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Inverse Source Problems for Diffusion Equations and Fractional Diffusion Equations

Inverse Source Problems for Diffusion Equations and Fractional Diffusion Equations
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Anisotropic fractional diffusion equation

Physica A: Statistical Mechanics and its Applications, 2005
Abstract We analyze an anisotropic fractional diffusion equation that extends some known diffusion equations by considering a diffusion coefficient with spatial and time dependence, the presence of external forces and time fractional derivatives.
G.A. Mendes, E.K. Lenzi, R.S. Mendes, L.R. da Silva   +3 more
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Discrete fractional diffusion equation

Nonlinear Dynamics, 2015
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Wu, Guo-Cheng, Baleanu, Dumitru, Zeng, Sheng-Da, Deng, Zhen-Guo   +3 more
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Fractional diffusion and fractional heat equation

Advances in Applied Probability, 2000
This paper introduces a fractional heat equation, where the diffusion operator is the composition of the Bessel and Riesz potentials. Sharp bounds are obtained for the variance of the spatial and temporal increments of the solution. These bounds establish the degree of singularity of the sample paths of the solution.
Angulo, J. M., Ruiz-Medina, M. D., Anh, V. V., Grecksch, W.   +3 more
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Nonlinear fractional diffusion equation: Exact results

Journal of Mathematical Physics, 2005
The nonlinear fractional diffusion equation ∂tρ=r1−d∂rμ′{rd−1D(r,t;ρ)∂rμρν}−r1−d∂r{rd−1F(r,t)ρ}+α¯(t)ρ is studied by considering the diffusion coefficient D(r,t;ρ)=D(t)r−θργ and the external force F(r,t)=−k1(t)r+kαrα. In addition, a rich class of diffusive processes, including normal and anomalous ones, is obtained from the study present in this work.
Lenzi, E. K., Mendes, R. S., Fa, Kwok Sau, Moraes, L. S., da Silva, L. R., Lucena, L. S.   +5 more
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On a fractional reaction–diffusion equation

Zeitschrift für angewandte Mathematik und Physik, 2017
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de Andrade, Bruno, Viana, Arlúcio
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The fractional diffusion equation

Journal of Mathematical Physics, 1986
In one space—and in one time—dimension a diffusion equation is solved, where the first time derivative is replaced by the λ-fractional time derivative, 0<λ≤1. The solution is given in closed form in terms of Fox functions.
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Fractional diffusion and wave equations

Journal of Mathematical Physics, 1989
Diffusion and wave equations together with appropriate initial condition(s) are rewritten as integrodifferential equations with time derivatives replaced by convolution with tα−1/Γ(α), α=1,2, respectively. Fractional diffusion and wave equations are obtained by letting α vary in (0,1) and (1,2), respectively.
W. R. Schneider, W. Wyss
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Fractional diffusion equations and applications

AIP Conference Proceedings, 2007
This paper presents a method for an explicit analysis of the equations with fractional derivatives that describe important physical processes in solar wind plasmas, in plasmas of thermonuclear devices, etc. Space‐time fractional diffusions account for anomalous features, which are observed in such physical processes.
Emil Popescu, Cristiana Dumitrache, Vasile Mioc, Nedelia A. Popescu   +3 more
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Fractional Number Operator and Associated Fractional Diffusion Equations

Mathematical Physics, Analysis and Geometry, 2017
The author studies the fractional number operator as an analog of the finite-dimensional fractional Laplacian and gives a relation with the Ornstein-Uhlenbeck process. Using a semigroup approach, the solution of the Cauchy problem associated to the fractional number operator is presented.
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