Retrieval of the optical properties of a layered medium based on an exact analytical solution of the time-dependent diffusion equation

2003 ◽  
Author(s):  
Fabrizio Martelli ◽  
Samuele del Bianco ◽  
Angelo Sassaroli ◽  
Giovanni Zaccanti
Keyword(s):  
Optical Properties ◽  
Analytical Solution ◽  
Diffusion Equation ◽  
Layered Medium ◽  
Exact Analytical Solution ◽  
Time Dependent

scholarly journals Exact Analytical Solution for 3D Time-Dependent Heat Conduction in a Multilayer Sphere with Heat Sources Using Eigenfunction Expansion Method

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Nemat Dalir
Keyword(s):  
Heat Conduction ◽  
Analytical Solution ◽  
Eigenfunction Expansion ◽  
Radial Direction ◽  
Exact Analytical Solution ◽  
Internal Heat ◽  
Expansion Method ◽  
Time Dependent ◽  
Heat Sources ◽  
Eigenfunction Expansion Method

An exact analytical solution is obtained for the problem of three-dimensional transient heat conduction in the multilayered sphere. The sphere has multiple layers in the radial direction and, in each layer, time-dependent and spatially nonuniform volumetric internal heat sources are considered. To obtain the temperature distribution, the eigenfunction expansion method is used. An arbitrary combination of homogenous boundary condition of the first or second kind can be applied in the angular and azimuthal directions. Nevertheless, solution is valid for nonhomogeneous boundary conditions of the third kind (convection) in the radial direction. A case study problem for the three-layer quarter-spherical region is solved and the results are discussed.

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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