On the solution of time-fractional KdV–Burgers equation using Petrov–Galerkin method for propagation of long wave in shallow water

Gupta, A. K.; Ray, S. Saha

doi:10.1016/j.chaos.2018.09.046

Cited by 22 publications

(12 citation statements)

References 25 publications

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“…, where F 0 and G i for i = 1, 2, 3 are known vectors. Now using Equations (2), (3), (38), and the above two relations, we get…”

Section: The Proposed Methodologymentioning

confidence: 99%

“…37 During the last decades, due to importance of this equation in real-world phenomena, numerous researchers have focused on the developing numerical methods for various fractional models of the KdV Burgers' equation, for instance, refer to previous studies. [38][39][40][41] According to the discussions presented above and since there is not any nonsingular VO fractional model for the KdV Burgers' equation, the major goals of this article are briefly reviewed as follows:…”

Section: Introductionmentioning

confidence: 99%

“…The Korteweg‐de Vries (KdV) Burgers' equation, as a well‐studied nonlinear differential equation, appears in several physical problems, such as the flow of liquids containing gas bubbles, the undular bores propagation in a shallow water, and the waves propagation in an elastic tube filled with a viscous fluid 37 . During the last decades, due to importance of this equation in real‐world phenomena, numerous researchers have focused on the developing numerical methods for various fractional models of the KdV Burgers' equation, for instance, refer to previous studies 38‐41 …”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Discrete Chebyshev polynomials for nonsingular variable‐order fractional KdV Burgers' equation

Heydari

Avazzadeh

Cattani

2020

Math Methods in App Sciences

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In this article, nonlinear variable-order (VO) fractional Korteweg-de Vries (KdV) Burgers' equation with nonsingular VO time fractional derivative is introduced and discussed. The approximate solution of the expressed problem is obtained in the form of a series expansion in terms of the shifted discrete Chebyshev polynomials (CPs) with great accuracy. The method is a computational procedure based on the collocation technique and the shifted discrete CPs together with their operational matrices (ordinary and VO fractional derivatives). The main advantage of the designed approach is that it provides a global solution for the problem. In order to examine the efficiency of the designed algorithm, some numerical problems have been provided. The obtained solutions confirm that the present method is computationally effective and sufficiently accurate in solving this equation.

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“…, where F 0 and G i for i = 1, 2, 3 are known vectors. Now using Equations (2), (3), (38), and the above two relations, we get…”

Section: The Proposed Methodologymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Discrete Chebyshev polynomials for nonsingular variable‐order fractional KdV Burgers' equation

Heydari

Avazzadeh

Cattani

2020

Math Methods in App Sciences

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“…Ucar and Feng used FEM for solution of Fractional diffusion equations [36,37]. In [38] the authors applied Petrov Galerkin method for time fractional KDV equation. In this work, we use cubic B-spline as a trial and basis function in FEM for numerical solution of MTTFPDEs.…”

Section: Introductionmentioning

confidence: 99%

Computational Study of Multi-Term Time-Fractional Differential Equations Using Cubic B-spline Finite Element Method

Arifeen¹,

Haq²,

Golkarmanesh

2022

Preprint

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Fractional differential equations precisely measure and describe biological and physical processes because of the symmetry feature in nature. Multi-term time-fractional (MTTF) introduced for modeling of complex processes in different physical phenomena. This article introduces a numerical method based on Cubic B-spline (CBS) Finite element method (FEM) for solution of MTTF partial differential equations(PDEs). Finite difference method is combined with theta-weighted scheme and simple quadrature rules are used for time-fractional discretization while CBS FEM is employed for space approximation. In addition, four-point Gauss Legendre quadrature is applied to evaluate the source term. Stability of the proposed scheme is discussed by Von Neumann method which verifies that the scheme is unconditionally stable. L 2 and L ∞ norms are used to check efficiency and accuracy of the proposed scheme. Computed results are compared with the exact and available methods in literature which show betterment of the proposed method.

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“…Yokus 31 proposed the extended finite difference method to solve the equation numerically. The fractional KdV–Burgers' equation also models very important physical phenomena such as the waves on shallow water surfaces, weakly nonlinear plasma waves, propagation of long waves in shallow water, and propagation of waves in elastic tube stuffed with a viscous fluid 32–34 . A lot of work has been done about classical nonlinear KdV–Burgers' model and outstanding numerical and analytical techniques have been proposed, but detailed studies of fractional models are required.…”

Section: Introductionmentioning

confidence: 99%

The numerical solution of fractional Korteweg‐de Vries and Burgers' equations via Haar wavelet

Zada

Aziz

2021

Math Methods in App Sciences

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In this article, Haar wavelet collocation technique is adapted to acquire the approximate solution of fractional Korteweg‐de Vries (KdV), Burgers', and KdV–Burgers' equations. The fractional order derivatives involved are described using the Caputo definition. In the proposed technique, the given nonlinear fractional differential equation is discretized with the help of Haar wavelet and reduced to the nonlinear system of equations, which are solved with Newton's or Broyden's method. The proposed method is semi‐analytic as it involves exact integration of Caputo derivative. The proposed technique is widely applicable and robust. The technique is tested upon many test problems. The results are computed and presented in the form of maximum absolute errors which show the accuracy, efficiency, and simple applicability of the proposed method.

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On the solution of time-fractional KdV–Burgers equation using Petrov–Galerkin method for propagation of long wave in shallow water

Cited by 22 publications

References 25 publications

Discrete Chebyshev polynomials for nonsingular variable‐order fractional KdV Burgers' equation

Discrete Chebyshev polynomials for nonsingular variable‐order fractional KdV Burgers' equation

Computational Study of Multi-Term Time-Fractional Differential Equations Using Cubic B-spline Finite Element Method

The numerical solution of fractional Korteweg‐de Vries and Burgers' equations via Haar wavelet

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