scholarly journals A second-order accurate numerical method for the space–time tempered fractional diffusion-wave equation

2017 ◽  
Vol 68 ◽  
pp. 87-93 ◽  
Author(s):  
Minghua Chen ◽  
Weihua Deng
Keyword(s):  
Wave Equation ◽  
Numerical Method ◽  
Fractional Diffusion ◽  
Space Time ◽  
Second Order ◽  
Diffusion Wave

scholarly journals Existence of Solution of Space–Time Fractional Diffusion-Wave Equation in Weighted Sobolev Space

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Kangqun Zhang
Keyword(s):  
Wave Equation ◽  
Sobolev Space ◽  
Fractional Derivatives ◽  
Weighted Sobolev Space ◽  
Fractional Diffusion ◽  
Space Time ◽  
Existence Of Solution ◽  
Multiplier Theorem ◽  
Diffusion Wave ◽  
Loss Of Regularity

In this paper, we consider Cauchy problem of space-time fractional diffusion-wave equation. Applying Laplace transform and Fourier transform, we establish the existence of solution in terms of Mittag-Leffler function and prove its uniqueness in weighted Sobolev space by use of Mikhlin multiplier theorem. The estimate of solution also shows the connections between the loss of regularity and the order of fractional derivatives in space or in time.

scholarly journals Numerical Algorithms for the Fractional Diffusion-Wave Equation with Reaction Term

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Hengfei Ding ◽  
Changpin Li
Keyword(s):  
Wave Equation ◽  
Difference Scheme ◽  
Fourier Method ◽  
Numerical Algorithms ◽  
Fractional Diffusion ◽  
Second Order ◽  
Diffusion Wave ◽  
Order Difference ◽  
Reaction Term ◽  
Liouville Derivative

Two numerical algorithms are derived to compute the fractional diffusion-wave equation with a reaction term. Firstly, using the relations between Caputo and Riemann-Liouville derivatives, we get two equivalent forms of the original equation, where we approximate Riemann-Liouville derivative by a second-order difference scheme. Secondly, for second-order derivative in space dimension, we construct a fourth-order difference scheme in terms of the hyperbolic-trigonometric spline function. The stability analysis of the derived numerical methods is given by means of the fractional Fourier method. Finally, an illustrative example is presented to show that the numerical results are in good agreement with the theoretical analysis.

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