A new algorithm based on Lucas polynomials for approximate solution of 1D and 2D nonlinear generalized Benjamin–Bona–Mahony–Burgers equation

Oru, mer

doi:10.1016/j.camwa.2017.07.046

Cited by 45 publications

(18 citation statements)

References 30 publications

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“…Here, we consider the 2‐D Equation (1.1) as following form [6, 17]

\begin{array}{l} \frac{\partial}{\partial t} v (x, y, t) - \frac{\partial}{\partial t} ((), \frac{\partial^{2} v (x, y, t)}{\partial x^{2}} + \frac{\partial^{2} v (x, y, t)}{\partial y^{2}}) - \frac{\partial^{2} v (x, y, t)}{\partial x^{2}} - \frac{\partial^{2} v (x, y, t)}{\partial y^{2}} \\ + \frac{\partial v (x, y, t)}{\partial x} + \frac{\partial v (x, y, t)}{\partial y} = \frac{\partial}{\partial x} ((), \frac{v^{2} false(x, y, t false)}{2}) + \frac{\partial}{\partial y} ((), \frac{v^{2} false(x, y, t false)}{2}) + g (x, y, t), \end{array}

with source term

g false(x, y, t false) = 2 t \cos false(x + y false) + false(3 + 2 t - 2 t^{2} \cos false(x + y false) false) \sin false(x + y false) .

…”

Section: Numerical Illustrationsmentioning

confidence: 99%

“…Table 6 lists the error norms, C‐order acquired and used CPU time of LSEM with N e = 10, p e = 4, and T = 2 on physical domain Ω = [0, 1] 2 . For

dt = \frac{1}{500}

and various values of dx , the obtained error norms of LSEM are compared with the results reported in [17] in Table 5. The error norms and C‐order obtained by LSEM and used CPU time with p e = 3 and dx = 1/10 at T = 1 on spatial domain Ω = [0, 1] 2 are listed in Table 7.…”

Section: Numerical Illustrationsmentioning

confidence: 99%

“…Authors of [6] discretized the time variable using forward-type finite difference, and they also approximated the spatial derivatives by the Kansa's method. Article [17] has been devoted to solving the 1-and 2-D versions of Equation (1.1) by using an algorithm based on Lucas polynomials. Dehghan et al [7] concentrated on solving the 2-D GBBMB equation by using the element free Galerkin approach.…”

Section: Introductionmentioning

confidence: 99%

“…and various values of dx, the obtained error norms of LSEM are compared with the results reported in [17] in Table 5. The error norms and C-order obtained by LSEM and used CPU time with p e = 3 and dx = 1/10 at T = 1 on spatial domain Ω = [0, 1] 2 are listed in Table 7.…”

mentioning

confidence: 99%

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Numerical and theoretical discussions for solving nonlinear generalized Benjamin–Bona–Mahony–Burgers equation based on the Legendre spectral element method

Dehghan

Shafieeabyaneh

Abbaszadeh

2020

Numerical Methods Partial

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This article is devoted to solving numerically the nonlinear generalized Benjamin-Bona-Mahony-Burgers (GBBMB) equation that has several applications in physics and applied sciences. First, the time derivative is approximated by using a finite difference formula. Afterward, the stability and convergence analyses of the obtained time semi-discrete are proven by applying the energy method. Also, it has been demonstrated that the convergence order in the temporal direction is O(dt). Second, a fully discrete formula is acquired by approximating the spatial derivatives via Legendre spectral element method. This method uses Lagrange polynomial based on Gauss-Legendre-Lobatto points. An error estimation is also given in detail for full discretization scheme. Ultimately, the GBBMB equation in the oneand two-dimension is solved by using the proposed method. Also, the calculated solutions are compared with theoretical solutions and results obtained from other techniques in the literature. The accuracy and efficiency of the mentioned procedure are revealed by numerical samples.

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“…Here, we consider the 2‐D Equation (1.1) as following form [6, 17]

\begin{array}{l} \frac{\partial}{\partial t} v (x, y, t) - \frac{\partial}{\partial t} ((), \frac{\partial^{2} v (x, y, t)}{\partial x^{2}} + \frac{\partial^{2} v (x, y, t)}{\partial y^{2}}) - \frac{\partial^{2} v (x, y, t)}{\partial x^{2}} - \frac{\partial^{2} v (x, y, t)}{\partial y^{2}} \\ + \frac{\partial v (x, y, t)}{\partial x} + \frac{\partial v (x, y, t)}{\partial y} = \frac{\partial}{\partial x} ((), \frac{v^{2} false(x, y, t false)}{2}) + \frac{\partial}{\partial y} ((), \frac{v^{2} false(x, y, t false)}{2}) + g (x, y, t), \end{array}

with source term

g false(x, y, t false) = 2 t \cos false(x + y false) + false(3 + 2 t - 2 t^{2} \cos false(x + y false) false) \sin false(x + y false) .

…”

Section: Numerical Illustrationsmentioning

confidence: 99%

“…Table 6 lists the error norms, C‐order acquired and used CPU time of LSEM with N e = 10, p e = 4, and T = 2 on physical domain Ω = [0, 1] 2 . For

dt = \frac{1}{500}

Section: Numerical Illustrationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Numerical and theoretical discussions for solving nonlinear generalized Benjamin–Bona–Mahony–Burgers equation based on the Legendre spectral element method

Dehghan

Shafieeabyaneh

Abbaszadeh

2020

Numerical Methods Partial

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“…The purpose of this article is to present a numerical method based on the Lucas multiwavelet functions (LMWFs), these functions are constructed from Lucas polynomials 25,26 . In addition, the proposed method includes the novel techniques for obtaining the modified operational matrix (MOM) of integration and POM of the VO‐fractional derivative.…”

Section: Introductionmentioning

confidence: 99%

A novel direct method based on the Lucas multiwavelet functions for variable‐order fractional reaction‐diffusion and subdiffusion equations

Dehestani

Ordokhani

Razzaghi

2020

Numerical Linear Algebra App

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In this article, we study the numerical technique for variable‐order fractional reaction‐diffusion and subdiffusion equations that the fractional derivative is described in Caputo's sense. The discrete scheme is developed based on Lucas multiwavelet functions and also modified and pseudo‐operational matrices. Under suitable properties of these matrices, we present the computational algorithm with high accuracy for solving the proposed problems. Modified and pseudo‐operational matrices are employed to achieve the nonlinear algebraic equation corresponding to the proposed problems. In addition, the convergence of the approximate solution to the exact solution is proven by providing an upper bound of error estimate. Numerical experiments for both classes of problems are presented to confirm our theoretical analysis.

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The numerical analysis of two linearized difference schemes for the Benjamin–Bona–Mahony–Burgers equation

Zhang

Liu

Zhang

2020

Numerical Methods Partial

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In the article, two linearized finite difference schemes are proposed and analyzed for the Benjamin-Bona-Mahony-Burgers (BBMB) equation. For the construction of the two-level scheme, the nonlinear term is linearized via averaging k and k + 1 floor, we prove unique solvability and convergence of numerical solutions in detail with the convergence order O(2 + h 2). For the three-level linearized scheme, the extrapolation technique is utilized to linearize the nonlinear term based on function. We obtain the conservation, boundedness, unique solvability and convergence of numerical solutions with the convergence order O(2 + h 2) at length. Furthermore, extending our work to the BBMB equation with the nonlinear source term is considered and a Newton linearized method is inserted to deal with it. The applicability and accuracy of both schemes are demonstrated by numerical experiments.

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A new algorithm based on Lucas polynomials for approximate solution of 1D and 2D nonlinear generalized Benjamin–Bona–Mahony–Burgers equation

Cited by 45 publications

References 30 publications

Numerical and theoretical discussions for solving nonlinear generalized Benjamin–Bona–Mahony–Burgers equation based on the Legendre spectral element method

Numerical and theoretical discussions for solving nonlinear generalized Benjamin–Bona–Mahony–Burgers equation based on the Legendre spectral element method

A novel direct method based on the Lucas multiwavelet functions for variable‐order fractional reaction‐diffusion and subdiffusion equations

The numerical analysis of two linearized difference schemes for the Benjamin–Bona–Mahony–Burgers equation

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