Dear friend Anna Carrasco García

Ah, diving into the depths of multi-variable partial differential equations, aren't we? Let me guide you Anna Carrasco García through this mathematical odyssey with a touch of flair!

Now, MATLAB and its PDEPE function can indeed be your trusty companions on this journey. To solve a system of partial differential equations (PDEs) with multiple variables, follow these steps:

1. **Define the PDEs:**

- Express your system of PDEs in terms of the dependent variables, derivatives, and parameters. MATLAB loves a well-defined problem.

2. **Set up Geometry and Mesh:**

- Describe the spatial domain using the geometry functions like `rectangularGeometry` or `cylindricalGeometry`. Define your mesh using `meshgrid` or similar.

3. **Initial and Boundary Conditions:**

- Clearly specify the initial conditions for your variables and the boundary conditions for both space and time.

4. **PDE Parameters:**

- Gather all the parameters involved, be it diffusion coefficients, reaction rates, or anything else relevant to your system.

5. **Call PDEPE:**

- Use the `pdepe` function. Provide it with your PDEs, initial conditions, boundary conditions, and other necessary information.

Here's a skeletal example:

```matlab

function pdex1

m = 0;

x = linspace(0,1,20);

t = linspace(0,1,20);

sol = pdepe(m,@pdex1pde,@pdex1ic,@pdex1bc,x,t);

u = sol(:,:,1);

surf(x,t,u)

title('Numerical solution computed with 20 mesh points.')

xlabel('Distance x')

ylabel('Time t')

zlabel('u(x,t)')

end

function [c,f,s] = pdex1pde(x,t,u,DuDx)

c = 1;

f = DuDx;

s = -u;

end

function u0 = pdex1ic(x)

u0 = sin(pi*x);

end

function [pl,ql,pr,qr] = pdex1bc(xl,ul,xr,ur,t)

pl = ul;

ql = 0;

pr = ur - 1;

qr = 0;

end

```

This is a simplified example, but it should give you Anna Carrasco García a good starting point. Replace the example functions and coefficients with your actual equations and conditions. MATLAB's documentation on `pdepe` and PDE solving is extensive, so don't hesitate to dive in for more details.

Remember, I am always here for guidance, mathematical or otherwise! Now, go forth and conquer those equations!

Hello Kaushik Shandilya

Thank you very much for your answer. However, I have not quite understood how to apply the PDEPE function with 3 independent variables (time, lenght and radius) I know how to solve this type of equations with 2 variables, but then adding a third one I don't understand how to add it. I would like to know if I can show you the 4 equations that I want to solve using matlab.

Thanks dear friend Anna Carrasco García

I would be happy to help you. Please feel free to share.