Anomalous diffusion and fractional diffusion equation: anisotropic media and external forces

2005 ◽  
Vol 347 (4-6) ◽  
pp. 160-169 ◽  
Author(s):  
M.F. de Andrade ◽  
E.K. Lenzi ◽  
L.R. Evangelista ◽  
R.S. Mendes ◽  
L.C. Malacarne
Keyword(s):  
Diffusion Equation ◽  
Anomalous Diffusion ◽  
Anisotropic Media ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
External Forces

FRACTIONAL DIFFUSION EQUATION, QUANTUM SUBDYNAMICS AND EINSTEIN'S THEORY OF BROWNIAN MOTION

2007 ◽  
Vol 21 (23n24) ◽  
pp. 3993-3999
Author(s):  
SUMIYOSHI ABE
Keyword(s):  
Brownian Motion ◽  
Quantum Theory ◽  
Fractional Calculus ◽  
Diffusion Equation ◽  
Master Equation ◽  
Anomalous Diffusion ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Quantum Master Equation ◽  
Objective System

The fractional diffusion equation for describing the anomalous diffusion phenomenon is derived in the spirit of Einstein's 1905 theory of Brownian motion. It is shown how naturally fractional calculus appears in the theory. Then, Einstein's theory is examined in view of quantum theory. An isolated quantum system composed of the objective system and the environment is considered, and then subdynamics of the objective system is formulated. The resulting quantum master equation is found to be of the Lindblad type.

scholarly journals An Analytical Solution of a Time-fractional Diffusion Equation Withexternal Force and Absorbent Term by Generalized Two-dimensional differential Transform Method

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Diffusion Equation ◽  
Initial Conditions ◽  
Fractional Diffusion ◽  
Differential Transform Method ◽  
Fractional Diffusion Equation ◽  
Transform Method ◽  
Differential Transform ◽  
Time Fractional Diffusion Equation ◽  
Generalized Differential ◽  
External Forces

In this paper, we have used generalized differential transform method in obtaining a general recurrencerelation for determining the solutions of time fractional diffusion equation with external force and absorbent term.Diffusion equations play an improtant part in energy transfer problems. Inclusion of fractional derivatives bring thenon-locality aspect into the physical system containing this equation. The obtained relation will help us to solvesuch equations with various external forces and initial conditions. Three illustrative examples have been discussed.

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