CHEBYSHEV WAVELET COLLOCATION METHOD FOR GINZBURG-LANDAU EQUATION

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1 CHEBYSHEV WAVELE COLLOCAION MEHOD FOR GINZBURG-LANDAU EQUAION Aydi SECER * ad Yasemi BAKIR *, Yildiz echical Uiversity, Faculty of Chemistry-Metallurgical, Departmet of Mathematical Egieerig, Davutpaşa Campus, 340, İstabul/URKEY * Correspodig author; aydisecer@gmail.com he mai aim of this paper is to ivestigate the efficiet Chebyshev wavelet collocatio method for Gizburg-Ladau equatio. he basic idea of this method is to have the approximatio of Chebyshev wavelet series of a oliear partial differetial equatio. We demostrate how to use the method for the umerical solutio of the Gizburg-Ladau Equatio with iitial ad boudary coditios. For this purpose, we have obtaied operatioal matrix for Chebyshev wavelets. By applyig this techique i Gizburg-Ladau equatio, the partial differetial equatio is coverted ito a algebraic system of o-liear equatios ad this system has bee solved usig Maple computer algebra system. We demostrate the validity ad applicability of this techique which has bee clarified by usig a example. Exact solutio is compared with a approximate solutio. Moreover, Chebyshev wavelet collocatio method is foud to be acceptable, efficiet, accurate ad computatioall for the o-liear or liear partial differetial equatio. Key Words: Chebyshev wavelet collocatio method, Gizburg-Ladau Equatio, operatioal matrices, No-liear PDE.. Itroductio Whe viewed from a historical perspective, wavelet aalysis is a ew method, although based o the work of the 9th cetury mathematical foudatio Joseph Fourier. Fourier has laid the foudatios for the theory of frequecy aalysis ad proved to be very importat ad effective. his approach is suitable for sigal filterig ad compressio with time-depedet waveform-like properties. However, it may ot be suitable for fuctios with regioal characteristics. I other words, series of Fourier gives oly frequecy resolutio but ot time resolutio. Although we ca idetify all the frequecies i a sigal, we ca ot determie whe those frequecies are. herefore, the researchers' iterest has become icreasigly scale-based from frequecy-based aalysis. he cocept of wavelet was first dealt with i 909 by Alfred Haar s master thesis. Sice that time, Wavelet aalysis methods have bee developed. After that, research o wavelets has reached iteratioal dimesio by Y.Meyer ad colleagues. he mai algorithm is based o the work of Stephae Mallat i 988. At the begiig of this researches are Igrid Daubechies, Roald Coifma ad Victor Wickerhauser, who pioeered others. he workig ad applicatio areas of wavelets are growig very rapidly. I geeral, wavelets have bee very used successful i may differet fields of sciece ad egieerig such as thermal sciece, sigal aalysis, data compressio, seismology. Wavelets ca be

2 used for algebraic maipulatios i the system of equatios obtaied which leads to better resultig system. For this reaso, wavelet methods have become a more preferred umerical method today. Wavelets have may differet families. hese are the most familiar ad simple Haar wavelets. Haar wavelets are used by may researchers because of their simplicity [7-]. he disadvatage of usig Haar's fuctios is the low accuracy of the umerical approach. herefore, it is thought that smooth Chebyshev wavelets have a more accurate approach to Haar wavelets [3]. Chebyshev polyomials ad their properties are used to obtai the geeral procedure for the formatio of matrices. he, operatioal matrices of Chebyshev wavelet expasios are applied for the solutio of o-liear partial differetial equatios. Due to their smoothess ad good iterpolatio properties, the correctess of Chebyshev wavelets is better tha that of Haar wavelets. ad therefore we use the chebyshev wavelet collocatio method to solve Gizburg-Ladau equatio. he complex Gizburg-Ladau equatio is kow as the first model of the amplitude equatios ad is give by: A A i A A i A () t I the complex Gizburg Ladau equatio, the ad parameters are real parameters, ad A x, t is a complex scalar domai called the order parameter. he qualitative dyamic behavioral solutios of equality chage with coefficiets ad. hese coefficiets ca be obtaied from the equatios foud at the ed as a result of exhaustive calculatios for a give system. For large values of ad, the complex Gizburg Ladau equatio is reduced to the oliear Schörediger equatio (Gabbay, 997). I recet years, the dyamics outside such boudaries have bee workig itesively i theory. he complex Gizburg Ladau equatio shows a broadly valid behavior. he equatio i two dimesios has spatial wave solutios ad is perhaps the simplest equatio solved i this form. he time depedet Gizburg Ladau heory is used to determie the o - equilibrium properties of supercoductors. More elemetary ad detailed itroductios to the cocepts uderlyig the equatio ca be foud i Maeville (990), va Saarloos (995), va Hecke et al. (994), Nicolis (995), ad Walgraaf (997). Lately, the Gizburg-Ladau Equatio has bee umerically solved by usig Galerki fiite elemet method [3], modified simple equatio method [4] ad decompositio method [4]. I this paper, we have discussed the Chebyshev wavelet collocatio method ad we have solved the Gizburg-Ladau equatio ad examied the accuracy of the results.. Properties of Chebyshev Wavelets ad Its Operatioal Matrix of Itegratio Wavelets costitute a family of fuctios costructed from dilatio ad traslatio of a sigle fuctio called the mother wavelet x. Whe the dilatio parameter a ad the traslatio parameter b vary cotiuously, we have the followig family of cotiuous wavelets as [] x b ab, x a, a, b, a 0 a m, x k,, m, x, have four argumets, where k 0,,, ad,,, k, m is degree of Chebyshev polyomials of the first kid, ad x deotes the ormalized time. hey are defied o the iterval 0, as Chebyshev wavelets are writte as ()

3 m, k x, x m k x m k k 0, otherwise (3) where m 0 m (4) m,, ad m0,,.., M,,,, k. Here, m x are the well-kow Chebyshev polyomials of order m, which are orthogoal with respect to the weight fuctio W x x o the iterval,, ad satisfy the followig recursive formula: x x x x x x (5) 0 x,, m m m k We should ote that m x k are orthogoal with respect to the weight fuctio W x W x 0, are Chebyshev polyomials of the first kid of degree m that. A fuctio f x L W may be expaded ito Chebyshev wavelets basis as f x x (6) where, m, m m0, m, m x f x, (7).,. deotes the ier product with weight fuctio W (x) i equatio (7). rucated form of equatio (6) ca be writte as ad k M, m, m. (8) m0 ΚΨ f x x x Κ,,,,,,,,,, 0 M 0 M k k 0 M x x x x x x k x k x Ψ,,,,,,,,,, For simplicity, we write (9) ad 0 M 0 M 0 M i i0, i,, i, M, x,,, k Κ,,, k, i x i 0 x, i x,, i, M x (9) (0) for i,,, k () Moreover, a arbitrary fuctio of two variables ux, t defied o 0, 0, Chebyshev wavelet basis as: ij k k, Κ, ca be explaied ito M M ij i j () i j u x t x t t x where Κ ad x, ux, t, t. akig the collocatio poits ij i j i,,,, k ti i M k M k k we defie the M M matrix Φ as: k 3 M,,, k k k M M M (3)

4 We eed to kow how to calculate the itegral of the operatioal matrix i order to implemet our method. hat s why, Kilicma ad Al Zhour [3] ivestigated the geeralized itegral operatioal matrix. he itegral of the vector x ca be represeted as x d Px (4) 0 k k where P is the M M operatioal matrix of the oe-time itegral of x [3]. Moreover, Kilicma ad Al Zhour [3] geeralized itegral operatioal matrices P of -times itegratio of x ca showed as follows: x 0 0 x... d... d P x (5) times Çelik [4] expressed a uiform method to obtai the correspodig itegral operatioal matrix of differet basis. herefore, the operatioal matrix of ad i geerally, where B Ψx ca showed followig form: P P (6) B P P B (7) P is the operatioal matrix of -time itegral of the block pulse fuctios m 0 m PB 0 0 m3 m! where i i i i. 3. Numerical Method Let s assume ad are costat ad equal to zero i equatio (). We cosider the followig Gizburg Ladau equatio described by u 3 u u u u x t x t t x with iitial coditio ad boudary coditios ux,0 0 (0) u0, t 0 () u, t t () he exact solutio of this problem ux, t xt [4]. Now, let's solve this problem usig the Chebyshev wavelet collocatio method with k, M 3, x x, x, x, x, x, x (3) obtai ad we have 0 x 0 4x, 0 x 0, otherwise 0 0 x 4x, 0 x 0, otherwise (8) (9)

5 we assume x x 4x, 0 x 0, otherwise 4x 3, x 0, otherwise ij mm 3 0 x x 0 4x 3, x 0, otherwise 4x 3, x 0, otherwise u x Κ t (4) t x Where Κ is a ukow matrix which should be foud ad t is the vector that defied i (0). By itegratio of (4) oe time with respect to t ad cosiderig (0), we obtai: u ( x, t) x ΚP t u ( x,0) (5) xx Also by itegratio of (4) two times with respect to x we get: u ( x, t) x P Κ t u (0, t) xu (0, t) (6) by puttig x ito (6). ad (7) substitute i (6) t t tx, Κ 0, 0, t t xt xx u t P t u t u t (7) Κ,, 0, Κ u ( x, t) x P t u 0 t x u t u t P t (8) t t t t Now by itegratio of (6) oe times with respect to t we get, Κ Κ, (, ),0 0, 0,0,,0 0, 0, 0 u x t x P P t x P P t G x t G x t u x u t u x u t u u t u Now by replacig (5), (9) ad (8) ito (9); Κ Κ x P t x P t x x P ΚP t x P ΚP t xt x P ΚP t x P ΚP t xt. x P ΚP t x P ΚP t xt 3 x ΚP t x t x t (30) We have the graphic ad error table for the approximate solutio of this equatio for k, M 3 is as follows: (9)

6 Figure : Gizburg-Ladau Equatio Exact Solutio Figure : CWCM Approximate Solutio for k, M 3 Figure 3: CWCM Error Aalysis for k, M 3

7 able : Error able for k, M 3 x t u umeric u x t exact u umeric u exact

8 Similarly, We have the graphic ad error table for the approximate solutio of this equatio for k, M 4 is as follows: Figure 4: CWCM Approximate solutio for k, M 4 Figure 5: CWCM Error Aalysis for k, M 4 able : Error able for k, M 4

9 x t uumeric uexact x t uumeric uexact Coclusio I this paper, Chebyshev wavelet collocatio method is applied to the Gizburg Ladau equatio ad also the umerical results are compared with exact solutio of the equatio. As show from the figures ad tables. It is clear that its umerical solutio approaches the exact solutio. Moreover, the

10 implemetatio of this method is very suitable for solvig iitial ad boudary value problems. I additio, the greatest advatage of this method is small computatio costs ad its simplicity. Ackowledgemets he authors thak the referees for their costructive commets ad suggestios. We also thak the aoymous reviewers for their valuable suggestios ad commets. Refereces [] Daubechies, I. e Lectures o Wavelet, SIAM, (99), Philadelphia, PA [] Razzaghi,M., Yousefi,S., Legedre wavelets operatioal matrix of itegratio, Iteratioal Joural of System Sciece, (00), 3(4): [3] Babolia,E., Fattahzadeh,F., Numerical solutio of differetial equatios by usig Chebyshev wavelet operatioal matrix of itegratio, Applied Mathematics ad Computatio, (007), 88:47-46 [4] Çelik,İ., Numerical solutio of differetial equatios by usig Chebyshev wavelet collocatio method, Çakaya Uiversity Joural of Sciece ad Egieerig,(03), 0(): [5] Fox,L., Parker,I.B., Chebyshev Polyomials i Numerical Aalysis, (968), Oxford Uiversity Press, Lodo [6] Yualu, L.I., Solvig a oliear fractioal differetial equatio usig Chebyshev wavelets, Commu Noliear Sci Numer Simulat, (00), 84-9 [7] Hooshmadasi, M.R., et al., Numerical Solutio of the Oe-Dimetio Heat Equatio by Usig Chebyshev Wavelets Method, Applied &Computatioal Mathematics, (0), [8] Heydari, M.H. et al., A ew approach of the Chebyshev wavelets method for partial differetial equatios with boudary coditios of the telegraph type, Applied Mathematical Modellig, (04), [9] Çelik,İ., Chebyshev Wavelet collocatio method for solvig geeralized Burgers Huxley equatio, Mathematical Methods i the Applied Scieces, (06), 39.3: [0] Çelik,İ., Gokme,G., Approximate solutio of periodic Sturm Liouville problems with Chebyshev collocatio method, Applied Mathematics ad Computatio, (005), 70.: [] IQBAL, M.A., ALI, A., et al., Chebyshev wavelets method for fractioal delay differetial equatios, Iteratioal Joural of Moder Applied Physics, (03), 4.: 49-6 [] ALI, A., IQBAL, M.A., et al., Chebyshev wavelets method for boudary value problems, Scietific Research ad Essays, (03), 8.46: 35-4 [3] Kilicma, A., Zeyad A.A.,Al Zhour., Kroecker operatioal matrices for fractioal calculus ad some applicatios, Applied Mathematics ad Computatio 87. (007): [4] Ic, M., Bildik, N., No-perturbative solutio of the Gizburg-Ladau equatio, Mathematical ad Computatioal Applicatios 5. (000): 3-7

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