The distributed-order fractional diffusion-wave equation of groundwater flow: Theory and application to pumping and slug tests

2015 ◽  
Vol 529 ◽  
pp. 1262-1273 ◽  
Author(s):  
Ninghu Su ◽  
Paul N. Nelson ◽  
Sarah Connor
Keyword(s):  
Wave Equation ◽  
Groundwater Flow ◽  
Flow Theory ◽  
Fractional Diffusion ◽  
Diffusion Wave ◽  
Slug Tests ◽  
Distributed Order

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 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.

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.

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