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*

1Department of Mathematical Science, Adamawa State University, Mubi-Nigeria

2Department of Mathematics and Statistics, Federal University, Wukari-Nigeria

Corresponding author

Raymond D
Department of Mathematics and Statistics
Federal University, Wukari-Nigeria
Tel. No: 0233207482596
E-mail: raymond@fuwukari.edu.ng

  • Received Date:13 February 2018
  • Accepted Date:16 April 2018
  • Published Date:08 May 2018

DOI:   10.31021/acs.20181109

Article Type:   Research Article

Manuscript ID:   ACS-1-109

Publisher:   Boffin Access Limited.

Volume:   1.2

Journal Type:   Open Access

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.


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

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.

Introduction

This paper solves second order initial value problems in the form

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

Derivation of the Method

We consider a power series approximate solution of the form
y( x )= j=0 2r+s1 a i ( x x n 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
y''( x )= j=2 2r+s1 a j j! h 2 ( j2 ) ( x x n h ) j2 =f( x,y,y' ),
y'''( x )= j=3 2r+s1 a j j! h 3 ( j3 ) ( x x n h ) j3 =g( x,y,y' ),                             (3)

Substituting (3) into (1) gives

f( x,y,y'' )= j=2 2r+s1 a j j! h 2 ( j2 ) ( x x n h ) j2 + j=3 2r+s1 a j j! h 3 ( j3 ) ( x x n h ) j3             (4)

Collocating (4) at all points x n+r ,r=0( 1 4 )1 and Interpolating Equation (2) at x n+s ,s=0, 1 4 gives a system of non linear equation of the form

AX=U                                           (5)
Where

A=[ a 0 , a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , a 8 , a 9 , a 10 , a 11 ] T ,

U=[ y n , y n+ 1 4 , f n , f n+ 1 4 , f n+ 1 2 , f n+ 3 4 , f n+1 , g n , g n+ 1 4 , g n+ 1 2 , g n+ 3 4 , g n+1 ] T , >
and

Figure 2

Solving (5) for a i 's using Gaussian elimination method, gives a continuous hybrid linear multistep method of the form
y( x )= j=0, 1 4 α j y n+j + h 2 [ j=0 1 β j f n+j + β k f n+k ]+ h 3 [ j=0 1 γ j g n+j + γ k g n+k ],k= 1 4 , 1 2 , 3 4      (6)
Differentiating (6) once yields
p'( x )= 1 h j=0, 1 4 α j y n+j +h[ j= 1 4 , 1 2 , 3 4 β j f n+j + j=0 1 β j f n+j ]+ h 2 [ j= 1 4 , 1 2 , 3 4 γ j g n+j + j=0 1 γ j g n+j ]    (7)
Where
α 0 =1 4(x x n ) h
α 1 4 =1 4(x x n ) h

Equation (6) is evaluated at the non-interpolating points and (7) at all points, produces the following general equations in block form
β 0 = 2602339 38320128 (x x n )h+ 1 2 (x x n ) 2 485 36 (x x n ) 4 h 2 + 4031 54 (x x n ) 5 h 3 85862 405 (x x n ) 6 h 4 + 208100 567 (x x n ) 7 h 5 75920 189 (x x n ) 8 h 6 + 66080 243 (x x n ) 9 h 7 - 126464 1215 (x x n ) 10 h 8 + 5120 297 (x x n ) 11 h 9
β 1 4 = 148231 11975040 (x x n )h 128 9 (x x n ) 4 h 2 + 7168 45 (x x n ) 5 h 3 270592 405 (x x n ) 6 h 4 + 841216 567 (x x n ) 7 h 5 - 364288 189 (x x n ) 8 h 6 + 358912 243 (x x n ) 9 h 7  - 753664 1215 (x x n ) 10 h 8 + 32768 297 (x x n ) 11 h 9
β 1 2 = 1807 80640 (x x n )h 12 (x x n ) 4 h 2 - 456 5 (x x n ) 5 h 3 4424 15 (x x n ) 6 h 4 10496 21 (x x n ) 7 h 5 + 3264 7 (x x n ) 8 h 6 2048 9 (x x n ) 9 h 7 + 2048 45 (x x n ) 10 h 8
β 3 4 = 243193 11975040 (x x n )h+ 128 9 (x x n ) 4 h 2 - 17408 135 (x x n ) 5 h 3 + 213248 405 (x x n ) 6 h 4 685568 567 (x x n ) 7 h 5 + 313088 189 (x x n ) 8 h 6 326144 243 (x x n ) 9 h 7 + 720896 1215 (x x n ) 10 h 8 32768 297 (x x n ) 11 h 9
β 1 = 382169 191600640 (x x n )h+ 53 36 (x x n ) 4 h 2 - 1241 90 (x x n ) 5 h 3 + 23758 405 (x x n ) 6 h 4 80356 567 (x x n ) 7 h 5 + 38992 189 (x x n ) 8 h 6 43552 243 (x x n ) 9 h 7 + 103936 1215 (x x n ) 10 h 8 5120 297 (x x n ) 11 h 9

γ 0 = 28343 18247680 (x x n ) h 2 + 1 6 (x x n ) 3 - 25 18 (x x n ) 4 h + 209 36 (x x n ) 5 h 2 398 27 (x x n ) 6 h 3 + 4546 189 (x x n ) 7 h 4 1600 63 (x x n ) 8 h 5 + 1360 81 (x x n ) 9 h 6 - 512 81 (x x n ) 10 h 7 + 512 495 (x x n ) 11 h 8 γ 1 4 = 551 49896 (x x n ) h 2 16 3 (x x n ) 4 h + 608 15 (x x n ) 5 h 2 18848 135 (x x n ) 6 h 3 + 7424 27 (x x n ) 7 h 4 20768 63 (x x n ) 8 h 5 + 19328 81 (x x n ) 9 h 6 38912 405 (x x n ) 10 h 7 + 8192 495 (x x n ) 11 h 8 γ 1 2 = 32027 3548160 (x x n ) h 2 6 (x x n ) 4 h + 264 5 (x x n ) 5 h 2 3124 15 (x x n ) 6 h 3 + 3224 7 (x x n ) 7 h 4 + 264 5 (x x n ) 5 h 2 3124 15 (x x n ) 6 h 3 + 3224 7 (x x n ) 7 h 4 608 (x x n ) 8 h 5 + 4288 9 (x x n ) 9 h 6 1024 5 (x x n ) 10 h 7 + 2048 55 (x x n ) 11 h 8
γ 3 4 = 3959 1596672 (x x n ) h 2 16 9 (x x n ) 4 h + 736 45 (x x n ) 5 h 2 9184 135 (x x n ) 6 h 3 + 30208 189 (x x n ) 7 h 4 14176 63 (x x n ) 8 h 5 + 15232 81 (x x n ) 9 h 6 34816 405 (x x n ) 10 h 7 + 8192 495 (x x n ) 11 h 8 γ 1 = 14339 127733760 (x x n ) h 2 1 12 (x x n ) 4 h + 47 60 (x x n ) 5 h 2 452 135 (x x n ) 6 h 3 + 1538 189 (x x n ) 7 h 4 752 63 (x x n ) 8 h 5 + 848 81 (x x n ) 9 h 6 2048 405 (x x n ) 10 h 7 + 512 495 (x x n ) 11 h 8 Equation (6) is evaluated at the non-interpolating points { x n+ 1 2 , x n+ 3 4 , x n+1 } and (7) at all points { x n , x n+ 1 4 , x n+ 1 2 , x n+ 3 4 , x n+1 } produces the following general equations in block form
A Y L =B R 1 +C R 2 +D R 3 +E R 4 +G R 5
Where,
       A=[ 2 1 0 0 0 0 0 0 3 0 1 0 0 0 0 0 4 0 0 1 0 0 0 0 4 h 0 0 0 0 0 0 0 4 h 0 0 0 1 0 0 0 4 h 0 0 0 0 1 0 0 4 h 0 0 0 0 0 1 0 4 h 0 0 0 0 0 0 1 ]Y=[ y n+ 1 4 y n+ 1 2 y n+ 3 4 y n+1 y ' n+ 1 4 y ' n+ 1 2 y ' n+ 3 4 y ' n+1 ]

       B=[ 1 0 2 0 3 0 4 h 1 4 h 0 4 h 0 4 h 0 4 h 0 ], R 1 =[ y n y ' n ]C=[ 37111 h 2 6967296 12679 h 2 1161216 19709 h 2 1161216 260233h 38320128 1961683h 95800320 1403419h 63866880 2182739h 95800320 5021969h 191600640 ], R 2 =[ f n ]

       D=[ 16463 h 2 435456 445 h 2 32256 2225 h 2 435456 3217 h 2 6967296 6149 h 2 72576 767 h 2 10752 1403 h 2 72576 1381 h 2 1161216 9925 h 2 72576 767 h 2 5376 5179 h 2 72576 8411 h 2 1161216 148231h 11975040 1807h 80640 243193h 11975040 382169h 191600640 83485h 1197504 1489h 80640 89279h 5987520 137161h 95800320 26921h 147840 767h 5376 103853h 3991680 5303h 2365440 1156801h 5987520 21521h 80640 165731h 1197504 358217h 95800320 2735353h 11975040 24817h 80640 2640391h 11975040 3530299h 38320128 ], R 3 =[ f n+ 1 4 f n+ 1 2 f n+ 3 4 f n+1 ]

       G=[ 17 h 3 9072 83 h 3 30720 871 h 3 1451520 599 h 3 23224320 3053 h 3 967680 83 h 3 15360 347 h 3 193536 127 h 3 1935360 29 h 3 7560 83 h 3 15360 269 h 3 241920 1117 h 3 3870720 551 h 2 49896 32027 h 2 3548160 3959 h 2 1596672 14339 h 2 127733760 23851 h 2 2280960 12053 h 2 1774080 5755 h 2 3193344 2567 h 2 31933440 95 h 2 16632 56677 h 2 3548160 7951 h 2 2661120 5311 h 2 42577920 72269 h 2 15966720 12053 h 2 1774080 123463 h 2 15966720 6439 h 2 31933440 61 h 2 249480 32027 h 2 3548160 15701 h 2 1140480 312317 h 2 127733760 ], R 5 =[ g n+ 1 4 g n+ 1 2 g n+ 3 4 g n+1 ]

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

       I Y L = B ¯ R 1 + C ¯ R 2 + D ¯ R 3 + E ¯ R 4 + G ¯ R 5

[ 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 ][ y n+ 1 4 y n+ 1 2 y n+ 3 4 y n+1 y ' n+ 1 4 y ' n+ 1 2 y ' n+ 3 4 y ' n+1 ]=[ 1 h 4 1 h 2 1 3h 4 1 h 0 1 0 1 0 1 0 1 ][ y n y ' n ]+[ 2602339 h 2 153280512 35 h 2 891 39015 h 2 630784 6353 h 2 74844 1539551h 17418240 24463h 272160 6501h 71680 1601h 17010 ][ f n ] +[ 148231 h 2 47900160 1807 h 2 322560 243193 h 2 47900160 382169 h 2 766402560 196 h 2 4455 h 2 40 68 h 2 4455 13 h 2 8910 18531 h 2 197120 3159 h 2 35840 6813 h 2 197120 8469 h 2 3153920 13952 h 2 93555 52 h 2 315 8578 h 2 93555 3457 h 2 374220 89371h 1088640 103h 2520 38341h 1088640 59681h 17418240 1654h 8505 52h 315 394h 8505 1153h 272160 921h 4480 81h 280 711h 4480 411h 71680 2048h 8505 104h 315 2048h 8505 1601h 17010 ][ f n+ 1 4 f n+ 1 2 f n+ 3 4 f n+1 ]+[ 28343 h 3 72990720 1277 h 3 1330560 9747 h 3 6307840 269 h 2 124740 26051 h 2 11612160 421 h 2 181440 339 h 2 143360 29 h 2 11340 ][ g n ]+ [ 551 h 3 199584 32027 h 3 14192640 3959 h 3 6386688 14339 h 2 510935040 41 h 3 5544 5 h 3 693 17 h 3 9240 109 h 2 1330560 4509 h 3 394240 19197 h 3 1576960 9 h 3 2464 27 h 2 180224 464 h 3 31185 10 h 3 693 16 h 3 4455 5 h 2 12474 31207 h 2 1451520 81 h 2 1520 1243 h 2 290304 2237 h 2 11612160 19 h 2 1134 h 2 40 31 h 2 5670 43 h 2 181440 279 h 2 17920 81 h 2 1520 183 h 2 17920 9 h 2 28672 32 h 2 2835 0 32 h 2 2835 29 h 2 11340 ][ g n+ 1 4 g n+ 1 2 g n+ 3 4 g n+1 ]

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:


C 11 = [ 6.1829× 10 14 ,1.752× 10 13 ,3.0454× 10 13 ,4.8542× 10 13 , 4.1791× 10 13 ,4.8542× 10 13 ,5.5292× 10 13 ,9.7083× 10 13 ] 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

A block method is said to be zero-stable if as h0 , the root z i ,i=1(1)k of the first characteristic polynomial ρ( z )=0 that is

ρ( z )=det[ j=0 k A (i) z ki ]=0

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



ρ( z )=| z[ 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 ][ 0 0 0 1 0 0 0 h 4 0 0 0 1 0 0 0 h 2 0 0 0 1 0 0 0 3h 4 0 0 0 1 0 0 0 h 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 0 0 0 0 1 ] |= z 6 ( z1 ) 2 ρ( z )= z 6 ( z1 ) 2 =0,

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).

Figure 1

Figure 1:

Absolute Stability Region.



= h 12 ( ( 7219 15107327262720000 ) w 4 ( 79 156959244288000 ) w 3 ) h 11 ( ( 3423139 139591703907532800 ) w 4 +( 1171 30519853056000 ) w 3 ) h 10 ( ( 4666483 3625758543052800 ) w 3 +( 585349 23930006384148480 ) w 4 )+ h 9 ( ( 73634299 2913555972096000 ) w 3 ( 2899283 1495625399009280 ) w 3 ) h 8 ( ( 12604072069 149562539900928000 ) w 4 +( 6897035147 8157956721868800 ) w 3 ) h 7 ( ( 775601 9442079539200 ) w 4 +( 707100649 109258348953600 ) w 3 ) + h 6 ( ( 1953881 256927334400 ) w 4 ( 26693444639 145677798604800 ) w 3 ) h 5 ( ( 2850443 36883123200 ) w 4 +( 357859 1077753600 ) w 3 ) h 4 ( ( 2587 4257792 ) w 4 +( 12221035 1034643456 ) w 3 )+ h 3 ( ( 25 1782 ) w 4 +( 269 114740 ) w 3 ) h 2 ( ( 16399 228096 ) w 4 +( 1009733 4790016 ) w 3 ) + w 4 w 3

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

Exact Solution: y( x )=exp( x )h= 1 10

Problem II: f( x,y,y' )=y',y( 0 )=1,y'( 0 )= 1 2 ,0x1.
Exact Solution: y( x )=1 e x withh= 1 100

x-values
Exact Solution
Computed Solution
Error in our method
Error in [13]
0.100
0.90483741803595957316
0.90483741803595957245
7.100E(-19)
1.054712E(-14)
0.200
0.81873075307798185867
0.81873075307798185795
7.200E(-19)
1.776357E(-14)
0.300
0.74081822068171786607
0.74081822068171786534
7.300E(-19)
2.342571E(-14)
0.400
0.67032004603563930074
0.67032004603563930001
7.300E(-19)
2.797762E(-14)
0.500
0.60653065971263342360
0.60653065971263342285
7.500E(-19)
3.130829E(-14)
0.600
0.54881163609402643263
0.54881163609402643185
7.800E(-19)
3.397282E(-14)
0.700
0.49658530379140951470
0.49658530379140951390
8.000E(-19)
3.563816E(-14)
0.800
0.44932896411722159143
0.44932896411722159058
8.500E(-19)
3.674838E(-14)
0.900
0.40656965974059911188
0.40656965974059911099
8.900E(-19)
3.730349E(-14)
1.00
0.36787944117144232160
0.36787944117144232065
9.500E(-19)
3.741452E(-14)

Table 1: Comparison of the proposed method

x-values
Exact Solution
Computed Solution
Error in our method
Error in [13]
0.100
-0.10517091807564762480
-0.10517091807564762481
1.000E(-20)
8.326679(-17)
0.200
-0.22140275816016983390
-0.22140275816016983391
1.000E(-20)
2.775557(-16)
0.300
-0.34985880757600310400
-0.34985880757600310397
-3.000E(-20)
5.551115(-16))
0.400
-0.49182469764127031780
-0.49182469764127031779
-1.000E(-20)
9.436896(-16)
0.500
-0.64872127070012814680
-0.64872127070012814679
-1.000E(-20)
2.109424(-15)
0.600
-0.82211880039050897490
-0.82211880039050897478
-1.000E(-19)
3.219647(-15)
0.700
-1.01375270747047652160
-1.01375270747047652150
-1.000E(-19)
4.440892(-15)
0.800
-1.22554092849246760460
-1.22554092849246760440
-2.000E(-19)
5.995204(-15)
0.900
-1.45960311115694966380
-1.45960311115694966350
-3.000E(-19)
7.771561(-15)
1.00
-1.71828182845904523540
-1.71828182845904523500
-4.000E(-19)
1.065814(-14)

Table 2

Conclusion

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.