numerically approximating the solution of third order initial value problems. This method has satisfied possessing properties that will prove its convergence when employed to solve third order ODEs as seen in the numerical results above. It is clear to see that, the new hybrid block method outperformed the existing methods in terms of error.

References [1] Abdelrahim, R., & Omar, Z. (2016). One step Block Method For The Direct Solution Of Third Order Initial value Problems Of Ordinary Differential Equation. Far East journal of Mathematicsl Sicnces, 99(6), 945–985. [2] Abdelrahim, R., & Omar, Z. (2015). Uniform Order One Step Hybrid Block Method with two Generalized Off step Points for Solving Third Order Ordinary Differential Equations Directly, Global journal of Pure and Applied Mathematic, 11(6), 4809– 4823.


103

[3] Adesanya, A.O., Udo, M.O. and Alkali, A.M. 2012, A new block-predictor cor rector algorithm for the solution of y = f (x, y, y ), The American Journal of Computational Mathematics, vol. 2, no. 4, pp. 341–344. [4] Awoyemi, D. O. and Idowu, O. M. 2005, A class of hybrid collocation methods for third-order ordinary differential equations, International Journal of Computer Mathematics, vol. 82, no. 10, pp. 1287–1293. [5] Awoyemi, D. O. 2003, A P-stable linear multistep method for solving general third order ordinary differential equations, International Journal of Computer Mathematics, vol. 80, no. 8, pp. 985–991. [6] Awoyemi, D. O., Udoh, M. O and Adesanya, A. O. 2006. Non-symmetric collocation method for direct solution of general third order initial value problems of ordinary differential equations, Journal of Natural and Applied Mathematics, vol. 7, pp. 31–37. [7] Henrici, P. (1962). Discrete variable methods in ordinary differential equations. [8] Langkah, K. B. D. T. E., Majid, Z. A., Azmi, N. A., Suleiman, M., and Ibrahaim, Z. B. (2012). Solving directly general third order ordinary differential equations using two-point four step block method. Sains Malaysiana, 41(5), 623–632. [9] Mechee, M., Senu, N., Ismail, F., Nikouravan, B., and Siri, Z. (2013). A three-stage fifth-order Runge-Kutta method for directly solving special third-order differential equation with application to thin film flow problem. Mathematical Problems in Engineering, 2013. [10] Omar, Z., & Abdelrahim, R. (2016). Application of single step with three generalized hybrid points block method for solving third order ordinary differential equations, J. Nonlinear Sci. Appl. 9, 2705–2717. [11] Yap, L. K., Ismail, F., and Senu, N. (2014). An accurate block hybrid collocation method for third order ordinary differential equations. Journal of Applied Mathematics, 2014.