Efficient Numerical Solution of the Multi-Term Time Fractional Diffusion-Wave Equation

2015 ◽  
Vol 5 (1) ◽  
pp. 1-28 ◽  
Author(s):  
Jincheng Ren ◽  
Zhi-Zhong Sun
Keyword(s):  
Wave Equation ◽  
Fractional Derivatives ◽  
Fractional Diffusion ◽  
Unconditional Stability ◽  
Caputo Fractional Derivatives ◽  
Diffusion Wave ◽  
One Dimensional ◽  
Compact Difference ◽  
L1 Approximation ◽  
Spatial Discretisation

AbstractSome efficient numerical schemes are proposed to solve one-dimensional and two-dimensional multi-term time fractional diffusion-wave equation, by combining the compact difference approach for the spatial discretisation and an L1 approximation for the multi-term time Caputo fractional derivatives. The unconditional stability and global convergence of these schemes are proved rigorously, and several applications testify to their efficiency and confirm the orders of convergence.

scholarly journals Efficient and Stable Numerical Methods for Multi-Term Time Fractional Sub-Diffusion Equations

2014 ◽  
Vol 4 (3) ◽  
pp. 242-266 ◽  
Author(s):  
Jincheng Ren ◽  
Zhi-zhong Sun
Keyword(s):  
Fractional Derivatives ◽  
Diffusion Equations ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
Caputo Fractional Derivatives ◽  
Numerical Examples ◽  
One Dimensional ◽  
Compact Difference ◽  
Spatial Discretisation ◽  
The Stability

AbstractSome efficient numerical schemes are proposed for solving one-dimensional (1D) and two-dimensional (2D) multi-term time fractional sub-diffusion equations, combining the compact difference approach for the spatial discretisation and L1 approximation for the multi-term time Caputo fractional derivatives. The stability and convergence of these difference schemes are theoretically established. Several numerical examples are implemented, testifying to their efficiency and confirming their convergence order.

Inversion of the Initial Value for a Time-Fractional Diffusion-Wave Equation by Boundary Data

2020 ◽  
Vol 20 (1) ◽  
pp. 109-120 ◽  
Author(s):  
Suzhen Jiang ◽  
Kaifang Liao ◽  
Ting Wei
Keyword(s):  
Inverse Problem ◽  
Wave Equation ◽  
Fractional Diffusion ◽  
Initial Value ◽  
Diffusion Wave ◽  
One Dimensional ◽  
Finite Dimensional Approximation ◽  
Finite Dimensional ◽  
Dimensional Approximation ◽  
Continuation Technique

AbstractIn this study, we consider an inverse problem of recovering the initial value for a multi-dimensional time-fractional diffusion-wave equation. By using some additional boundary measured data, the uniqueness of the inverse initial value problem is proven by the Laplace transformation and the analytic continuation technique. The inverse problem is formulated to solve a Tikhonov-type optimization problem by using a finite-dimensional approximation. We test four numerical examples in one-dimensional and two-dimensional cases for verifying the effectiveness of the proposed algorithm.

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.

Cauchy and Signaling Problems for the Time-Fractional Diffusion-Wave Equation

2014 ◽  
Vol 136 (5) ◽  
Author(s):  
Yuri Luchko ◽  
Francesco Mainardi
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Diffusion Equation ◽  
Fundamental Solutions ◽  
Fractional Diffusion ◽  
The Other ◽  
Diffusion Wave ◽  
One Dimensional ◽  
The One ◽  
Propagation Velocities

In this paper, some known and novel properties of the Cauchy and signaling problems for the one-dimensional time-fractional diffusion-wave equation with the Caputo fractional derivative of order β,1≤β≤2 are investigated. In particular, their response to a localized disturbance of the initial data is studied. It is known that, whereas the diffusion equation describes a process where the disturbance spreads infinitely fast, the propagation velocity of the disturbance is a constant for the wave equation. We show that the time-fractional diffusion-wave equation interpolates between these two different responses in the sense that the propagation velocities of the maximum points, centers of gravity, and medians of the fundamental solutions to both the Cauchy and the signaling problems are all finite. On the other hand, the disturbance spreads infinitely fast and the time-fractional diffusion-wave equation is nonrelativistic like the classical diffusion equation. In this paper, the maximum locations, the centers of gravity, and the medians of the fundamental solution to the Cauchy and signaling problems and their propagation velocities are described analytically and calculated numerically. The obtained results for the Cauchy and the signaling problems are interpreted and compared to each other.

scholarly journals Analytical Solutions of the Diffusion–Wave Equation of Groundwater Flow with Distributed-Order of Atangana–Baleanu Fractional Derivative

2021 ◽  
Vol 11 (9) ◽  
pp. 4142
Author(s):  
Nehad Ali Shah ◽  
Abdul Rauf ◽  
Dumitru Vieru ◽  
Kanokwan Sitthithakerngkiet ◽  
Poom Kumam
Keyword(s):  
Wave Equation ◽  
Fractional Derivative ◽  
Groundwater Flow ◽  
Analytical Solutions ◽  
Fractional Derivatives ◽  
Integral Transforms ◽  
Fractional Diffusion ◽  
Diffusion Wave ◽  
Time Fractional Derivative ◽  
Distributed Order

A generalized mathematical model of the radial groundwater flow to or from a well is studied using the time-fractional derivative with Mittag-Lefler kernel. Two temporal orders of fractional derivatives which characterize small and large pores are considered in the fractional diffusion–wave equation. New analytical solutions to the distributed-order fractional diffusion–wave equation are determined using the Laplace and Dirichlet-Weber integral transforms. The influence of the fractional parameters on the radial groundwater flow is analyzed by numerical calculations and graphical illustrations are obtained with the software Mathcad.

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