Abstract
The study of partial differential wave equations arising from shallow water theory has been a discipline of augmenting passion in disparate fields of mathematics and physics. One of the acclaimed models is named as Korteweg de Vries (KdV) equation, which was first imported by Boussinesq in around 1877 and again derived by Korteweg and de Vires in around 1895. Herein, a numerical spectral technique to obtain an approximate solution of nonlinear KdV partial differential equation is implemented. A certain combination of the shifted Chebyshev polynomials of the first kind is used as basis functions. The main idea behind the proposed technique is established on converting the governed boundary-value problem into a system of nonlinear algebraic equations via the application of the spectral tau method. The resulting nonlinear system can be efficiently solved by expedients of Newton’s procedure. The convergence and error analysis of the Tchebyshev expansion are carefully investigated. Finally, a numerical example is given to demonstrate the applicability and accuracy of the suggested scheme.
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Bibliography
Abdelhakem M, Ahmed A, El-Kady M (2021) Spectral monic Chebyshev approximation for higher order differential equations. Math Sci Lett 8(2):11–17
Abd-Elhameed WM, Machado JAT, Youssri YH (2021) Hypergeometric fractional derivatives formula of shifted Chebyshev polynomials: tau algorithm for a type of fractional delay differential equations. Int J Nonlinear Sci Numer Simul 10:1–21
Abrahamsen D, Fornberg B (2021) Solving the Korteweg-de Vries equation with Hermite-based finite differences. Appl Math Comput 401:126101
Abramowitz M, Stegun IA (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables, vol 55. U.S. Government Printing Office, Washington, DC
Abumaryam S (2018) The convergence of the approximated derivative function by Chebyshev polynomials. IOSR J Math 14(2):05–11
Ahmadian A, Salahshour S, Chan CS (2016) Fractional differential systems: a fuzzy solution based on operational matrix of shifted Chebyshev polynomials and its applications. IEEE Trans Fuzzy Syst 25(1):218–236
Aksan EN, Özdeş A (2006) Numerical solution of Korteweg-de Vries equation by Galerkin B-spline finite element method. Appl Math Comput 175(2):1256–1265
Andrews GE, Askey R, Roy R (1999) Special functions, number 71. Cambridge University Press, Cambridge, UK
Atta AG, Moatimid GM, Youssri YH (2019) Generalized Fibonacci operational collocation approach for fractional initial value problems. Int J Appl Comput Math 5(1):1–11
Atta AG, Moatimid GM, Youssri YH (2020) Generalized Fibonacci operational tau algorithm for fractional Bagley-Torvik equation. Prog Fract Differ Appl 6:215–224
Atta AG, Abd-Elhameed WM, Moatimid GM, Youssri YH (2021) Shifted fifth-kind Chebyshev Galerkin treatment for linear hyperbolic first-order partial differential equations. Appl Numer Math 167:237–256
Başhan A (2021) Modification of Quintic B-spline differential quadrature method to nonlinear Korteweg-de Vries equation and numerical experiments. Appl Numer Math 167:356–374
Canuto C, Hussaini MY, Quarteroni A, Zhang TA (2012) Spectral methods in fluid dynamics. Springer, New York
Cao J, Chen Y, Wang Y, Cheng G, Barrière T, Wang L (2021) Numerical analysis of fractional viscoelastic column based on shifted Chebyshev wavelet function. Appl Math Model 91:374–389
Coffey MW (1996) Nonlinear dynamics of vortices in ultraclean type-II superconductors: integrable wave equations in cylindrical geometry. Phys Rev B 54(2):1279
Das GC, Sarma J (1999) Response to “comment on ‘a new mathematical approach for finding the solitary waves in dusty plasma’” [Phys Plasmas 6, 4392 (1999)]. Phys Plasmas 6(11):4394–4397
Dehghan M, Shokri A (2007) A numerical method for KdV equation using collocation and radial basis functions. Nonlinear Dyn 50(1):111–120
Doha EH (1991) The coefficients of differentiated expansions and derivatives of ultraspherical polynomials. Comput Math Appl 21(2–3):115–122
Doha EH, Abd-Elhameed WM, Youssri YH (2019) Fully Legendre spectral Galerkin algorithm for solving linear one-dimensional telegraph type equation. Int J Comput Methods 16(8):1850118
El Bahi MI, Hilal K (2021) Lie symmetry analysis, exact solutions, and conservation laws for the generalized time-fractional KdV-like equation. J Funct Spaces 2021:6628130
Finlayson BA (2013) The method of weighted residuals and variational principles. SIAM, Philadelphia
Giordano C, Laforgia A (2003) On the Bernstein-type inequalities for ultraspherical polynomials. J Comput Appl Math 153(1–2):243–248
Gottlieb D, Orszag SA (1977) Numerical analysis of spectral methods: theory and applications. SIAM, Philadelphia
Kong D, Xu Y, Zheng Z (2019) A hybrid numerical method for the KdV equation by finite difference and sinc collocation method. Appl Math Comput 355:61–72
Ludu A, Draayer JP (1998) Nonlinear modes of liquid drops as solitary waves. Phys Rev Lett 80(10):2125
Luke YL (1969) Special functions and their approximations, vol 2. Academic Press, London
Mason JC, Handscomb DC (2002) Chebyshev polynomials. CRC Press, Boca Raton
Osborne AR (1995) The inverse scattering transform: tools for the nonlinear Fourier analysis and filtering of ocean surface waves. Chaos, Solitons Fractals 5(12):2623–2637
Pandir Y, Sahragül E (2021) Exact solutions of the two dimensional KdV-Burger equation by generalized Kudryashov method. J Inst Sci Technol 11(1):617–624
Pourbabaee M, Saadatmandi A (2021) Collocation method based on Chebyshev polynomials for solving distributed order fractional differential equations. Comput Methods Differ Equ 9(3):858–873
Rehman MFU, Gu Y, Yuan W (2021) Exact analytical solutions of generalized fifth-order KdV equation by the extended complex method. J Funct Spaces 2021:5549288
Shen J, Tang T, Wang L (2011) Spectral methods: algorithms, analysis and applications, vol 41. Springer, Berlin
Singh K, Gupta RK (2006) Lie symmetries and exact solutions of a new generalized Hirota–Satsuma coupled KdV system with variable coefficients. Int J Eng Sci 44(3–4):241–255
Türk Ö, Codina R (2019) Chebyshev spectral collocation method approximations of the stokes eigenvalue problem based on penalty techniques. Appl Numer Math 145:188–200
Youssri YH, Abd-Elhameed WM (2018) Numerical spectral Legendre-Galerkin algorithm for solving time fractional telegraph equation. Rom J Physiol 63(107):1–16
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Youssri, Y.H., Atta, A.G. (2022). Double Tchebyshev Spectral Tau Algorithm for Solving KdV Equation, with Soliton Application. In: Helal, M.A. (eds) Solitons. Encyclopedia of Complexity and Systems Science Series. Springer, New York, NY. https://doi.org/10.1007/978-1-0716-2457-9_771
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