How can I solve a 4D PDE using numerical methods in MATLAB or Python?

[sks1983]

Hello,

I want to solve a 4-dimensional PDE problem using some numerical code. Possibly MATLAB or Python.

I have a solved a simple version of the PDE in 2D using MATLAB PDETool.
Also I solved a simplified pde in 3D using FiPy library in Python.

However, most MATLAB existing tools allow 2D problem solution, while FiPy allowed me to go upto 3D.

I am looking for some help for a Finite Difference Method to solve following PDE.
4 variables - x_1 to x_4

In Latex Notation:

[itex]\frac{\partial^2 u}{\partial x_3^2} + \frac{\partial^2 u}{\partial x_4^2} -
x_3 \frac{\partial u}{\partial x_1} - x_4 \frac{\partial u}{\partial x_2} = 0 [/itex]

Boundary Condition: (boundary is circles in x_1 - x_2 plane.)(x_3 x_4 in R plane)

[itex]u = 0 \ \text{for} \ x_1^2 + x_2^2 = 1 \ \text{and} \ (x_1 x_3 + x_2 x_4) \geq 0 [/itex]
[itex]u = 1 \ \text{for} \ x_1^2 + x_2^2 = 0.1 \ \text{and} \ (x_1 x_3 + x_2 x_4) < 0 [/itex]

Appreciate any help and comments.

Thanks.

 

[jvicens]

It is great to hear that you have already made some progress in solving your PDE using MATLAB and FiPy. Both of these tools are commonly used for solving PDE problems, so you are on the right track.

In terms of solving your 4-dimensional PDE, there are a few options you can consider. One approach is to use a multi-dimensional extension of the finite difference method, such as the Alternating Direction Implicit (ADI) method. This method allows for efficient solving of PDEs in multiple dimensions and can be implemented using MATLAB or Python. Another option is to use a spectral method, which involves representing the solution as a sum of basis functions and solving for the coefficients using a matrix equation. This method can also be implemented using MATLAB or Python.

In terms of the boundary conditions, you can consider using a combination of Dirichlet and Neumann boundary conditions to represent the circles and the inequality constraints, respectively. These types of boundary conditions are commonly used in PDE solving and can be easily implemented in both MATLAB and Python.

I hope this helps and please let me know if you have any further questions. Best of luck in your PDE solving journey!