Generalized One-Step Third Derivative Implicit Hybrid Block Method for the Direct Solution of Second Order Ordinary Differential Equation

Transcription

1 Appled Mathematcal Sceces, Vol. 1, 16, o. 9, HIKARI Ltd, Geeralzed Oe-Step Thrd Dervatve Implct Hybrd Block Method for the Drect Soluto of Secod Order Ordary Dfferetal Equato Z. Omar Departmet of Mathematcs School of Quattatve Sceces College of Art ad Sceces Uversty Utara Malaysa, Malaysa M. F. Alkasassbeh Departmet of Mathematcs School of Quattatve Sceces College of Art ad Sceces Uversty Utara Malaysa, Malaysa Copyrght c 15 Z. Omar ad M. F. Alkasassbeh. Ths artcle s dstrbuted uder the Creatve Commos Attrbuto Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, provded the orgal work s properly cted. Abstract I ths artcle, a mplct hybrd method of order sx s developed for the drect soluto of secod order ordary dfferetal equatos usg collocato ad terpolato approach. To derve ths method, the approxmate soluto power seres s terpolated at the frst ad off-step pots ad ts secod ad thrd dervatves are collocated at all pots the gve terval. Besdes havg good umercal method propertes, the ew developed method s also superor to the exstg methods terms of accuracy whe solvg the same problems. Keywords: Collocato, Iterpolato, Hybrd Block Methods, Secod Order Ordary Dfferetal Equato, Drect Soluto, Thrd dervatve

2 418 Z. Omar ad M. F. Alkasassbeh 1 Itroducto Ths artcle proposes a geeral oe-step thrd dervatve mplct hybrd block method (GOHBM) for the drect soluto of the secod order ODEs the form y = f(x, y, y ), y(a) = y, y (a) = y, a x b (1) wth the assumpto that f s dfferetable ad satsfes Lpchtz s codto whch guaratees the exstece ad uqueess of the soluto ([1]). Block methods whch are wdely used by may scholars for solvg (1) were frst troduced by [14] ad later by [9] maly to provde startg values for predctor-corrector algorthms. Those methods produced better accuracy tha the usual step by step methods. [8], o the other had, exteded Mle s dea to develop block methods for solvg ODEs. I order to obta hgher order methods ad hece to crease the accuracy of the approxmate soluto, [4] proposed hybrd block methods whch cluded off-step pot(s) the dervato of the algorthms. Furthermore, hybrd block methods were used to crcumvet Dahlqusts barrer codtos whch stpulate that the order of a k-step Lear Multstep Method (LMM) caot exceed k + 1 for k s odd or k + for k s eve for the method to be zero-stable ([6]). I addto, hybrd block methods are also kow to share wth Ruge-Kutta methods ther favourable advatage of beg self startg ad more accurate sce they are mplemeted as a block. I hybrd block methods, step ad off-step pots are combed to form a sgle block for solvg ODEs ( see [4], [15], [1]). I addto, [16] troduced secod dervatve methods whch are specal types of hybrd methods (referred by [14] as Obrechkoff methods) to ehace the accuracy of the approxmato whch show to reach a order k +. Meawhle, some scholars such as [5], [11] proposed a Smpso s-type secod dervatve method for the soluto of stff system of frst order IVPs. Ther work motvated us to propose a ew geeralzed oe step thrd dervatve mplct hybrd block method for solvg secod order ODEs drectly usg terpolato ad collocato the form α t y +t = h [ β t f +t +β 1 f +1 ]+h [ γ t g +t +γ 1 g +1 ], x [x, x +1 ] =, 1,,..., N 1, h = x x 1 s the costat step sze for the partto π N of the terval [a, b] whch s gve by π N = [a = x < x 1 <... < x N 1 < x N = b], α t, β t ad γ t are ukow coeffcets, g +t = f +t ad g +1 = f +1.

3 Geeralzed oe-step thrd dervatve mplct hybrd block method 419 Developmet of the Method Let us assume the followg power seres be the approxmate soluto to (1) y(x) = s+r 1 j= a j x j () r ad s are the umber of terpolato ad collocato pots respectvely. Dfferetatg () twce ad thrce yelds y (x) = y (x) = s+r 1 j= s+r 1 j= a j j(j 1)x j = f(x, y, y ) () a j j(j 1)(j )x j = g(x, y, y ) (4) Iterpolatg () at x +r, r = {, t} ad collocatg () ad (4) at x +s, s = {, t, 1} t (, 1), ad o combg gves a system of equatos matrx form AX = U (5) A = [ a a 1 a a a 4 a 5 a 6 a 7 ] T, ad U = [ y y +t f f +t f +1 g g +t g +1 ] T, 1 x x x x 4 x 5 x 6 x 7 1 x +t x +t x +t x 4 +t x 5 +t x 6 +t x 7 +t 6x 1x x x 4 4x 5 6x X = +t 1x +t x +t x 4 +t 4x 5 +t 6x +1 1x +1 x +1 x x x 6x 1x 1x 4 6 4x +t 6x +t 1x +t 1x 4 +t 6 4x +1 6x +1 1x +1 1x 4 +1 Solvg (5) for the ukow costat a js usg matrx mapulato ad substtutg them back to () gves a cotuous hybrd lear mult-step method the form y(x) = α t y +t + h [ β t f +t + β 1 f +1 ] + h [ γ t g +t + γ 1 g +1 ] (6)

4 4 Z. Omar ad M. F. Alkasassbeh whose frst dervatve s y (x) = 1 h α ty +t + h[ β tf +t + β 1f +1 ] + h γ tg +t + γ 1g +1 ] (7) Evaluatg (6) at x = x +1 ad (7) at x = x +, = {, t, 1} produces the followg geeral equatos block form A () Y m+1 = A (t) Y m + B () F m+ + D () G m+ (8) A () s a 4 4 detty matrx, Y m+1 = [ y +t, y +1, y +t, y +1] T,Ym = [ ] y t, y, y t, y T, F m = [ ] T f t, f t, f t, f, Fm+1 = [ ] T f +t, f +1, Gm = [ ] T g t, g t, g t, g, G m+1 = [ g +t, g +1 ] T. The matrces A (t), B (), D () wll be descrbed later. To obta the specfc equatos of (8), let us cosder the followg three cases for demostrato. Case I : t = 1 Substtutg t = 1 ad z = x x + 1 h (6) ad (7) we get y(z) = α y + + h [ β f + + β 1 f +1 ] + h [ γ g + + γ 1 g +1 ] α = z α 1 = 1 + z β = (z(1 + z)(79 17z + 681z z 15147z z 5 ))/171 β 1 = ((z(1 + z)( 66 64z + 196z + 87z 41z 4 + 4z 5 ))/54) β 1 = ((z(1 + z)( z 954z 94z 745z z 5 ))/168) γ = (z(1 + z)(44 1z + 96z + 7z 16z z 5 ))/171 γ 1 = (z(1 + z)( z z 675z 78z 4 + 4z 5 ))/151 γ 1 = (z(1 + z)( z 15z 141z 81z 4 + 4z 5 ))/168

5 Geeralzed oe-step thrd dervatve mplct hybrd block method 41 ad y (z) = 1 h α y + + h[ β f + + β 1f +1 ] + h γ tg + + γ 1g +1 ] α = α 1 = β = ( z 51z 4 146z z 6 )/171 β 1 = ( z 646z + 855z z 5 51z 6 )/54 β 1 = ( z z z z 6 )/168 γ = ( z 618z z 6 )/171 γ 1 = ( z + 151z 5515z 4 484z z 6 )/151 γ 1 = ( z 5515z z 6 )/168 Now, equato (8) ca be wrtte as A () Y m+1 = A ( 1 ) Y m + B () F m+ + D () G m+ 1 h 61h 171 A ( 1 ) = 1 h h 1 B() = 14 18h B(1) = h 5 5h 11h h D () h 9h = 7 h 17h D(1) = 8 1 9h h h 9h h h 54 7h 11 11h 7 7h 8 7h 168 5h 11 1h h 8 Case II : t = 1 Smlarly, replacg t = 1 ad z = x x + 1 h (6) ad (7) produces y(z) = α y + + h [ β f + + β 1 f +1 ] + h [ γ g + + γ 1 g +1 ]

6 4 Z. Omar ad M. F. Alkasassbeh ad α = z α 1 = 1 + z β = (z(z + 1)(96z 5 98z 4 76z + 748z 74z + 187))/6 β 1 = z(z 5 4z + z + 11)/6 β 1 = ((z(z + 1)(96z 5 z 4 84z 148z + 74z 7))/6) γ = z(z + 1)(4z 5 48z 4 + z + 16z 8z + 4)/84 γ 1 = z(z 6 6z z 19)/84 γ 1 = (z(z 6 + 4z 5 168z 4 14z 5))/6 y (z) = 1 h α y + + h[ β f + + β 1f +1 ] + h γ g + + γ 1g +1 ] α = α 1 = β = (144z 6 576z 5 84z z + 187)/6 β 1 = (19z 5 16z + 6z + 11)/6 β 1 = ( 144z 6 576z z z + 7)/6 γ = (8z 6 168z 5 15z 4 + 7z + )/4 γ 1 = (4z 6 168z 4 + 4z 19)/84 γ 1 = (4z z 5 84z 4 56z 5)/6 Thus, equato (8) becomes A () Y m+1 = A ( 1 ) Y m + B () F m+ + D () G m+ 1 h 1h 168 A ( 1 ) = 1 h 1 B() = 79h 4 11h B(1) = h h 4 4h 15 4h 15 8h 15 h h 4 11h 48 7h

7 Geeralzed oe-step thrd dervatve mplct hybrd block method 4 59h 144 D () h = 84 1h D(1) = 96 h 6 h 15 h 15 h 4 11h 144 h 1 h h 6 Case III : t = Fally, puttg t = ad z = x x + h (6) ad (7) we have y(z) = α = z/ α = 1 + z/ α y + + h [ β f + + β 1 f +1 ] + h [ γ g + + γ 1 g +1 ] β = (z( + z)(44 66z z 8775z 4698z z 5 ))/168 β = (z( + z)(8 + 69z 15z 1755z + 178z 4 + 4z 5 ))/54 β 1 = ((z( + z)( z 5z 7z + 547z z 5 ))/171) γ = (z( + z)( z + 144z 61z 16z 4 + 4z 5 ))/168 γ = (z( + z)( z + 16z 1z + 648z 4 + 4z 5 ))/151 γ 1 = (z( + z)( z 4z 594z + 891z z 5 ))/171 ad y (z) = 1 h α = / α 1 = / α y + + h[ β f + + β 1f +1 ] + h β = ( z 17865z z z 6 )/168 β γ g + + γ 1g +1 ] = ( z 646z 855z z z 6 )/54 β 1 = ( z z z 5 16z 6 )/855 γ = ( z 5515z z 6 )/168 γ = ( z 151z 5515z z z 6 )/151 γ 1 = ( 8 756z + 618z z 6 )/171

8 44 Z. Omar ad M. F. Alkasassbeh Hece, we ca wrte equato (8) as below A () Y m+1 = A ( ) Y m + B () F m+ + D () G m+ 1 h 1111h 855 A ( ) = 1 h 61h 1 B() = 8 17h B(1) = h 8 6h 85 D () 9h = 56 5h D(1) = 4 h h 85 9h 11 17h 15 9h 8 16h 171 h 15 h 115 h h 15 7h 8 7h 45 7h 8 75h 855 1h 7 4h 115 h 5 Aalyss of the Method Order of the method The lear operator ˆL assocated wth the hybrd block methods formula (8) accordg to [1] ad [7] s sad to be of order p f ˆL{y(x); h} = A () Y m A (t) Y m+1 B () F m+ expadg Taylor seres ad combg lke terms D () G m+ ˆL{y(x); h} = C h y () = (9) C = C 1 =... = C p+1 = ad C p+ The term C p+ s called the error costat ad the local trucato error s gve by : t +k = C p+ y p+ h p+ (x ) + O(h p+ ) For Case (I), substtutg t = 1 (9), we get

9 Geeralzed oe-step thrd dervatve mplct hybrd block method 45 ( h! )y() ( h! )y() ( h! )y(+1) y h y 61h y () 171 ( h! )y(+1) y hy h y () y 18hy() 115 y hy() 5 h + y (+) (619( 1 168! ) + 7) 5h y () 4 h + y (+) (44( 1 68! ) + 1) h + y (+) 14 (7( 1 11! ) + 5) h y () 7 h + y (+) (7( 1 84! ) 4) = h +1 y (+) (57( ! ) + 1) 17h y () 4 h + y (+) (61( ! ) + 5) h +1 y (+) (7( 1 8! ) + 1) h+ y () h + y (+) (7( 1 4! ) 5) Comparg the coeffcets of y ad h produces C = C 1 =... = C 7 = wth vector of error costats C 8 = [ T ] whch mples the order (p) of ths method s 6. For Case (II), substtutg t = 1 (9), we have ( h! )y() ( h! )y() ( h! )y(+1) y h y 1h y () y hy 79h y () 4 ( h! )y(+1) y 11hy() h + y (+) (18( 1 144! ) + 11) h + y (+) (7( 1 168! ) + 1) 59h y () 144 (11( 1 4! ) + 19) 5h y () 4 = (18( 1 4! ) + 11) 1h y () 96 (16( 1! ) + 7) h+ y () 6 h + y (+) h + y (+) 4! (8( 1 ) + ) y 7hy() h +1 y (+) h + y (+) 96! (4( 1 ) + ) h +1 y (+) h + y (+) 6! Assocatg the coeffcets of y ad h yelds C = C 1 =... = C 7 = wth vector of error costats C 8 = [ T ] whch also mples that the order (p) of ths method s 6.

10 46 Z. Omar ad M. F. Alkasassbeh For Case (III), substtutg t = (9), we get ( (h)! )y() y h y 1111h y () 855 h + y (+) (7( 855! ) + 75) 6h y () 85 h + y (+) (7( 8! ) + 5) 9h y () 56 = (189( 115! ) + 4) 5h y () 4 (7( 8! ) + ) h+ y () 48 4h + y (+) 855! (87( ) + ) ( h! )y() y hy 61h y () 8 ( (h)! )y(+1) ( h! )y(+1) y 17hy() 115 y 1hy() 8 h + y (+) 168! (15( ) + ) h +1 y (+) h + y (+) 115! (15( ) + ) h +1 y (+) h + y (+) 4! (7( ) + 8) Matchg the coeffcets of y ad h yelds C = C 1 =... = C 7 = wth vector of error costats C 8 = [ T ] whch aga mples that the order (p) of ths method s 6..1 Cosstecy Defto.1. A block method s sad to be cosstet f ts order s greater tha oe. We coclude from the three cases above that the order (p) of the hybrd block methods formula (8) s greater tha 1 hece the cosstecy property s satsfed.. Zero Stablty Defto.. The hybrd block method formula (8) s sad to be zero stable f o root of the frst characterstc equato ρ(r) has modulus greater tha oe.e R s 1 ad f R s = 1 the the multplcty of R s must ot exceed two. To show that the roots of the frst characterstc equato satsfes the pror defto we assume that t (, 1) ad hece ρ(r) = det[ra () A (t) ] =

11 Geeralzed oe-step thrd dervatve mplct hybrd block method 47 R 1 th ρ(r) = det[ R R 1 h 1 ] = R 1 R (R 1) = R 1 = R = R = R 4 = 1 As a result, the developed method s zero stable.. Covergece Theorem.1. (Herc,196) Cosstecy ad zero stablty are suffcet codtos for a lear mult step method to be coverget The hybrd block method (8) s coverget sce t satsfes both the cosstecy ad zero stablty codtos. 4 Numercal Examples I ths secto accuracy of the geeral oe-step mplct hybrd block method (8) wth order 6 s tested o three expermetal problems for the three cases smultaeously, wth a fxed step sze h = 5 1 for the frst problem h = 1 1 for the secod ad h =.1 for the thrd. The computed results are the compared wth recet methods ad the ew methods s foud to have advatages as show Tables I-III. Problem (1) : f(x, y, y ) = y + 8e x, y() = 1, y () = 1. Exact Soluto : y = 4e x + e x + wth h = 5. 1 Source : []. Table I : Comparso of the proposed method wth A.M. Badmus (14). X VALUE ERROR FOR ERROR FOR ERROR FOR ERROR t = 1 t = 1 t = AMB E(-16) E(-16).446E(-16).159 E(-7) E(-16) E(-16) E(-16) 1.79 E(-6) E(-16).446E(-16) E(-16) E(-6) E(-16) E(-16) E(-16) E(-5) E(-16) E(-16) E(-16).9558 E(-5). 1.68E(-15) E(-15).66455E(-15) E(-5) E(-15) E(-15) E(-15) E(-5) Remark: AMB s the error [].

12 48 Z. Omar ad M. F. Alkasassbeh Problem () : f(x, y, z) = x(y ), y() = 1, y () = 1. Exact Soluto : y = 1 + l( +x) wth h = 1. x 1 Source : [1]. Table II : Comparso of the proposed method wth Adetola Olade (1). X VALUE ERROR FOR ERROR FOR ERROR FOR ERROR t = 1 t = 1 t = FOR EAO.1.446E(-16).446E(-16) E(-16) 9.99 E(-15)..E(+).446E(-16) 1.68E(-15) E(-14) E(-16).446E(-16) E(-16) 4.7 E(-1) E(-15) E(-16) 1.68E(-15) 1.67 E(-1).5.669E(-15).44491E(-15) E(-15) E(-1) E(-15) E(-15) E(-14) E(-11) E(-14) E(-14).641E(-14).51 E(-11) E(-14) E(-14) E(-14) 5.15 E(-11) E(-14) E(-14) E(-1) 1.76 E(-11) E(-1) E(-1) E(-1).17 E(-1) Remark: EAO s the error [1]. Problem() : y + 6 x y + 6 y =, y(1) = 1, y (1) = 1. x Exact Soluto : y = 5 wth h =.1. x x 4 Source : []. Table III : Comparso of the proposed method wth A.M.Badmus (14). X VALUE ERROR FOR ERROR FOR ERROR FOR ERROR t = 1 t = 1 t = FOR EAM E(-16).446E(-16).446E(-16) 8.E(-8) 1.65.E(+).446E(-16) E(-16) 1.16E(-6) E(-16) E(-16) E(-16) 6.68E(-6) E(-16) 1.11E(-15) E(-15) 9.491E(-6) E(-15) E(-15).1864E(-15) 1.955E(-6) E(-15).66455E(-15) E(-15) 9.416E(-6) E(-15).9968E(-15) 6.888E(-15) 4.655E(-5) E(-15) 5.176E(-15) E(-15) 4.71E(-5) E(-15) E(-15) E(-14 ) E(-4) E(-15) 7.747E(-15) E(-14) 4.41E(-4) Remark: EAM s the error [] 5 Cocluso A geeral oe-step hybrd (GOHBM) block method wth oe off step pot of order 6 has bee successfully developed for the drect soluto of geeral secod order IVP. The developed method s tested o t = { 1, 1, }. Numercal aalyss shows that the developed method s cosstet ad zero stable whch mples ts covergece. Apart from havg good propertes of the umercal

13 Geeralzed oe-step thrd dervatve mplct hybrd block method 49 method, the umercal results suggest that the ew method has ot oly out performed the exstg methods, but also crcumvet Dahlqusts barrer. Refereces [1] Adetola Olade, Mattew Remleku Odekule ad Mfo Odo Udoh, Four Steps Cotuous Method for the Soluto of y =f(x,y,y ), Amerca Joural of Computatoal Mathematcs, (1), o., [] A.M. Badmus, A Effcet Seve-pot Hybrd Block Method for the Drect Soluto of y = f (x, y, y ), Brtsh Joural of Mathematcs ad Computer Scece, 4 (14), o. 19, [] A.M. Badmus, A New Eghth Order Implct Block Algorthms for the Drect Soluto of Secod Order Ordary Dfferetal Equatos, Amerca Joural of Computatoal Mathematcs, 4 (14), o. 4, [4] C.W. Gear, Hybrd methods for tal value problems ordary dfferetal equatos, SIAM Joural of Numercal Aalyss, (1965), o. 1, [5] F. Ngwae, S. Jator, Block hybrd-secod dervatve method for stff systems, It. J. Pure Appl. Math, 8 (1), o. 4, [6] G.G. Dahlqust, Numercal tegrato of ordary dfferetal equatos, Math. Scad., 4 (1956), [7] J.D. Lambert, Computatoal Methods Ordary Dfferetal Equatos, Joh Wley, New York, 197. [8] J.B. Rosser, A Ruge-kutta for all seasos, SIAM, Rev., 9 (1967), o., [9] L.F. Shampe, H. A. Watts, Block mplct oe-step methods, Mathematcs of Computato, (1969), o. 18, [1] P. Herc, Dscrete Varable Methods ODEs, Wley, New York, 196. [11] R.K. Sah, S.N. Jator, N. A. Kha, A smpso s-type secod dervatve method for stff systems, Iteratoal Joural of Pure ad Appled Mathematcs, 81 (1), o. 4,

14 4 Z. Omar ad M. F. Alkasassbeh [1] S.N. Jator, Solvg secod order tal value problems by a hybrd multstep method wthout predctors, Appled Mathematcs ad Computato, 17 (1), o. 8, [1] S.O. Fatula, Numercal Methods for Ital Value Problems Ordary Dfferetal Equato, Academc Press, New-York, [14] W.E. Mle, Numercal Soluto of Dfferetal Equatos, Joh Wley ad Sos, 195. [15] W. Gragg, H.J. Stetter, Geeralzed multstep predctor-corrector methods, J. Assoc. Comput. Mach., 11 (1964), o., [16] W.H. Erght, Secod dervatve multstep methods for stff ordary dfferetal equatos, SIAM J. Numer. Aal., 11 (1974), o., Receved: November 6, 15; Publshed: February 8, 16