Modeling a system of partial differential equations that contain spatial and time-dependent variables

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R'mani Haulcy 2019-10-14

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https://ww2.mathworks.cn/matlabcentral/answers/485243-modeling-a-system-of-partial-differential-equations-that-contain-spatial-and-time-dependent-variable

编辑： R'mani Haulcy 2019-10-14

UPDATE: I know this may all seem overwhelming. The shortened version of my question is: given that the model was simulated in the rectangular domain below and that the six variables that have spatial derivatives in their time derivative equations satisfy periodic boundary conditions, can the finite difference method be used to simulate this model in MATLAB? It seems like some information is missing, particularly information that explains how to calculate the value of each variable at position x,y. Is that information somehow embedded in the fact that those equations satisfy periodic boundary conditions?

I am trying to replicate the plots in this paper using forward euler in MATLAB. The model consists of 18 partial differential equations (equations 1 - 18 in the paper). According to the paper, the model was simulated in a rectangular domain

and 6 of the unknown variables satisfy periodic boundary conditions (these are the variables that have spatial derivatives in the corresponding derivative equations).

I am having trouble simulating the Δ terms. I have tried to incorporate them into my code by using finite difference to generate a matrix that represents the delta terms but I'm not sure how to incorporate the periodic boundary conditions. I read online that this can be reflected in the matrix by shifting the rows but that hasn't lead to the correct results.

Can someone explain how I can approximate the delta terms in equations 9, 10, and 15-18 using a matrix and while satisfying periodic boundary conditions and the initial conditions provided in the paper? I attached all of my code. test_Alzheimer.m is the file that should be run.

These are my current results:

Desired results:

Only 5 of the plots are currently correct. I have been working really hard on this and haven't been able to fix the remaining plots because I don't know how to properly calculate the delta terms. I have tried to contact the authors of the paper to get more information but have not had any luck so far. Any help is greatly appreciated. I think I'm close. I just need a little help. The majority of the code has been written. I think I just have bugs somewhere because I don't fully understand how the delta terms were calculated.