Analytical solution of the time-dependent photon diffusion equation for a layered medium

1999 ◽  
Author(s):  
Fabrizio Martelli ◽  
Angelo Sassaroli ◽  
Yukio Yamada
Keyword(s):  
Analytical Solution ◽  
Diffusion Equation ◽  
Layered Medium ◽  
Time Dependent ◽  
Photon Diffusion ◽  
Photon Diffusion Equation

Analytical Solution of One Dimension Time Dependent Advection- Diffusion Equation

2016 ◽  
Vol 4 (2) ◽  
pp. 67-73
Author(s):  
A. A. Marrouf ◽  
Maha S. El-Otaify ◽  
Adel S. Mohamed ◽  
Galal Ismail ◽  
Khaled S. M. Essa
Keyword(s):  
Analytical Solution ◽  
Diffusion Equation ◽  
Time Dependent ◽  
One Dimension ◽  
Advection Diffusion Equation ◽  
Advection Diffusion

scholarly journals Stable, Explicit, Leapfrog-Hopscotch Algorithms for the Diffusion Equation

Computation ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 92
Author(s):  
Ádám Nagy ◽  
Issa Omle ◽  
Humam Kareem ◽  
Endre Kovács ◽  
Imre Ferenc Barna ◽  
...  
Keyword(s):  
Analytical Solution ◽  
Diffusion Equation ◽  
Initial Conditions ◽  
Numerical Algorithms ◽  
Time Dependent ◽  
Random Parameters ◽  
Order Of Accuracy ◽  
Kummer Functions ◽  
New Methods ◽  
Large Systems

In this paper, we construct novel numerical algorithms to solve the heat or diffusion equation. We start with 105 different leapfrog-hopscotch algorithm combinations and narrow this selection down to five during subsequent tests. We demonstrate the performance of these top five methods in the case of large systems with random parameters and discontinuous initial conditions, by comparing them with other methods. We verify the methods by reproducing an analytical solution using a non-equidistant mesh. Then, we construct a new nontrivial analytical solution containing the Kummer functions for the heat equation with time-dependent coefficients, and also reproduce this solution. The new methods are then applied to the nonlinear Fisher equation. Finally, we analytically prove that the order of accuracy of the methods is two, and present evidence that they are unconditionally stable.

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