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

Vong, Seakweng; Lyu, Pin; Chen, Xu; Lei, Siu‐Long

doi:10.1007/s11075-015-0041-3

Cited by 72 publications

(27 citation statements)

References 18 publications

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“…One of the most well‐known way to approximate the Caputo fractional derivative is the ℒ2‐1 σ formula proposed by Alikhanov in , with the truncation error

O (τ^{3 - α})

for 0 < α < 1. Recently, most of the work done by the authors were based on the ℒ2‐1 σ formula, such as two‐dimensional time–space FDEs , fractional diffusion wave equations , multiterm and distributed‐order fractional subdiffusion equations , fourth‐order fractional subdiffusion equations , and nonlinear Klein‐Gordon equations .…”

Section: Introductionmentioning

confidence: 99%

Fast high order difference schemes for the time fractional telegraph equation

Liang

Yao

Wang

2019

Numerical Methods Partial

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In this paper, a fast high order difference scheme is first proposed to solve the time fractional telegraph equation based on the ℱℒ2‐1σ formula for the Caputo fractional derivative, which reduces the storage and computational cost for calculation. A compact scheme is then presented to improve the convergence order in space. The unconditional stability and convergence in maximum norm are proved for both schemes, with the accuracy order Oτ2+h2 and Oτ2+h4, respectively. Difficulty arising from the two Caputo fractional derivatives is overcome by some detailed analysis. Finally, we carry out numerical experiments to show the efficiency and accuracy, by comparing with the ℒ2‐1σ method.

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“…One of the most well‐known way to approximate the Caputo fractional derivative is the ℒ2‐1 σ formula proposed by Alikhanov in , with the truncation error

O (τ^{3 - α})

Section: Introductionmentioning

confidence: 99%

Fast high order difference schemes for the time fractional telegraph equation

Liang

Yao

Wang

2019

Numerical Methods Partial

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“…Deng et al introduced a second‐order shifted Grünwald difference method to solve the space fractional equations. Vong et al proposed a numerical method to the fractional Caputo/Riemann‐Liouville differential equation.…”

Section: Introductionmentioning

confidence: 99%

Numerical analysis for Klein‐Gordon equation with time‐space fractional derivatives

Zhang

Wang

Zhou

2019

Math Methods in App Sciences

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We present and analyze two numerical schemes for solving a nonlinear Klein‐Gordon equation with time‐space fractional derivatives. Numerical methods are base on finite difference scheme in fractional derivative and Fourier‐spectral method in spatial variable. It is proved that the linearized method is conditionally stable while the nonlinearized one is unconditionally stable. In addition, the error estimate shows that the linearized method is in the order of scriptOfalse(normalΔt+Nβ−rfalse), and the nonlinearized method converge with the order scriptOfalse(normalΔt3−α+Nβ−rfalse), where normalΔt, N, β, and r are, respectively, step of time, polynomial degree, the fractional derivative in space, and regularity of u. Some numerical experiments are performed to demonstrate the theoretical results.

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“…These motivated us to study (1.1)–(1.2) in numerical way. A lot of researches have proposed different efficient numerical methods to study time fractional differential equations, in which the time fractional derivatives are usually modelled in Caputo's sense, interested readers can refer to and the references therein. There are many favorable approximations for the Caputo fractional derivative, here we review several researches which provide some recent progress related to our work.…”

Section: Introductionmentioning

confidence: 99%

A linearized and second‐order unconditionally convergent scheme for coupled time fractional Klein‐Gordon‐Schrödinger equation

Lyu

Vong

2018

Numerical Methods Partial

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In this work, we study finite difference scheme for coupled time fractional Klein‐Gordon‐Schrödinger (KGS) equation. We proposed a linearized finite difference scheme to solve the coupled system, in which the fractional derivatives are approximated by some recently established discretization formulas. These formulas approximate the solution with second‐order accuracy at points different form the grid points in time direction. Taking advantage of this property, our proposed linearized scheme evaluates the nonlinear terms on the previous time level. As a result, iterative method is dispensable. The coupled terms in the scheme bring difficulties in analysis. By carefully studying these effects, we proved that the proposed scheme is unconditionally convergent and stable in discrete L 2 norm with energy method. Numerical results are included to justify the theoretical statements.

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

Cited by 72 publications

References 18 publications

Fast high order difference schemes for the time fractional telegraph equation

Fast high order difference schemes for the time fractional telegraph equation

Numerical analysis for Klein‐Gordon equation with time‐space fractional derivatives

A linearized and second‐order unconditionally convergent scheme for coupled time fractional Klein‐Gordon‐Schrödinger equation

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