Time distributed-order diffusion-wave equation. II. Applications of Laplace and Fourier transformations

2009 ◽  
Vol 465 (2106) ◽  
pp. 1893-1917 ◽  
Author(s):  
Teodor M. Atanackovic ◽  
Stevan Pilipovic ◽  
Dusan Zorica
Keyword(s):  
Wave Equation ◽  
Fractional Derivative ◽  
Fractional Diffusion ◽  
Diffusion Wave ◽  
The Maximum Principle ◽  
Dimensional Diffusion ◽  
First Case ◽  
Special Cases ◽  
Telegraph Equations ◽  
Distributed Order

A Cauchy problem for a time distributed-order multi-dimensional diffusion-wave equation containing a forcing term is reinterpreted in the space of tempered distributions, and a distributional diffusion-wave equation is obtained. The distributional equation is solved in the general case of weight function (or distribution). Solutions are given in terms of solution kernels (Green's functions), which are studied separately for two cases. The first case is when the order of the fractional derivative is in the interval [0, 1], while, in the second case, the order of the fractional derivative is in the interval [0, 2]. Solutions of fractional diffusion-wave and fractional telegraph equations are obtained as special cases. Numerical experiments are also performed. An analogue of the maximum principle is also presented.

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.

scholarly journals Time-Fractional Diffusion-Wave Equation with Mass Absorption in a Sphere under Harmonic Impact

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 433 ◽  
Author(s):  
Bohdan Datsko ◽  
Igor Podlubny ◽  
Yuriy Povstenko
Keyword(s):  
Fourier Transform ◽  
Wave Equation ◽  
Fractional Derivative ◽  
Graphical Representation ◽  
Fractional Diffusion ◽  
Diffusion Wave ◽  
Mass Absorption ◽  
The Laplace Transform ◽  
Time Fractional Diffusion Equation ◽  
Time Fractional Derivative

The time-fractional diffusion equation with mass absorption in a sphere is considered under harmonic impact on the surface of a sphere. The Caputo time-fractional derivative is used. The Laplace transform with respect to time and the finite sin-Fourier transform with respect to the spatial coordinate are employed. A graphical representation of the obtained analytical solution for different sets of the parameters including the order of fractional derivative is given.

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.

scholarly journals Analysis of two Legendre spectral approximations for the variable-coefficient fractional diffusion-wave equation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Wenping Chen ◽  
Shujuan Lü ◽  
Hu Chen ◽  
Lihua Jiang
Keyword(s):  
Wave Equation ◽  
Fractional Derivative ◽  
Variable Coefficient ◽  
Equivalent Form ◽  
Fractional Diffusion ◽  
Stability And Convergence ◽  
Diffusion Wave ◽  
Fully Discrete ◽  
Spectral Approximations ◽  
Time Fractional Derivative

Abstract In this paper, we solve the variable-coefficient fractional diffusion-wave equation in a bounded domain by the Legendre spectral method. The time fractional derivative is in the Caputo sense of order $\gamma \in (1,2)$ γ ∈ ( 1 , 2 ) . We propose two fully discrete schemes based on finite difference in temporal and Legendre spectral approximations in spatial discretization. For the first scheme, we discretize the time fractional derivative directly by the $L_{1}$ L 1 approximation coupled with the Crank–Nicolson technique. For the second scheme, we transform the equation into an equivalent form with respect to the Riemann–Liouville fractional integral operator. We give a rigorous analysis of the stability and convergence of the two fully discrete schemes. Numerical examples are carried out to verify the theoretical results.

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 Asymptotic properties of solutions of the fractional diffusion-wave equation

2014 ◽  
Vol 17 (3) ◽  
Author(s):  
Anatoly Kochubei
Keyword(s):  
Wave Equation ◽  
Heat Equation ◽  
Fractional Derivative ◽  
Parabolic Equations ◽  
Hyperbolic Equations ◽  
Asymptotic Properties ◽  
Fractional Diffusion ◽  
Time Variable ◽  
Diffusion Wave

AbstractFor the fractional diffusion-wave equation with the Caputo-Djrbashian fractional derivative of order α ∈ (1, 2) with respect to the time variable, we prove an analog of the principle of limiting amplitude (well-known for the wave equation and some other hyperbolic equations) and a pointwise stabilization property of solutions (similar to a well-known property of the heat equation and some other parabolic equations).

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