A high‐order compact scheme for the nonlinear fractional <scp>K</scp>lein–<scp>G</scp>ordon equation

Vong, Seakweng; Wang, Zhen

doi:10.1002/num.21912

Cited by 31 publications

(20 citation statements)

References 30 publications

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“…If we put κ 1 =0, κ 2 = κ 3 =1 and f ( u ( x , t )) =− u ( x , t ) 2 − u ( x , t ) 3 , we have the following problem (nonlinear Klein–Gordon equation):

\begin{array}{l} D_{t}^{α} u + u_{t} + u = u_{xx} - u^{2} - u^{3} + ((), \frac{normalΓ (3 + α)}{2} t^{2} + (2 + α) t^{1 + α} + (1 + π^{2}) t^{2 + α}) \sin (πx) \\ + t^{4 + 2 α} \overset{2}{\sin} (πx) + t^{6 + 3 α} \overset{3}{\sin} (πx), 0 \leq x \leq 1, 0 \leq t \leq 1, \end{array}

with boundary conditions

u (0, t) = 0, u (1, t) = 0, 0 \leq t \leq 1,

and initial conditions

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

The exact solution of the preceding test problem is given by

u (x, t) = t^{2 + α} \sin (πx) .

We solve this test problem with the scheme presented in the current paper with different values of α , M , and τ . The computational results of the scheme for different values of α and M are reported in Table .…”

Section: Numerical Resultsmentioning

confidence: 99%

“…The author of used H e ′ s variational iteration method for solving linear and nonlinear Klein–Gordon equations. S. Vong and Z. Wang presented a high‐order finite‐difference scheme for solving nonlinear fractional Klein–Gordon equation. They also used this scheme for the numerical solution of two‐dimensional fractional Klein–Gordon equation with Neumann boundary conditions .…”

Section: Introductionmentioning

confidence: 99%

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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 put κ 1 =0, κ 2 = κ 3 =1 and f ( u ( x , t )) =− u ( x , t ) 2 − u ( x , t ) 3 , we have the following problem (nonlinear Klein–Gordon equation):

\begin{array}{l} D_{t}^{α} u + u_{t} + u = u_{xx} - u^{2} - u^{3} + ((), \frac{normalΓ (3 + α)}{2} t^{2} + (2 + α) t^{1 + α} + (1 + π^{2}) t^{2 + α}) \sin (πx) \\ + t^{4 + 2 α} \overset{2}{\sin} (πx) + t^{6 + 3 α} \overset{3}{\sin} (πx), 0 \leq x \leq 1, 0 \leq t \leq 1, \end{array}

with boundary conditions

u (0, t) = 0, u (1, t) = 0, 0 \leq t \leq 1,

and initial conditions

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

The exact solution of the preceding test problem is given by

u (x, t) = t^{2 + α} \sin (πx) .

Section: Numerical Resultsmentioning

confidence: 99%

Section: Introductionmentioning

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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“…A compact scheme is developed. We remark that compact difference schemes were successfully applied to improve the spatial accuracy of fractional diffusion equations in recent years (see [9][10][11][12][13] and the references therein). As a whole, we established a scheme which converges with O(τ 2 + h 4 1 + h 4 2 ), where τ is the temporal step size and h 1 , h 2 are the spatial step sizes respectively.…”

Section: Introductionmentioning

confidence: 99%

High order finite difference method for time-space fractional differential equations with Caputo and Riemann-Liouville derivatives

et al. 2015

Self Cite

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We consider high order finite difference methods for two-dimensional fractional differential equations with temporal Caputo and spatial Riemann-Liouville derivatives in this paper. We propose a scheme and show that it converges with second order in time and fourth order in space. The accuracy of our proposed method can be improved by Richardson extrapolation. Approximate solution is obtained by the generalized minimal residual (GMRES) method. A preconditioner is proposed to improve the efficiency for the implementation of the GMRES method.Keywords Two-dimensional fractional differential equation · High order difference scheme · Discrete energy method · Preconditioned GMRES method Mathematics Subject Classification (2010) 35R11 · 65M06 · 65M12 · 65M15

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“…Throughout this article, we suppose that the considered problem (1.1)-(1.3) has a unique solution with sufficiently smooth properties defined in [0, 1] 2 × [0, T ] (the solvability of the problem's scheme may be considered by using the Leray-Schauder fixed point theorem applied in [62,63]) and g(x, t) and u 0 (x) are sufficiently smooth to meet the need of our theoretical analysis. Furthermore, we assume that u tt is continuous in [0, 1] 2 × [0, T ].…”

Section: Introductionmentioning

confidence: 99%

Finite difference/finite element method for a nonlinear time-fractional fourth-order reaction–diffusion problem

Yang

et al. 2015

Computers & Mathematics with Applications

142

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Available online xxxx Keywords:Time-fractional fourth-order reaction-diffusion problem Finite element method Finite difference scheme Caputo-fractional derivative Unconditional stability Optimal a priori error analysis a b s t r a c tIn this article, a finite difference/finite element algorithm, which is based on a finite difference approximation in time direction and finite element method in spatial direction, is presented and discussed to cast about for the numerical solutions of a time-fractional fourth-order reaction-diffusion problem with a nonlinear reaction term. To avoid the use of higher-order elements, the original problem with spatial fourth-order derivative need to be changed into a second-order coupled system by introducing an intermediate variable σ = ∆u. Then the fully discrete finite element scheme is formulated by using a finite difference approximation for time fractional and integer derivatives and finite element method in spatial direction. The unconditionally stable result in the norm, which just depends on initial value and source item, is derived. Some a priori estimates of L 2 -norm with optimal order of convergence O(∆ 2−α t +h m+1 ), where ∆ t and h are time step length and space mesh parameter, respectively, are obtained. To confirm the theoretical analysis, some numerical results are provided by our method.

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A high‐order compact scheme for the nonlinear fractional Klein–Gordon equation

Cited by 31 publications

References 30 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

High order finite difference method for time-space fractional differential equations with Caputo and Riemann-Liouville derivatives

Finite difference/finite element method for a nonlinear time-fractional fourth-order reaction–diffusion problem

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