## THERMAL SCIENCE

International Scientific Journal

### CHEBYSHEV WAVELET COLLOCATION METHOD FOR GINZBURG-LANDAU EQUATION

**ABSTRACT**

The main aim of this paper is to investigate the efficient Chebyshev wavelet collocation method for Ginzburg-Landau equation. The basic idea of this method is to have the approximation of Chebyshev wavelet series of a non-linear partial differential equation. We demonstrate how to use the method for the numerical solution of the Ginzburg-Landau Equation with initial and boundary conditions. For this purpose, we have obtained operational matrix for Chebyshev wavelets. By applying this technique in Ginzburg-Landau equation, the partial differential equation is converted into an algebraic system of non-linear equations and this system has been solved using Maple computer algebra system. We demonstrate the validity and applicability of this technique which has been clarified by using an example. Exact solution is compared with an approximate solution. Moreover, Chebyshev wavelet collocation method is found to be acceptable, efficient, accurate and computationall for the non-linear or linear partial differential equation.

**KEYWORDS**

PAPER SUBMITTED: 2018-09-20

PAPER REVISED: 2018-10-17

PAPER ACCEPTED: 2018-10-30

PUBLISHED ONLINE: 2018-12-16

**THERMAL SCIENCE** YEAR

**2019**, VOLUME

**23**, ISSUE

**Supplement 1**, PAGES [S57 - S65]

- Daubechies, I. Ten Lectures on Wavelet, SIAM, (1992), Philadelphia, PA
- Razzaghi,M., Yousefi,S., Legendre wavelets operational matrix of integration, International Journal of System Science, (2001), 32(4):495-502
- Babolian,E., Fattahzadeh,F., Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration, Applied Mathematics and Computation , (2007), 188:417-426
- Çelik,İ., Numerical solution of differential equations by using Chebyshev wavelet collocation method, Çankaya University Journal of Science and Engineering ,(2013), 10(2): 169-184
- Fox,L., Parker,I.B., Chebyshev Polynomials in Numerical Analysis, (1968), Oxford University Press, London
- Yuanlu, L.I., Solving a nonlinear fractional differential equation using Chebyshev wavelets, Commun Nonlinear Sci Numer Simulat, (2010), 2284-2292
- Hooshmandasi, M.R., et al., Numerical Solution of the One-Dimention Heat Equation by Using Chebyshev Wavelets Method, Applied &Computational Mathematics, (2012), 2168-9676
- Heydari, M.H. et al., A new approach of the Chebyshev wavelets method for partial differential equations with boundary conditions of the telegraph type, Applied Mathematical Modelling, (2014), 1597-1606
- Çelik,İ., Chebyshev Wavelet collocation method for solving generalized Burgers-Huxley equation, Mathematical Methods in the Applied Sciences, (2016), 39.3: 366-377
- Çelik,İ., Gokmen,G., Approximate solution of periodic Sturm - Liouville problems with Chebyshev collocation method, Applied Mathematics and Computation, (2005), 170.1: 285-295
- IQBAL, M.A., ALI, A., et al., Chebyshev wavelets method for fractional delay differential equations, International Journal of Modern Applied Physics, (2013), 4.1: 49-61
- ALI, A., IQBAL, M.A., et al., Chebyshev wavelets method for boundary value problems, Scientific Research and Essays, (2013), 8.46: 2235-2241
- Kilicman, A., Zeyad A.A.,Al Zhour., Kronecker operational matrices for fractional calculus and some applications, Applied Mathematics and Computation 187.1 (2007): 250-265
- Inc, M., Bildik, N., Non-perturbative solution of the Ginzburg-Landau equation, Mathematical and Computational Applications 5.2 (2000): 113-117
- Gabbay M., Edward D., Guzdar N. P., Reconnection of Vortex Filaments in the Complex Ginzburg - Landau Equation, Physical Review E, (1997), 2576 p.
- Saarloos W., The Complex Ginzburg Landau Equation for Beginners, (1994), 1-2 p.
- Lepik U., Application of the Haar wavelet transform to solving integral and differential equations, Proc Estonian Acad Sci Phys Math, (2007), 56(1):28-46
- Islam S, Aziz I, Sarler B., The numerical solution of second-order boundary value problems by collocation method with the Haar wavelets, Math Comput Model, (2010), 52:1577-90
- Bujurke NM, Salimath CS, Shiralashetti SC., Numerical solution of stiff systems from nonlinear dynamics using single-term Haar wavelet series, Nonlinear Dyn, (2008), 51:595-605
- Bujurke NM, Salimath CS, Shiralashetti SC., Computation of eigenvalues and solutions of regular Sturm-Liouville problems using Haar wavelets, J Comput Appl Math, (2008), 219:90-101
- Bujurke NM, Shiralashetti SC, Salimath CS., An application of single-term Haar wavelet series in the solution of nonlinear oscillator equations, J Comput Appl Math, (2009), 227:234-44
- Shiralashetti SC, Deshi AB., An efficient Haar wavelet collocation method for the numerical solution of multi-term fractional differential equations, Nonlinear Dyn, (2016), 83:293-303
- Li, Meng, Chengming Huang, and Nan Wang.,Galerkin finite element method for the nonlinear fractional Ginzburg-Landau equation, Applied Numerical Mathematics 118 (2017), 131-149
- Arnous, Ahmed H., et al. Optical solitons with complex Ginzburg-Landau equation by modified simple equation method, Optik-International Journal for Light and Electron Optics, 144 (2017), 475-480
- Karabacak, M., Çelik E., The Numerical Solution of Fractional Differential-Algebraic Equation (FDAEs) by Haar Wavelet Functions, Int. J. Eng. Appl. Sci., (2015), 2(10):65-71