Need help in solving 2D Parabolic PDE through PDE toolbox.

Variables Description:

h is film thickness

p is pressure (Variable)

grid is written in x and z

got the coefficient of pde through ‘pdeCoefficients’

Formatted as a simplified equation.

Code Snippet:

syms x

h = 1 + cos(x);

syms p(x,z)

% pdeeq = diff(p,x,x) + diff(p,z,z) + (diff(h,x)/h)*diff(p,x) - diff(h,x)/h;

pdeeq = laplacian(p,[x, z]) + (diff(h,x)/h)*diff(p,x) - diff(h,x)/h;

coeffs = pdeCoefficients(pdeeq,p)

coeffs.f('show')

coeffs.c('show')

Error:

Warning: After extracting m, d, and c, some gradients remain. Writing all remaining terms to f.

Note:

Seems like this equation can’t be translated in divergence form.

Can this PDE be solved, if yes through PDE toolbox can anyone please guide?

Answer 1

[Udit06]

Hi Aditya,

I understand that you are trying to solve a 2-D parabolic PDE that cannot be translated in a form which can be solved using PDE toolbox as specified in the following documentation.

https://www.mathworks.com/help/pde/ug/equations-you-can-solve.html

You can solve such equations using the numerical methods for solving PDE such as finite difference method or finite element method. You can refer to the following resources to understand more about the finite difference method and finite element method respectively.

• https://people.maths.ox.ac.uk/suli/fem.pdf
• https://www.mathworks.com/help/pde/ug/basics-of-the-finite-element-method.html

I hope this helps.