Chebyshev Wavelet Method for Numerical Solutions of Coupled Burgers  Equation

Oruç, Ömer; Bulut, Fatih; Esen, Alaattin

doi:10.15672/hjms.2018.642

Cited by 13 publications

(14 citation statements)

References 45 publications

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“…where Y m and y m denote the approximate and exact solutions to the unknown functions at m th knot, respectively. The computational outcomes are obtained and compared with those methods already available in literature like the Chebyshev spectral collocation method (CSCM) (Khater et al, 2008), Fourier pseudo-spectral method (FPM) (Rashid and Ismail, 2009), cubic B-spline collocation method (CBSCM) (Mittal and Arora, 2011), differential quadrature method (DQM) (Mittal and Jiwari, 2012), CBS-based differential quadrature method (CBS-DQM) (Mittal and Tripathi, 2014), quintic B-spline collocation method (QBSCM) (Raslan et al, 2017), Fourier expansion basis differential quadrature method (FEB-DQM) (Jima et al, 2018), Chebyshev wavelets method (CVM) (Oruç et al, 2019) and septic B-spline collocation method (SpBSM) (Shallal et al, 2019).…”

Section: Numerical Resultsmentioning

confidence: 99%

“…The error norms L 1 and L 2 for Example 3, corresponding to different choices of spatial grid size at t = 1, by setting Dt = 0.01 and l = 0.1,0.4, are listed in Table 7. In Tables 8 and 9, a comparison of computational outcomes with CVM (Oruç et al, 2019) is presented for different values of M at t = 3 and 5 using fixed Dt and l . It is revealed that our commuted results exhibit a better agreement with the analytical exact solution as compared to CVM.…”

Section: New Cubic Bspline Approximation Techniquementioning

confidence: 99%

“…Kutluay et al (2019) presented a Galerkin finite element method based on the Hopf-Cole transformation for solving a system of two-dimensional (2D) CVBEs. Oruç et al (2019) used the Chebyshev wavelets method (CVM) to solve 1D timedependent coupled Burgers' equations. Angstmann et al (2020) proposed an explicit numerical approach for solving non-linear Burgers' equations.…”

Section: Introductionmentioning

confidence: 99%

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New cubic B-spline approximation technique for numerical solutions of coupled viscous Burgers equations

Nazir

Abbas

Iqbal

2020

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Purpose The purpose of this paper is to present a new cubic B-spline (CBS) approximation technique for the numerical treatment of coupled viscous Burgers’ equations arising in the study of fluid dynamics, continuous stochastic processes, acoustic transmissions and aerofoil flow theory. Design/methodology/approach The system of partial differential equations is discretized in time direction using the finite difference formulation, and the new CBS approximations have been used to interpolate the solution curves in the spatial direction. The theoretical estimation of stability and uniform convergence of the proposed numerical algorithm has been derived rigorously. Findings A different scheme based on the new approximation in CBS functions is proposed which is quite different from the existing methods developed (Mittal and Jiwari, 2012; Mittal and Arora, 2011; Mittal and Tripathi, 2014; Raslan et al., 2017; Shallal et al., 2019). Some numerical examples are presented to validate the performance and accuracy of the proposed technique. The simulation results have guaranteed the superior performance of the presented algorithm over the existing numerical techniques on approximate solutions of coupled viscous Burgers’ equations. Originality/value The current approach based on new CBS approximations is novel for the numerical study of coupled Burgers’ equations, and as far as we are aware, it has never been used for this purpose before.

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Section: Numerical Resultsmentioning

confidence: 99%

Section: New Cubic Bspline Approximation Techniquementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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New cubic B-spline approximation technique for numerical solutions of coupled viscous Burgers equations

Nazir

Abbas

Iqbal

2020

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“…Oruc et al [25,26] applied a unified finite difference Chebyshev wavelet approach for time fractional Burger equations. The same authors studied the Chebyshev wavelet method for approximation of coupled Burgers equations [27]. Ali et al [4] applied a meshfree collocation method based on the Crank-Nicolson method for time discretization and radial basis function for space discretization to solve two-dimensional coupled Burger equations, whereas the meshless method of radial basis functions (RBFs) and local RBFs were described in [28,29] for approximate solutions of Burger-type equations.…”

Section: Introductionmentioning

confidence: 99%

An efficient numerical scheme based on Lucas polynomials for the study of multidimensional Burgers-type equations

et al. 2021

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We propose a polynomial-based numerical scheme for solving some important nonlinear partial differential equations (PDEs). In the proposed technique, the temporal part is discretized by finite difference method together with θ-weighted scheme. Then, for the approximation of spatial part of unknown function and its spatial derivatives, we use a mixed approach based on Lucas and Fibonacci polynomials. With the help of these approximations, we transform the nonlinear partial differential equation to a system of algebraic equations, which can be easily handled. We test the performance of the method on the generalized Burgers–Huxley and Burgers–Fisher equations, and one- and two-dimensional coupled Burgers equations. To compare the efficiency and accuracy of the proposed scheme, we computed $L_{\infty }$ L ∞ , $L_{2}$ L 2 , and root mean square (RMS) error norms. Computations validate that the proposed method produces better results than other numerical methods. We also discussed and confirmed the stability of the technique.

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“…A.K. Gupta and S.S. Ray studied the solution of fractional fifth-order Sawada-Kotera equation using second kind Chebyshev wavelet method [14] and many others [16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

New Analytical Solutions for Time-Fractional Kolmogorov-Petrovsky-Piskunov Equation with Variety of Initial Boundary Conditions

2019

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The generalized time fractional Kolmogorov-Petrovsky-Piskunov equation (FKPP),, which plays an important role in engineering, chemical reaction problem is proposed by Caputo fractional order derivative sense. In this paper, we develop a framework wavelet, including shift Chebyshev polynomial of the first kind as a mother wavelet, and also construct some operational matrices that represent Caputo fractional derivative to obtain analytical solutions for FKPP equation with three different types of Initial Boundary conditions (Dirichlet, Dirichlet-Neumann, and Neumann-Robin). Our results shown that the Chebyshev wavelet is a powerful method, due to its simplicity, efficiency in analytical approximations, and its fast convergence. The comparison of the Chebyshev wavelet results indicates that the proposed method not only gives satisfactory results but also do not need large amount of CPU times.

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Chebyshev Wavelet Method for Numerical Solutions of Coupled Burgers Equation

Cited by 13 publications

References 45 publications

New cubic B-spline approximation technique for numerical solutions of coupled viscous Burgers equations

New cubic B-spline approximation technique for numerical solutions of coupled viscous Burgers equations

An efficient numerical scheme based on Lucas polynomials for the study of multidimensional Burgers-type equations

New Analytical Solutions for Time-Fractional Kolmogorov-Petrovsky-Piskunov Equation with Variety of Initial Boundary Conditions

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