Some properties of the fundamental solution to the signalling problem for the fractional diffusion-wave equation

Open Physics ◽  
2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Yuri Luchko ◽  
Francesco Mainardi
Keyword(s):  
Wave Equation ◽  
Fundamental Solution ◽  
Wave Equations ◽  
Propagation Speed ◽  
Fractional Diffusion ◽  
Maximum Point ◽  
Diffusion Wave ◽  
Constant Coefficients ◽  
The One ◽  
Signalling Problem

AbstractIn this paper, the one-dimensional time-fractional diffusion-wave equation with the Caputo fractional derivative of order α, 1 ≤ α ≤ 2 and with constant coefficients is revisited. It is known that the diffusion and the wave equations behave quite differently regarding their response to a localized disturbance. Whereas the diffusion equation describes a process where a disturbance spreads infinitely fast, the propagation speed 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 and investigate the behavior of its fundamental solution for the signalling problem in detail. In particular, the maximum location, the maximum value, and the propagation velocity of the maximum point of the fundamental solution for the signalling problem are described analytically and calculated numerically.

scholarly journals Subordination in a class of generalized time-fractional diffusion-wave equations

2018 ◽  
Vol 21 (4) ◽  
pp. 869-900 ◽  
Author(s):  
Bazhlekova Emilia
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Fractional Diffusion ◽  
Bernstein Function ◽  
Present Survey ◽  
Diffusion Wave ◽  
Survey Paper ◽  
Completely Monotone ◽  
Generalized Time ◽  
Distributed Order

Abstract Motivated by recently proposed generalizations of the diffusion-wave equation with the Caputo time fractional derivative of order α ∈ (1, 2), in the present survey paper a class of generalized time-fractional diffusion-wave equations is introduced. Its definition is based on the subordination principle for Volterra integral equations and involves the notion of complete Bernstein function. Various members of this class are surveyed, including the distributed-order time-fractional diffusion-wave equation and equations governing wave propagation in viscoelastic media with completely monotone relaxation moduli.

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 On Some New Properties of the Fundamental Solution to the Multi-Dimensional Space- and Time-Fractional Diffusion-Wave Equatio

2017 ◽  
Author(s):  
Yuri Luchko
Keyword(s):  
Wave Equation ◽  
Fundamental Solution ◽  
Open Problem ◽  
Special Functions ◽  
Dimensional Space ◽  
Fractional Diffusion ◽  
Two Dimensional ◽  
Diffusion Wave ◽  
Space And Time ◽  
Mellin Convolution

In this paper, some new properties of the fundamental solution to the multi-dimensional space- and time-fractional diffusion-wave equation are deduced. We start with the Mellin-Barnes representation of the fundamental solution that was derived in the previous publications of the author. The Mellin-Barnes integral is used to get two new representations of the fundamental solution in form of the Mellin convolution of the special functions of the Wright type. Moreover, some new closed form formulas for particular cases of the fundamental solution are derived. In particular, we solve an open problem of representation of the fundamental solution to the two-dimensional neutral-fractional diffusion-wave equation in terms of the known special functions.

scholarly journals FUNGSI WRIGHT SEBAGAI SOLUSI ANALITIK PERSAMAAN DIFUSI-GELOMBANG FRAKSIONAL PADA MEDIA VISKOELASTIS

2020 ◽  
Vol 3 (1) ◽  
pp. 19-33
Author(s):  
Ray Novita Yasa ◽  
Agus Yodi Gunawan
Keyword(s):  
Wave Equation ◽  
Analytical Solution ◽  
Fractional Derivative ◽  
Wave Equations ◽  
Equations Of Motion ◽  
Fractional Diffusion ◽  
Form Solution ◽  
Wright Function ◽  
Diffusion Wave ◽  
Viscoelastic Media

A fractional diffusion-wave equations in a fractional viscoelastic media can be constructed by using equations of motion and kinematic equations of viscoelasticmaterial in fractional order. This article concerns the fractional diffusion-wave equations in the fractional viscoelastic media for semi-infinite regions that satisfies signalling boundary value problems. Fractional derivative was used in Caputo sense. The analytical solution of the fractional diffusion-wave equation in the fractional viscoelastic media was solved by means of Laplace transform techniques in the term of Wright function for simple form solution. For general parameters, Numerical Inverse Laplace Transforms (NILT) was used to determine the solution.

scholarly journals Approximate Solutions of Time Fractional Diffusion Wave Models

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 923 ◽  
Author(s):  
Abdul Ghafoor ◽  
Sirajul Haq ◽  
Manzoor Hussain ◽  
Poom Kumam ◽  
Muhammad Asif Jan
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Quadrature Rule ◽  
Approximate Solutions ◽  
Fractional Diffusion ◽  
Haar Wavelets ◽  
Test Problems ◽  
Algebraic Equations ◽  
Diffusion Wave ◽  
Wave Models

In this paper, a wavelet based collocation method is formulated for an approximate solution of (1 + 1)- and (1 + 2)-dimensional time fractional diffusion wave equations. The main objective of this study is to combine the finite difference method with Haar wavelets. One and two dimensional Haar wavelets are used for the discretization of a spatial operator while time fractional derivative is approximated using second order finite difference and quadrature rule. The scheme has an excellent feature that converts a time fractional partial differential equation to a system of algebraic equations which can be solved easily. The suggested technique is applied to solve some test problems. The obtained results have been compared with existing results in the literature. Also, the accuracy of the scheme has been checked by computing L 2 and L ∞ error norms. Computations validate that the proposed method produces good results, which are comparable with exact solutions and those presented before.

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