An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein–Gordon equations

Dehghan, Mehdi; Abbaszadeh, Mostafa; Mohebbi, Akbar

doi:10.1016/j.enganabound.2014.09.008

Cited by 119 publications

(60 citation statements)

References 87 publications

(89 reference statements)

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“…If we set κ 1 = κ 2 = κ 3 =0 and f ( u ( x , t ))=− sin( u ( x , t )) in Eq. , we have the following problem :

\begin{matrix} D_{t}^{α} u (x, t) = u_{xx} (x, t) - \sin (u (x, t)) + ((), \frac{2 t^{2 - α}}{normalΓ (3 - α)} + t^{2}) \sin (x) + \sin (t^{2} \sin (x)), \\ 0 \leq x \leq 1, 0 \leq t \leq 1, \end{matrix}

with boundary conditions

u (0, t) = 0, u (1, t) = t^{2} \sin (1), 0 \leq t \leq 1,

and initial conditions

u (x, 0) = 0, ψ (x) = 0, 0 \leq x \leq 1 .

The exact solution of this test problem is

u (x, t) = t^{2} \sin (x) .

We solve this problem with the scheme that presented in this paper with different values of α , M , and τ . The RMS and L ∞ errors for different values of α , M , and τ are given in Tables and at final time T = 1.…”

Section: Numerical Resultsmentioning

confidence: 99%

The dual reciprocity boundary integral equation technique to solve a class of the linear and nonlinear fractional partial differential equations

Dehghan

Safarpoor

2015

Math Methods in App Sciences

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In this paper, we apply the boundary integral equation technique and the dual reciprocity boundary elements method (DRBEM) for the numerical solution of linear and nonlinear time-fractional partial differential equations (TFPDEs). The main aim of the present paper is to examine the applicability and efficiency of DRBEM for solving TFPDEs . We employ the time-stepping scheme to approximate the time derivative, and the method of linear radial basis functions is also used in the DRBEM technique. This method is improved by using a predictor-corrector scheme to overcome the nonlinearity that appears in the nonlinear problems under consideration. To confirm the accuracy of the new approach, several examples are presented. The convergence of the DRBEM is studied numerically by comparing the exact solutions of the problems under investigation.

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“…If we set κ 1 = κ 2 = κ 3 =0 and f ( u ( x , t ))=− sin( u ( x , t )) in Eq. , we have the following problem :

\begin{matrix} D_{t}^{α} u (x, t) = u_{xx} (x, t) - \sin (u (x, t)) + ((), \frac{2 t^{2 - α}}{normalΓ (3 - α)} + t^{2}) \sin (x) + \sin (t^{2} \sin (x)), \\ 0 \leq x \leq 1, 0 \leq t \leq 1, \end{matrix}

with boundary conditions

u (0, t) = 0, u (1, t) = t^{2} \sin (1), 0 \leq t \leq 1,

and initial conditions

u (x, 0) = 0, ψ (x) = 0, 0 \leq x \leq 1 .

The exact solution of this test problem is

u (x, t) = t^{2} \sin (x) .

Section: Numerical Resultsmentioning

confidence: 99%

The dual reciprocity boundary integral equation technique to solve a class of the linear and nonlinear fractional partial differential equations

Dehghan

Safarpoor

2015

Math Methods in App Sciences

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“…Now, consider irregular domains of Figure from with criterion

r = \frac{1}{n^{2}} [1 + 2 n + n^{2} - (n + 1) \cos (nθ)],

where n = 4 in Ω 1 and n = 8 in Ω 2 . Also, the center of this criterion is moved from origin to the point [ π /2, π /2].…”

Section: Numerical Examplesmentioning

confidence: 99%

“…In addition, Hussain et al , investigated numerical solution of the 1‐D KG and 2‐D SG equations by meshless method of lines using RBFs. Also, Dehghan et al , used an implicit RBF meshless approach for solving the time‐fractional nonlinear SG and KG equations.…”

Section: Introductionmentioning

confidence: 99%

A moving Kriging‐based MLPG method for nonlinear Klein–Gordon equation

Shokri

Habibirad

2016

Math Methods in App Sciences

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In this paper, the meshless local Petrov–Galerkin approximation is proposed to solve the 2‐D nonlinear Klein–Gordon equation. We used the moving Kriging interpolation instead of the MLS approximation to construct the meshless local Petrov–Galerkin shape functions. These shape functions possess the Kronecker delta function property. The Heaviside step function is used as a test function over the local sub‐domains. Here, no mesh is needed neither for integration of the local weak form nor for construction of the shape functions. So the present method is a truly meshless method. We employ a time‐stepping method to deal with the time derivative and a predictor–corrector scheme to eliminate the nonlinearity. Several examples are performed and compared with analytical solutions and with the results reported in the extant literature to illustrate the accuracy and efficiency of the presented method. Copyright © 2016 John Wiley & Sons, Ltd.

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“…Recently, Mohebbi et al [41] proposed a combination of a collocation framework and radial basis functions for providing an accurate and efficient numerical solution of the TFSE, and they also developed this method to improve the accuracy of the algorithm greatly for solving nonlinear sineGordon and Klein-Gordon equations with fractional orders [42]. For other numerical schemes for fractional Schrödinger equations, the interested reader is referred to ([43]- [48] and the references therein).…”

Section: Introductionmentioning

confidence: 98%

A fully spectral collocation approximation for multi-dimensional fractional Schrödinger equations

Bhrawy

Abdelkawy

2015

Journal of Computational Physics

162

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An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein–Gordon equations

Cited by 119 publications

References 87 publications

The dual reciprocity boundary integral equation technique to solve a class of the linear and nonlinear fractional partial differential equations

The dual reciprocity boundary integral equation technique to solve a class of the linear and nonlinear fractional partial differential equations

A moving Kriging‐based MLPG method for nonlinear Klein–Gordon equation

A fully spectral collocation approximation for multi-dimensional fractional Schrödinger equations

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