Numerical Solution of Deformation Equations. in Homotopy Analysis Method

Transcription

1 Appled Mathematcal Scences, Vol. 6, 2012, no. 8, Nmercal Solton of Deformaton Eqatons n Homotopy Analyss Method J. Izadan and M. MohammadzadeAttar Department of Mathematcs, Faclty of Scences, Mashhad Branch, Islamc Azad Unversty, Mashhad, Iran M. Jall Department of Mathematcs, Neyshabr Branch, Islamc Azad Unversty, Neyshabr, Iran Jall.maryam@yahoo.com Abstract In ths paper, we consder the homtopy analyss method (HAM) for solvng nonlnear ordnary dfferental eqaton wth bondary condtons. The partal sm of solton seres s determned sng fnte dfference method and spectral method. These methods are sed for solvng deformaton eqaton. For mprovng the rate of convergence, we apply the methods of Marnca and N-Wang. The reslts are compared wth fnte dfference method and spectral method. Unlke HAM, these methods don t need convergence control parameter. Nmercal eperments show effcency and performance of proposed methods. Keywords: Homotopy analyss method, Deformaton eqaton, Spectral method, Fnte dfference method 1 Introdcton Ordnary dfferental eqatons wth bondary condtons (BVP) are very mportant n varos domans of scences and engneerng. In most cases, t s very complcated to acheve analytc soltons of these eqatons. Nmercal fnte dfference methods and spectral methods can be sed for solvng certan types of these problems wth applyng nonlnear eqaton solvers as Newton's methods

2 358 J. Izadan, M. MohammadzadeAttar and M. Jall [7,8], bt these technqes are vald only for weakly nonlnear problems. Recently, homotopy analyss method (HAM) has been sccessflly appled to varos types of these problems. In 2003, Lao pblshed the book [1] n whch he smmarzed the basc deas of the homotopy analyss method and gave the detals of hs approach both n the theory and on a large nmber of practcal eamples and n a seres of papers [2-4,9], he developed HAM. Homotopy seres solton s depended to convergence control parameter. For nstance Lao has sed the fnctonal seres that ther coeffcents are the control parameters. These parameters can be sed for acceleratng the convergence of soltons seres, or fndng a homotopy seres that s convergent for certan vale of homotopy parameters. Some athors lke Marnca et al. [5], Yabshta et al. [10], and N et al. [6] have consdered the optmzaton convergence control parameters that can be obtaned by mnmzng the sqare resdal error. In ths paper, the known methods lke fnte dfference method and spectral method are appled to solvng deformaton eqaton, net, Marnca, and N-Wang methods are sed to mnmze the task of determnng the solton, and these two methods are compared by presentng some nmercal eperments. Ths paper s organzed as follows. In secton 2 the basc dea of HAM s presented. In the secton 3 the applcaton of fnte dfference and Chebyshev collocaton method for solvng deformaton eqaton s descrbed. In secton 4 Marnca and N-Wang methods are combned wth nmercal methods of secton 3, net n the secton 5 the nmercal reslts are presented. The reslts of Marnca and N- Wang methods are compared for two bondary vale problems. Fnally, we dscss and analyse the reslts. 2 The basc dea of HAM for solvng BVP We consder the followng bondary vale problems:,,,,,, where s a nonlnear dfferental operator of second order, s ndependent varable and s an nknown fncton. For ths problem the homotopy eqaton can be wrtten as follows :,,, where,,, s an embeddng parameter, s an ntal gess of, s lnear dfferental operator of second order, and, s the homotopy seres,

3 Nmercal solton of deformaton eqatons 359 That s assmed to be convergent on [0,1]. It easly dedces that!,. (3) When 1, t holds,, (4) that s a solton of (2), and conseqently the solton of (1), see [1]. The ntal gess satsfes two bondary condtons of (1), see [1,2,4]. Referrng the homotopy lteratre [1-4], one can fnd the deformaton eqatons of order m s fond as follows: where and, m 1, (5)!,,,,,, 1, Then, the eqaton (5) s a second order lnear ODEs wth bondary condtons,, By acceptng s a solton of,,,, The homotopy seres solton (4) can be determned step by step by solvng followng lnear problem :, (6) In fact, one can only comptes partal sm of seres solton. However, n general these eqatons are solved analytcally or wth pade homotopy method [1]. In these methods the control parameters are sed to garantee convergence of homotopy seres, for more nformaton, the reader s refered the approprate lteratre. Ths partal sm can be determned nmercally by solvng (6), wth a proper nmercal method. Here the fnte dfference method and the spectral method are appled to solve deformaton eqaton.

4 360 J. Izadan, M. MohammadzadeAttar and M. Jall 3 Nmercal solton of deformaton eqaton 3.1 Fnte dfference method Consder a partton of [a,b], as follows:,,,,...,, and take (for smplcty), by omttng the trncaton error, eqaton (6) yelds :,,,,, (7) where,,,,,,, Now by applyng bondary condtons, we have a lnear system of eqatons that can be solved by one of sted methods. Then we have an appromaton of on [a,b]. By repeatng M-tmes ths procedre, fnally we obtan an appromaton of partal sm of seres solton: where,. 3.2 Spectral method, Another method that can be sed for solvng deformaton eqatons, s spectral method wth Chebyshev ponts. Here, we consder an appromaton of, that s solton of deformaton eqaton (5) as gven:, (8) where s Lagrange polynomals, and, 0,1,2,, are Chebyshev ponts [8]. If we denote, for =0,1,2,,n and, then (8) can be wrtten as:, By twce dfferentaton the eqaton (9) wth respect to, we have,

5 Nmercal solton of deformaton eqatons 361. Or by sng the spectral dfferentaton matr D = ( d j ) n+ 1, n+ 1 see[7,8], we have,,,,,,. By sbstttng these dervatves n deformaton eqaton, a lnear system of (n+1) eqatons wth (n-1) nknowns,,,,, s obtaned:, (10),,, The ntal solton s consdered as followng :,,, that can be easly transformed to [-1,1] by a lnear transformaton to se Chebyshev ponts. For, the bondary condtons are sed. Then (7) s redced to a lnear system wth n-1 nknowns that can be easly solved. One knows that the coeffcent matr of ths system s not sparse, however regardng the spectral precson of method ths matr can be of an acceptable order [8]. 4 Fndng optmal control parameters To have a solton seres that s convergent for q=1, sometmes we need to apply convergence control parameters. In sch cases sng the so called h-crve can be sefl for determnng the convergence control parameter [4]. However there are the methods that allow s to fnd the control parameter by a mnmzaton process. Here we apply two methods to compte comptng optmal control parameters de to Marnca and N-Wang [6]. For ths prpose one chooses the followng zero-th order deformaton eqaton (or homotopy eqaton): where,,,.

6 362 J. Izadan, M. MohammadzadeAttar and M. Jall Then the homotopy solton s the fncton of,, we can determne, sch that,, can be appromated by a partal sm of small order (,,,,. For determnng more precse appromate solton one note that for the eact solton (),the followng eqatons s satsfed:,,,,,, Then to fnd of a good appromate solton, each mst become very small. Ths we defne as [6],the followng fncton,,,, (11) Where s M-th partal sm of seres solton. In method of Marnca, one mnmze (11) to fnd an optmal appromaton of at end of M-th step of comptng, =1,2,,M n method of N-wang n every step one mnmzes a fncton of one varable that s defned by,,,,, where,,, have been determned, n prevos step. For more nformaton see [4,6]. These two methods are appled for two nonlnear eamples and ther reslts are compared n the net secton. 5 Nmercal eperments In ths secton we present the nmercal reslts for fnte dfference method and spectral method, then Marnca method and N-Wang method are appled to two bondary vale problems, Compter codes have been prepared by sng MATLAB 7.6.(2008) wth a PC of model Pentm IV. Denotes and are appromate solton and eact solton n, respectvely, and n N ( % ) = ( ( N( ( )) ) = % Eample Followng lnear dfferental eqaton s consdered :,,, sbject to the bondary condtons (1),, wth.

7 Nmercal solton of deformaton eqatons 363 We know that the eact solton s, the program has eected wth M=20 and n=40. Nmercal reslts are shown n table 5.1 and 5.2. The graph of solton s presented n fg Table 5.1. The reslts of ftnte dfference method for M=20 and n= CPU Tme 2.03 s Table 5.2. The reslts of spectral method for M=20 and n= CPU Tme s Fg 5.1. The reslts of fnte dfference method for m=20 and n= Eample We consder a nonlnear dfferental eqaton: sbject to bondary condtons,

8 364 J. Izadan, M. MohammadzadeAttar and M. Jall The eact solton s nknown. For M=20 and n=40, nmercal reslts are presented n table 5.2 and 5.3. The graph of solton s presented n fg Table 5.3. The reslts of fnte dfference method for M=20 and n= CPU Tme 1.67 s Table 5.4. The reslts of spectral method for M=20 and n= CPU Tme s Fg 5.2. The reslt of fnte dfference method for M=20 and n= Eample In ths part the reslts of the two prevos eamples sng spectral method and N-Wang method s presented n tables 5.5 and 5.6, then the comparson of these methods are accomplshed n tables 5.8.

9 Nmercal solton of deformaton eqatons 365 Table 5.5 The reslts of Marnca method for eample(1) M=4 and n= CPU Tme s Ma of relatve error 10 Table 5.6. The reslts of N-Wang method eample(1) M=4 and n= CPU Tme s Ma of relatve error 10 table 5.7. The reslts of Marnca method for eample (2) M=7 and n= CPU Tme Table 5.8. The reslts of N-Wang method for for eample (2) M=7 and n= CPU Tme s

10 366 J. Izadan, M. MohammadzadeAttar and M. Jall Table 5.9. Comparson N Wang and Marnca reslts for eample (1) n=20and m=1,2,,10. Marnca N Wang M CPU Tme CPU Tme / / table 5.10 Comparson N Wang and Marnca reslts for eample(2) n=20 and m=1,2,,10. Marnca N Wang M CPU Tme CPU Tme () n table 5.9 denote that eecton of program was not sccessfl becase of memory overflow. 6 Conclson and Dscsson In ths paper, HAM s mplemented to solve the nonlnear ODE s wth bondary condtons. For solvng deformaton eqaton, we appled fnte dfference method and spectral method. The reslts show that the spectral method s more accrate, bt t s tme consmng. The fnte dfference reslts are satsfactory. The comparson of methods Marnca and N-Wang show that the latter s more rapd, and more precse n majorty of cases, bt for certan problem s not stable. Marnca method s tme consmng bt t s more precse.

11 Nmercal solton of deformaton eqatons 367 References [1] Lao S.J., Beyond pertrbaton: ntrodcton to the homotopy analyss method. Boca Raton: Chapman & Hall / CRC Press; (2003). [2] Lao S.J., On The Homotopy Analyss method for Non-Lnear Problems, Appl. Math. Compt. 147 (2004) [3] Lao S.J., Tan Y., A general approach to obtan seres soltons of nonlnear dfferental eqatons. Stdes n Appled Mathematcs, 119 (2007) [4] Lao S.J., Notes on the homotopy analyss method: Some defntons and theorems, Commn. Nonlnear Sc. Nmer. Smlat. 14 (2009) [5] Marnca V., Hersan N., et al., An optmal homotopy asymptotc method appled to the steady flow of a forth-grade fld past a poros plate, Appled Mathematcs Letters, 22 (2009) [6] N Zh., Wang, Ch., A One-Step Optmal Homotopy Analyss Method for Nonlnear Dfferental Eqaton, Commn. Nonlnear Sc. Nmer. Smlat. 1 (2010) [7] Stoer J., Bblrsch R., Introdcton to Nmercal Analyss, Sprnger Verlag, (1983). [8] Trefethen N., Spectral methods n Matlab SIAM ed., Phladelpha, (2000). [9] W YY., Cheng KF., Homotopy solton for nonlnear dfferental eqatons n wave propagaton problems. Wave Moton 46 (2009) [10] Yabshta K., Yamashta M., Tsbo K., An analytc solton of projectle moton wth the qadratc resstance law sng the homotopy analyss method, J. Phys. A: Math. Theor. 40 (2007) Receved: Jne, 2011