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

In summary, the conversation is about solving an equation in a PDF that describes anisotropic diffusion in 3D with an additional term for hydrogen bonding. The person has considered using Laplace transforms and solving in the Laplace domain, but is unsure how to approach it and wonders if solving numerically in three dimensions would be computationally expensive. They are looking for suggestions on how to arrive at an analytical solution or an efficient way to implement it numerically. The equation can be simplified by separating variables and leads to a system in s, s' and S, S'. However, it is unclear how to proceed without making assumptions about S' or n/N.
  • #1
landon244
1
0
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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  • #2
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.
 

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

1. What is a second order coupled partial differential equation (PDE)?

A second order coupled PDE is a type of mathematical equation that involves multiple variables and their partial derivatives up to the second order. It describes the relationship between these variables and how they change with respect to each other.

2. Why is it important to solve second order coupled PDEs?

Solving second order coupled PDEs is important in many fields of science and engineering, as it allows us to model and understand complex systems and phenomena. This can lead to insights and advancements in various areas, such as physics, chemistry, and biology.

3. How is a solution to a second order coupled PDE in x, y, z, and time obtained?

There are various methods for obtaining solutions to second order coupled PDEs. Some common approaches include separation of variables, Fourier transforms, and numerical methods. The specific method used will depend on the nature of the PDE and the available tools and resources.

4. What are some real-world applications of second order coupled PDEs?

Second order coupled PDEs have a wide range of applications in science and engineering. For example, they can be used to model heat transfer, fluid dynamics, and electromagnetic fields. They are also used in the design and analysis of structures, such as bridges and buildings.

5. What are the limitations of using a second order coupled PDE to model a system?

While second order coupled PDEs can be powerful tools for modeling and understanding complex systems, they also have some limitations. These equations may not always accurately reflect the behavior of a system, as they rely on simplifying assumptions and may not take all factors into account. Additionally, obtaining a solution to a second order coupled PDE can be challenging and time-consuming, especially for highly complex systems.

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