Solution to Second Order Coupled PDE in x,y,z, and time

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SUMMARY

The discussion focuses on solving a second-order coupled partial differential equation (PDE) that models anisotropic diffusion in three dimensions, incorporating hydrogen bonding dynamics. The user has explored Laplace transforms and numerical inversion methods but seeks an analytical solution or a more efficient numerical approach, potentially using the Crank-Nicolson method. The equation can be simplified by separating variables, leading to a system of equations that may be solvable using analytic methods under certain assumptions about the variables.

PREREQUISITES
  • Understanding of partial differential equations (PDEs)
  • Familiarity with Laplace transforms and numerical inversion techniques
  • Knowledge of numerical methods, specifically Crank-Nicolson
  • Concepts of anisotropic diffusion and hydrogen bonding in materials
NEXT STEPS
  • Research methods for solving second-order coupled PDEs analytically
  • Explore advanced numerical techniques for PDEs, focusing on Crank-Nicolson implementation
  • Study variable separation methods in the context of PDEs
  • Investigate linearization techniques for simplifying complex equations
USEFUL FOR

Mathematicians, physicists, and engineers working on diffusion processes, particularly those dealing with complex materials and requiring advanced analytical and numerical methods for PDEs.

landon244
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I'm trying to solve equation in the attached pdf, which describes anistropic diffusion in 3D with an additional term to account for hydrogen bonding and unbonding of the diffusing substance to the medium. I've considered Laplace transforms, then solving in the Laplace domain, then inverting numerically. Ideally I would get a fully analytical solution, but I'm not sure how to approach it. Solving it numerically in three dimensions (using Crank-Nicolson or something similar) would undoubtedly be very computationally expensive, correct?


If anyone has any suggestions on how to arrive at an analytical solution to this, or an efficient way to implement it numerically, I would greatly appreciate your input. I’ve been struggling with this for some time now, and just haven’t gotten very far. Thanks in advance.
 

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The equation simplifies considerably if we separate the variables, i.e., n=p(x)q(y)r(z)s(t)
and N= M(x,y,z)S(t). On substitution & elimination of n/N , we end up with a system in
s,s' and S,S'.
I can't presume to advise you on modelling the diffusion. Yet, the resulting equations are amenable to analytic methods if you either assume S' <<beta S(to linearise ) or assume something about n/N.
 

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