Comparison Solutions Between Lie Group Method and Numerical Solution of (RK4) for Riccati Differential Equation

Transcription

1 Appled and Computatonal Mathematcs 06; 5(: do: 0.648/j.acm ISSN: (Prnt; ISSN: 8-56 (Onlne Comparson Solutons Between Le Group Method and Numercal Soluton of (RK4 for Rccat Dfferental Equaton Sam H. Altoum, Salh Y. Arbab Department of Mathematcs, Unverst College of Qunfudha, Umm Alqura Unverst, Makkah, KSA Engneerng College, Albaha Unverst, Albaha, KSA Emal address: (S. H. Altoum, (S. Y. Arbab To cte ths artcle: Sam H. Altoum, Salh Y. Arbab. Comparson Solutons Between Le Group Method and Numercal Soluton of (RK4 for Rccat Dfferental Equaton. Appled and Computatonal Mathematcs. Vol. 5, No., 06, pp do: 0.648/j.acm Receved: Februar 7, 06; Accepted: March 5, 06; Publshed: Aprl 5, 06 Abstract: Ths paper ntroduced Le group method as a analtcal method and then compared to RK4 and Euler forward method as a numercal method. In ths paper the general Rccat equaton s solved b smmetr group. Numercal comparsons between eact soluton, Le smmetr group and RK4 on these equatons are gven. In partcular, some eamples wll be consdered and the global error computed numercall. Kewords: Rccat Equaton, Smmetr Group, Infntesmal Generator, Runge-Kutta. Introducton f (, = P( + Q( + R ( ( where P(,Q(,R ( functons of The Rccat dfferental equaton s named after the Italan nobleman Count Jacopo Francesco Rccat ( The book of Red [] contans the fundamental theores of Rccat equaton. Ths equaton s perhaps one of the smplest nonlnear frst ODE whch plas a ver mportant role n soluton of varous non-lnear equatons ma be found n numerous scentfc felds. The soluton of ths equaton can be obtaned numercall b usng classcal numercal method such as the forward Euler method and Runge-Kutta method. An analtc soluton of the non-lnear Rccat equaton reached see [] usng A doman Decomposton Method. [5] used dfferental Transform Method(DTM to solve Rccat dfferental equatons wth varable co-effcent and the results are compared wth the numercal results b (RK4 method.[4] solved Rccat dfferental equatons b usng (ADM method and the numercal results are compared wth the eact solutons. B.Batha [5] solved Rccat dfferental equatons usng Varatonal Iteraton Method (VIM and numercal comparson between VIM, RK4 and eact soluton on these equatons are gven. Le smmetres are utlzed to solve both lnear and non-lnear frst order ODE, and the majort of ad hoc methods of ntegraton of ordnar dfferental equaton could be eplaned and deduced smpl b means of the theor of Le group. Moreover Le gave a classfcaton of ordnar dfferental equaton n terms of ther smmetr groups. Le's classfcaton shows that the second order equatons ntegrable b hs method can be reduced to merel four dstnct canoncal forms b changes of varables. Subjectng these four canoncal equatons to changes of varables alone, one obtans all known equatons ntegrated b classcal methods as well as nfntel man unknown ntegrable equatons. We wll consder n our test onl a frst order dfferental equaton and the numercal soluton to second-order and hgher-order dfferental equatons s formulated n the same wa as t s for frst-order equatons. We have attempted to gve some eamples to the use of Le group methods for the soluton of frst-order ODEs. The Le group method of solvng hgher order ODEs and sstems of dfferental equatons s more nvolved, but the basc dea s the same: we fnd a coordnate sstem n whch the equatons are smpler and eplot ths smplfcaton.,,, s Defnton: A change of varables, called an equvalence transformaton of Rccat dfferental equaton f an equaton of the form ( transformed nto an

2 65 Sam H. Altoum and Salh Y. Arbab: Comparson Solutons Between Le Group Method and Numercal Soluton of (RK4 for Rccat Dfferental Equaton equaton of same tpe wth possbl dfferent coeffcents. Equatons related b an equvalence transformaton are sad to be equvalent. Theorem: The general equvalence transformaton of Rccat equaton comprses of ϕ 0. Change of ndependent varable = ϕ,. Lnear-ratonal transformatons of the dependent varable ( + β( ( ( α =, αδ βγ 0 γ + δ Eample: Fnd the transformaton of the coeffcents of the Rccat equaton under change of ndependent varable ( Soluton: Accordng to the chan rule of dfferentaton, d d = ϕ ( d d d ϕ = + + d P Q R. Upon dvson b ϕ ( and substtuton becomes = ϕ, t where P d P Q R d = + +, P( ϕ (, Q( ϕ ( ϕ ( R ( ϕ ( R ( = ϕ ϕ ( = ( ϕ ( ( Q =, ϕ ϕ. General Soluton of Rccat Equaton The general soluton ( of equaton (, we suppose s( s a partcular soluton for ( and also assume the general soluton s ( = s( +, ( z where z ( s unknown functon and we seek about t b substtuton the general soluton ( from ( n ( we deduce: s Q s ( z ( = P ( s ( + Q( s( + R ( + P( + + z ( z ( z z ( The partcular soluton s( s soluton of ( so satsfng t, consequentl substtute nstead of n equaton ( we, get s = P s + Q s + R, (4 so b substtuton (4 n ( we deduce: z + P s + Q z = P (5 It's clear equaton (5 s lnear equaton from frst order and b choosng partcular after soluton we obtan s( va (Tral and Error we obtan the general soluton = s +. z. Soluton of Le Group Method O.D.E A set G of nvertble pont transformaton n the (, plane R = f (,,a, g (,,a =, (6 dependng on a parameter a s called one-parameter contnuous group, f G Contans the dentt transformaton (e.g. f =, g = at a = 0 as well as the nverse of elements and ther composton. It s sad that transformatons (6 form asmmetr group of dfferental equaton ( n f the equaton s form nvarant; Whenever F,,,..., = 0 (7 ( n F,,,..., = 0 ( n F,,,..., = 0. A smmetr group of dfferental equaton s also termed a group admtted b ths equaton. Le's theor reduced the constructon of the largest smmetr group G to the determnaton of t's nfntesmal transformatons: Where + ξ, a, ξ ( = f (,a + aη,. (8 a a= 0 defned as a lnear part (n the group parameter a n the Talor epanson of the fnte transformatons (6 of G. It's convenent represent nfntesmal transformaton (8 b the.

3 Appled and Computatonal Mathematcs 06; 5(: lnear dfferental operator. (9 X = ξ, + η, Called the nfntesmal transformaton or nfntesmal operator of the group G wth smmetr condton η ξ h + ( η ξ h ( ξ h + η h = 0 (0 The generator X of the group admtted b dfferental equaton s also termed an operator admtted b ths equaton. An essental feature of a smmetr group G s that t conserves the set of solutons of the dfferental equaton admttng ths group. Namel, the smmetr transformatons merel permute the ntegral curves among themselves. It ma happen that some of the ntegral curves are ndvduall unaltered under group G. Such ntegral curves are termed nvarant solutons 4. Canoncal Varables Equaton (7 furnshes us wth a smple eample for ehbtng the smmetr of dfferental equatons. Snce ths equaton does not eplctl contan the ndependent varable, t does not alter after an transformaton = + a wth an arbtrar parameter a. The latter transformaton from a group known as the group of translaton along the as. Equaton (7 s n fact the general nth order ordnar dfferental equaton admttng the group of translatons along the as. Moreover, an one parameter group reduces n proper varables to the group of translatons. These new varables, canoncal varables t and u are obtaned b solvng the equaton X ( t =, X u = 0 ( Where X s the generator (9 of group G. It follows that an nth order ordnar dfferental equaton admttng a oneparameter group reduces to the form (7 n the canoncal varables, and hence one can reduce ts order to n. In partcular, an frst-order equaton wth a known oneparameter smmetr group can be ntegrated b quadrature usng canoncal varables. 5. Numercal Method We shall consder the soluton of sets of frst-order dfferental equatons onl. Users nterested n solvng hgher order ODES can reduce ther problem to a set of frst order equatons. Fortunatel, the method of Le group gves sometmes the soluton of ODEs, but t s dffcult to obtan an eact soluton. For that, we wll use numercal methods to obtan a soluton. Snce the numercal method used to compute an appromaton at each step of the sequence, errors are compounded at ever step. We ntroduce the numercal method wll be compared to Le group method and we don't forget the comparson of the error at each eample wll be presented. In ths contet, we wll use Runge Kutta-4 and Euler forward methods whch s the most wdel one step methods to solve d (, = f (, ( d nvolves defnng 4 quanttes k n as follows: m k = f α + c h, + h α k n n n,m m m= here, ~ ( represents our numerc appromaton to soluton of the dfferental equaton at =. The coeffcents ~ ( and c α,m are constants that we are gong to choose shortl, and h = = + s the step sze. Once one calculates k n gven at (, the value of + ~ ( + s gven b M = + h b k n n n= In ths epresson, the b n are constants. Then we have k = f (, k = f + h, + hk k = f + h, + hk k4 = f ( + h, + hk + = + h k + k + k + k4 6 6 We recall that usng M = 4, (RK4 the global error s 4 Ο (h. 6. Eact and Numercal Soluton In ths secton we solve the followng eamples b two methods, frstl smmetr group soluton and secondl numercall-forward Euler method and Runge-Kutta methods. Fnall we see the prvlege and more effectve of mentoned methods. Eample : Frst: Smmetr soluton Consder Rccat equaton d + = 0. ( d In ths eample we see the functon s not defne n 0 =. Ths equaton admts the one-parameter group of non-

4 67 Sam H. Altoum and Salh Y. Arbab: Comparson Solutons Between Le Group Method and Numercal Soluton of (RK4 for Rccat Dfferental Equaton a homogeneous dlatons (scalng transformaton = e, = e a. Indeed the dervatve s wrtten n the new varables as = e a, whence d d a + = e 0 + =. d d (b, Fgure (c and Fgure (d confrm our predctons that there s a small error between analtcal and numercal soluton usng RK4 but there s no-convergence when one use Forward Euler method. Hence, the equaton s form nvarant. The smmetr group has the generator X = + The soluton of the equaton wth the above operator X provdes canoncal varables, from equaton ( we, get t = ln( and u =. In these varables, the orgnal Rccat equaton takes the followng ntegrable form: du u u 0 dt + = ( Fg. (a. Case N=50. Then we, get the fnal soluton of equaton ( s = and =. Second: Classcal Soluton Here we usng gven ntal condton ( =. If = s a partcular soluton, then = u + s the general soluton. Satsfes the Bernoull equaton 4 + = u u u. On the natural wa, denote further lnear equaton 4 vc v = v = u to obtan the Fg. (b. Case N=00. Now we derve the soluton of the lnear equaton v (c = c c = cons tan ts So the soluton s + c = c ( +, f c = 0 then the general soluton =. Fgure (a gve the behavor of the soluton usng RK4, Forward Euler and Le group technque. We notce that there s a bg error occurs usng forward Euler. Moreover Fgure Fg. (c. Case N=00.

5 Appled and Computatonal Mathematcs 06; 5(: Fg. (d. Case N=900. Fg.. Soluton and error of frst order ode. Table. Error between RK4 and Euler forward. N ε Euler.5E- 6.6E-.7E-.6E-.E-.0E- 9.6E-4 8E-4 7E-4 ε 9.9E-7 5.5E E-0.74E E-.8E-.44E-.E- 6.4E- RK4 We see for even a moderate number of steps, the agreement between the Runge-Kutta method and the analtc soluton s remarkable. We can quantf just how much better the Runge-Kutta stencl does b defnng a measure of the global error as the k ε k ( 0, N, Method = Method ( 0N LG = Where the method {Euler, RK4}, k =,,. Clearl, the error for the Runge-Kutta method s several orders of magntude lower. Furthermore, the global error curves both look lnear on the log-log plot, whch suggests that there s a power law dependence of on N : ε ~ ε N 0 α α αh Eample : Consder the Rccat equaton = (4 correspondng to the choce R( =,Q( =,P( =. In equaton ( Show that the transformaton X = s a one parameter Le group of scalng smmetres for ths Rccat equaton. Use the one parameter Le group to show that t has two nvarant solutons = ± Use the method of characterstcs to solve the defnng Eqs. ( for canoncal coordnates. Show that (t,u = (,ln s a soluton for 0. Show that n these canoncal coordnates the Rccat Eq. (4 reduces to du = dt t Integrate the last equaton to show that n the orgnal (, coordnates, du = dt t u + c = arctan(t where c real parameter wll be determnate usng ntal condton Substtute (t,u = (,ln, we get ln + c = tanh ( ln + c = tanh ( For dfferent to zero we have tanh ( ln + c = = tanh ln + c Usng (0. =, we can determne the value of c

6 69 Sam H. Altoum and Salh Y. Arbab: Comparson Solutons Between Le Group Method and Numercal Soluton of (RK4 for Rccat Dfferental Equaton tanh ( ln + c = (0. tanh ( ln + c = (0. = 0.0 ln + c = arctan h( 0.0 c = arctan( 0.0 ln 0 usng arctan( = arctan( we, get Fnall the soluton s c = ln(0 + arctan 50. tanh ln( + ln(0 arctan h 50 = Fg. (c. Soluton RK4 and Le Group case N=00. Fg. (a. Soluton RK4 and Le Group case N=50. Fg. (d. Soluton RK4 and Le Group case N=900. Fg. (b. Soluton RK4 and Le Group case N=00. Fg.. Global Error.

7 Appled and Computatonal Mathematcs 06; 5(: Table. Error between RK4 and Euler forward. N ε.7e-.e- 4.E-.E-.5E-.E-.8E-.6E-.4E- Euler ε 67.98E-6.4E-7.90E-9.E-9 4.5E-0,4E-0.E E- 4.48E- RK4 Eample Consder the Bernoull equaton ' e = +, when substtuted nto condton (0, leads to ( ( ( ( ( η ξ + e + η ξ + e ξ e + η e = 0 Ths, agan, s too dffcult as t sts, so we tr a few smplfng assumptons before we dscover that ξ =, η = η ( elds η + e + e η e = 0 Because some terms depend onl on, we solve η η = 0 to obtan η = c. Insertng ths form of nto the remanng equaton η + η = 0, we arrve at η =. Now that we have settled on the smbols ( ξ, η = (,, we fnd canoncal coordnates b solvng d η = = to get d ξ u and t. Remember that we seek famles of functons that reman constant for u so u = c = e. The second d coordnate s s found b ntegratng du = to get u =. The net step s to fnd the dfferental equaton n the canoncal coordnates b computng; We learn that du u + u h =. dt t + t h du = = dt e + e ( + e e + e. ( c e e = Now from ntal condton (0. = we, get Fnall we, get = c e e = 4 e = e 0. c = 0. e = + e 0.c 0. e = e + e c ( c = ln e + e c = ln e ( + e c = ln + e 5 0. c 5 0. e = e + e = e e e ( + e e e e + e + e e 5 e Epressng n t +, whence t Ths ntegrates to e e + terms of du t = dt t + t and u leads to t u = ln + + c Returnng to the orgnal coordnates, we obtan e = ln + + c Fg. (a. Soluton RK4 and Le Group case N=50.

8 7 Sam H. Altoum and Salh Y. Arbab: Comparson Solutons Between Le Group Method and Numercal Soluton of (RK4 for Rccat Dfferental Equaton Fg. (b. Soluton RK4 and Le Group case N=00. Fg.. Global Error. Fg. (c. Soluton RK4 and Le Group case N=00. Here, we compare the soluton of the Bernoull equaton obtaned usng Le group and the numercal soluton gven b RK4, see Fgure (a. We see for even a moderate number of steps, the agreement between the Runge- Kutta method and the analtc soluton s remarkable. We can quantf just how much better the Runge-Kutta method does b defnng a measure of the global error e as the magntude of the dscrepanc between the numercal and actual values of (, see Fgure (b. Clearl, the error for the Runge-Kutta method s several orders of magntude lower. Furthermore, the global error curves both look lnear on the log-log plot, whch suggests that there s a power law dependence of ε on N. We can determne the power a b fttng a power law to the data obtaned n our Maple code ths leads that the global error usng RK4 s O( h ths match our epectaton 5 that the one step error s O( h. 7. Conclusons In ths work we conclude that, frstl the general soluton of Rccat equaton s facng the problem of (Tral and Error and t s lead to the classcal method. Secondl we used Numercal soluton (Forward Euler, RK4 method and we fnd the Le group method s better than classcal method but t fal at the sngulart. The numercal soluton s accurac than smmetr group soluton and classcal method. In the last, we note that the Euler method are tested numercall and we have not convergence and the determned the global error s about O( h and ths don t match our epectaton. The graphs n ths have been performed usng Maple 8. References Fg. (d. Soluton RK4 and Le Group case N=900. [] W. T. Red, Rccat Dfferental Equatons (Mathematcs n scence and engneerng, New York: Academc Press, 97.

9 Appled and Computatonal Mathematcs 06; 5(: [] F. Dubos, A. Sad, Uncondtonall Stable Scheme for Rccat Equaton, ESAIM Proceedng. 8(000, 9-5. [] A. A. Bahnasaw, M. A. El-Tawl and A. Abdel-Nab, Solvng Rccat Equaton usng Adomans Decomposton Method, App. Math. Comput. 57(007, [4] T. Allahvraloo, Sh. S. Bahzad. Applcaton of Iteratve Methods for Solvng General Rccat Equaton, Int. J. Industral Mathematcs, Vol. 4, ( 0 No. 4, IJIM [5] Supra Mukherjee, Banamal Ro. Soluton of Rccat Equaton wth Varable Co-effcent b Dfferental Transform Method, Int. J. of Nonlnear Scence Vol.4, (0 No., pp [6] Tawo, O. A., Oslagun J. A. Appromate Soluton of Generalzed Rccat Dfferental Equaton b Iteratve Decomposton Algorthm, Internatonal Journal of Engneerng and Innvatve Technolog(IJEIT Vol. (0 No., pp [7] J. Bazar, M. Eslam. Dfferental Transform Method for Quadratc Rccat Dfferental Equaton, vol. 9 (00 No.4, pp [8] Crstnel Mortc. The Method of the Varaton of Constants for Rccat Equatons, General Mathematcs, Vol. 6(008 No., pp. -6. [9] B. Gbadamos, O. adebmpe, E. I. Aknola, I. A. I. Olopade. Solvng Rccat Equaton usng Adoman Decomposton Method, Internatonal Journal of Pure and Appled Mathematcs, Vol. 78(0 No., pp [0] Olever. P. J. Applcaton of Le Groups to Dfferental Equatons. New York Sprnger-Verlag, (99. [] Al Fred Gran. Modern Dfferental Geometr of Curves and Surfaces, CRC Press, (998. [] Aubn Therr. Dfferental Geometr, Amercan Mathematcal Socet, (00. [] Nal. H. Ibragmov, Elementr Le Group Analss and Ordnar Dfferental Equatons, John Wle Sons New York, (996. [4] T. R. Ramesh Rao, "The use of the A doman Decomposton Method for Solvng Generalzed Rccat Dfferental Equatons" Proceedngs of the 6 th IMT-GT Conference on Mathematcs, Statstcs and ts Applcatons (ICMSA00 Unverst Tunku Abdul Rahman, Kuala Lumpur, Malasa pp [5] B. Batha, M. S. M. Nooran and I. Hashm, " Applcaton of Varatonal Iteraton Method to a General Rccat Equaton" Internatonal Mathematcal Forum, Vol., no. 56, pp , 007.