Multi-time fractional diffusion equation

Published on Oct 7, 2013in European Physical Journal-special Topics2.707
· DOI :10.1140/EPJST/E2013-01975-Y
A. V. Pskhu1
Estimated H-index: 1
(RAS: Russian Academy of Sciences)
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Abstract
We construct a fundamental solution of a multi-time diffusion equation with the Dzhrbashyan-Nersesyan fractional differentiation operator with respect to the time variables. We give a representation for a solution of the Cauchy problem and prove the uniqueness theorem in the class of functions of fast growth. The corresponding results for equations with Riemann-Liouville and Caputo derivatives are obtained as particular cases of the proved assertions.
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#1Francesco MainardiH-Index: 66
Essentials of Fractional Calculus Essentials of Linear Viscoelasticity Fractional Viscoelastic Media Waves in Linear Viscoelastic Media: Dispersion and Dissipation Waves in Linear Viscoelastic Media: Asymptotic Methods Pulse Evolution in Fractional Viscoelastic Media The Eulerian Functions The Bessel Functions The Error Functions The Exponential Integral Functions The Mittag-Leffler Functions The Wright Functions.
#1Vasily E. Tarasov (MSU: Moscow State University)H-Index: 57
The calculus of derivatives and integrals of non-integer order go back to Leibniz, Liouville, Grunwald, Letnikov and Riemann. The fractional calculus has a long history from 1695, when the derivative of order α = 0.5 was described by Leibniz (Oldham and Spanier, 1974; Samko et al., 1993; Ross, 1975). The history of fractional vector calculus (FVC) is not so long. It has only 10 years and can be reduced to the papers (Ben Adda, 1997, 1998a, b, 2001; Engheta, 1998; Veliev and Engheta, 2004; Ivakhn...
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Abstract In the paper, a maximum principle for the generalized time-fractional diffusion equation over an open bounded domain G × ( 0 , T ) , G ⊂ R n is formulated and proved. The proof of the maximum principle is based on an extremum principle for the Caputo–Dzherbashyan fractional derivative that is given in the paper, too. The maximum principle is then applied to show that the initial-boundary-value problem for the generalized time-fractional diffusion equation possesses at most one classical...
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#1Jocelyn SabatierH-Index: 35
#2Om P. AgrawalH-Index: 36
Last. J. A. Tenreiro MachadoH-Index: 68
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In the last two decades, fractional (or non integer) differentiation has played a very important role in various fields such as mechanics, electricity, chemistry, biology, economics, control theory and signal and image processing. For example, in the last three fields, some important considerations such as modelling, curve fitting, filtering, pattern recognition, edge detection, identification, stability, controllability, observability and robustness are now linked to long-range dependence pheno...
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ductor, where the conductivity is sufficiently high, the displacement current density can be neglected. In this case, the conduction current density is given by the product of the electric field and the conductance. One of the aspects of with SE is it attenuates the higher frequency components of a signal. The SE was first verified by Kelvin in 1887. Since then many researchers developed work on the subject and presently a comprehensive physical model, based on the Maxwell equations, is well est...
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#1Anatoly A. KilbasH-Index: 26
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1. Preliminaries. 2. Fractional Integrals and Fractional Derivatives. 3. Ordinary Fractional Differential Equations. Existence and Uniqueness Theorems. 4. Methods for Explicitly solving Fractional Differential Equations. 5. Integral Transform Methods for Explicit Solutions to Fractional Differential Equations. 6. Partial Fractional Differential Equations. 7. Sequential Linear Differential Equations of Fractional Order. 8. Further Applications of Fractional Models. Bibliography Subject Index
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#1A. V. Pskhu (RAS: Russian Academy of Sciences)H-Index: 5
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#1A. V. Pskhu (RAS: Russian Academy of Sciences)H-Index: 5
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#1Anatoly A. Alikhanov (RAS: Russian Academy of Sciences)H-Index: 10
Solutions of the Dirichlet and Robin boundary value problems for the multi-term variable-distributed order diffusion equation are studied. A priori estimates for the corresponding differential and difference problems are obtained by using the method of the energy inequalities. The stability and convergence of the difference schemes follow from a priory estimates. The credibility of the obtained results is verified by performing numerical calculations for test problems.
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In this paper, a class of multi-term time fractional advection diffusion equations (MTFADEs) is considered. By finite difference method in temporal direction and finite element method in spatial direction, two fully discrete schemes of MTFADEs with different definitions on multi-term time fractional derivative are obtained. The stability and convergence of these numerical schemes are discussed. Next, a V-cycle multigrid method is proposed to solve the resulting linear systems. The convergence of...
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#1Vassili N. Kolokoltsov (Warw.: University of Warwick)H-Index: 27
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The purpose is to study the Cauchy problem for non-linear in time and space pseudo- differential equations. These include the fractional in time versions of Hamilton-Jacobi-Bellman (HJB) equations governing the limits of controlled scaled Continuous Time Random Walks (CTRWs). As a preliminary step which is of independent interest we analyse the corresponding linear equation proving its well-posedness and smoothing properties.
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