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1

If you had used an indexing notation, probably you could have noticed the error yourself, and it was very hard for me to see it too. Take the line J11 = phi*eye(N) + (3*r.^2 + m.^2); for example. You are adding a matrix phi*eye(N) and a vector (3*r.^2 + m.^2) together. This is an allowed operation in MATLAB for a few years now as it has good uses in data ...

3

There are some unknowns in what you are doing but for simplicity, suppose we want to find $u(t)$ as discrete times $t_1, t_2, \cdots, t_n$. Let $\textbf{F} = [F(t_1), F(t_2), \cdots, F(t_n)]^T$ and $\textbf{u} = [u(t_1), u(t_2), \cdots, u(t_n)]^T$ be column vectors representing $F$ and $u$ evaluated at the desired times. From your problem statement, you wish ...

7

Julia's DifferentialEquations.jl has a lot of tooling for automatically deriving (sparse) matrices. For more information, see the JuliaCon 2020 video on Auto-Optimization and Parallelism in DifferentialEquations.jl. Combined with the orders of magnitude acceleration commonly seen over the MATLAB solvers, this might be a good option for you and is a quick ...

4

It looks like you have a advection diffusion PDE discretized with finite differences. This gives an ODE of the form $$y' = f(y) = A y + D y,$$ where $A$ is the discretized advection operator and $D$ is the discretized diffusion operator. In your case, it seems you have zero Dirchlet and Neumann boundary conditions which can be encoded in the $A$ and $D$ ...

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