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.

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.

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 Inverse problem for fractional diffusion equation

2011 ◽  
Vol 14 (1) ◽  
Author(s):  
Vu Tuan
Keyword(s):  
Inverse Problem ◽  
Diffusion Coefficient ◽  
Lower Bound ◽  
Diffusion Equation ◽  
Spectral Data ◽  
A Priori ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
One Dimensional ◽  
Initial Distributions

AbstractWe prove that by taking suitable initial distributions only finitely many measurements on the boundary are required to recover uniquely the diffusion coefficient of a one dimensional fractional diffusion equation. If a lower bound on the diffusion coefficient is known a priori then even only two measurements are sufficient. The technique is based on possibility of extracting the full boundary spectral data from special lateral measurements.

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