How to solve diffusion equation in spherical coordinate using Matlab?

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Yashar 2019-2-18

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/445678-how-to-solve-diffusion-equation-in-spherical-coordinate-using-matlab

评论： Rohit Jain 2023-10-3

PDE toolbox in Matlab expects the equations in Cartesian coordinate. In this page it says what the general shape of the equations should be: https://nl.mathworks.com/help/pde/pde-coefficients.html

that is:

I want to solve the diffusion equations:

Thus, in my case m, a, and f are zero. If we are in Cartesian coordinate then d is one and c, the diffusion constant, is for example 0.001. However, I want to solve the equations in spherical coordinates. Here is a link to Laplacian in spherical coordinate:

http://mathworld.wolfram.com/LaplacesEquationSphericalCoordinates.html

How can I solve the diffusion equation in the spherical coordinates in Matlab? or How can I determine the matrix of coefficienct "c" in spherical coordinates?

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Yashar 2019-2-20

At the end I want to solve this problem:

One molecue diffuses through a sphere, which represents a cell, and reaches the surface of the sphere. At the surface this molecule converts to another molecule which attaches to membrane that is stays at the surface and diffuses only at the surface (like a 2D diffusion however on the surface of the sphere). This problem is the simplification of a larger problem. If I can solve this simplified problem, I hope I can solve the system of PDEs in this paper ( it is open acess and the equations are in the methods sections):

https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1003396

Here is a snapshot of the equations that at the end I hope to solve: