The numerical solution of nonlinear high dimensional generalized Benjamin–Bona–Mahony–Burgers equation via the meshless method of radial basis functions

Dehghan, Mehdi; Abbaszadeh, Mostafa; Mohebbi, Akbar

doi:10.1016/j.camwa.2014.05.019

Cited by 141 publications

(66 citation statements)

References 44 publications

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“…Figure 5: Graphs of approximation solution and absolute error using the present method with h = 1/20, τ = 1/100, r = 3 and ρ = 3h on rectangular domain for Test problem 2. Table 6 Errors and computational order obtained on Ω 1 for present method with h = 1/20 and r = 3 for Test problem 2 τ L ∞ with ρ = 2.15h C-order L ∞ with ρ = 3h C-order Table 7 Comparison between errors of present method and method of [20] for Test problem 2…”

Section: Test Problemmentioning

confidence: 98%

“…Table 2 Errors and computational order obtained for present method with τ = 0.01 Table 3 Comparison between errors of present method and method of [20] for Test problem 1…”

Section: Test Problemmentioning

confidence: 98%

“…Method of [20] Present method Table 4 shows the errors and computational order obtained for present method with h = 1/20 and different value of ρ. Table 5 presents that the errors and computational order obtained for the new method with τ = 0.01 and different value of ρ.…”

Section: Test Problemmentioning

confidence: 98%

“…Also, Tables 4, 5 and 6 show that the computational orders of time and space discrete scheme are close to the theoretical orders. The obtained errors using method of [20] and present method are compared in Table 7 for Test problem 1. Figure 5 shows the graphs of approximation solution and absolute error using the new method with h = 1/20, τ = 1/100, r = 3 and ρ = 3h on rectangular domain for Test problem 2.…”

Section: Test Problemmentioning

confidence: 98%

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The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations on non-rectangular domains with error estimate

Dehghan

Abbaszadeh

Mohebbi

2015

Journal of Computational and Applied Mathematics

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Please cite this article as: M. Dehghan, M. Abbaszadeh, A. Mohebbi, The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin-Bona-Mahony-Burgers and regularized long-wave equations on non-rectangular domains with error estimate, Journal of Computational and Applied Mathematics (2015), http://dx. AbstractIn this paper a numerical technique is proposed for solving the nonlinear generalized Benjamin-Bona-Mahony-Burgers and regularized long-wave equations. Firstly, we obtain a time discrete scheme by approximating time derivative via a finite difference formula, then we use the interpolating element-free Galerkin approach to approximate the spatial derivatives. The element-free Galerkin method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the element-free Galerkin method test and trial functions are moving least squares approximation shape functions. Also, in the element-free Galekin method, we do not use any triangular, quadrangular or other type of meshes. It is a global method while finite elements method is a local one. The element free Galerkin method is not a truly meshless method and for integration employs a background mesh. We prove that the time discrete scheme is unconditionally stable and convergent in time variable using the energy method. We show that convergence order of the time discrete scheme is O(τ ). Since the shape functions of moving least squares approximation don't have Kronecker delta property, we can not implement the essential boundary condition, directly. Thus, the improved moving least squares shape functions that have the mentioned property are employed . An error estimate for the method proposed in the current paper is obtained. Also, the two-dimensional version of both equations on different complex geometries is solved. The aim of this paper is to show that the meshless method based on the weak form is also suitable for the treatment of the nonlinear partial differential equations and to obtain an error bound for the new method. Numerical examples confirm the efficiency of the proposed scheme. (M. Dehghan), a mohebbi@kashanu.ac.ir (A. Mohebbi), m.abbaszadeh@aut.ac.ir (M. Abbaszadeh).

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Section: Test Problemmentioning

confidence: 98%

“…Table 2 Errors and computational order obtained for present method with τ = 0.01 Table 3 Comparison between errors of present method and method of [20] for Test problem 1…”

Section: Test Problemmentioning

confidence: 98%

Section: Test Problemmentioning

confidence: 98%

Section: Test Problemmentioning

confidence: 98%

See 2 more Smart Citations

The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations on non-rectangular domains with error estimate

Dehghan

Abbaszadeh

Mohebbi

2015

Journal of Computational and Applied Mathematics

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“…This suggests us to append the dissipative term (δu xx ) into the RLW equation, which results in the BBMB equation (see [31,7]). Many work have been devoted to the development of numerical methods for the BBMB equation, that include finite difference methods [32][33][34], Adomian's decomposition method [35], and meshfree method [36,37]. Mittal et al [38,39] used cubic B-spline approximation to develop numerical methods for Burgers equation (α = β = 0 and η = 1 in (5)) (also see [40] for a nonlinear parabolic equation).…”

Section: Introductionmentioning

confidence: 98%

B-spline quasi-interpolation based numerical methods for some Sobolev type equations

Kumar

Baskar

2016

Journal of Computational and Applied Mathematics

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a b s t r a c tIn this article, we intend to use quadratic and cubic B-spline quasi-interpolants to develop higher order numerical methods for some Sobolev type equations in one space dimension. Our aim is also to compare the performance of the proposed methods in terms of the accuracy and the rate of convergence. We also discuss another approach to the cubic B-spline quasi-interpolation based method, where we achieve fourth order of accuracy in space. We theoretically establish the order of accuracy for the three proposed methods and also establish the L 2 -stability in the linear case using von Neumann analysis. As a particular case of the Sobolev type equations, we take the equal width and the Benjamin-Bona-Mahony-Burgers equations, and perform several numerical experiments to support our theoretical results.

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A new approach for solving highly nonlinear partial differential equations by successive differentiation method

Khalid

Khan

2017

Math Methods in App Sciences

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In this work successive differentiation method is applied to solve highly nonlinear partial differential equations (PDEs) such as Benjamin–Bona–Mahony equation, Burger's equation, Fornberg–Whitham equation, and Gardner equation. To show the efficacy of this new technique, figures have been incorporated to compare exact solution and results of this method. Wave variable is used to convert the highly nonlinear PDE into ordinary differential equation with order reduction. Then successive differentiation method is utilized to obtain the numerical solution of considered PDEs in this paper. Copyright © 2017 John Wiley & Sons, Ltd.

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The numerical solution of nonlinear high dimensional generalized Benjamin–Bona–Mahony–Burgers equation via the meshless method of radial basis functions

Cited by 141 publications

References 44 publications

The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations on non-rectangular domains with error estimate

The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations on non-rectangular domains with error estimate

B-spline quasi-interpolation based numerical methods for some Sobolev type equations

A new approach for solving highly nonlinear partial differential equations by successive differentiation method

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