system of 2 pdes

I have to solve a system of 2 parabolic non linear partial differential equations in 2 dimensions with variables u1(x,y,t) and u2(x,y,t) . The system is :

du1/dt=k*(du1^2/dx^2 +du2^2/dy^2) +A*u2*exp((-b)/(u1-Ta))-A*C*(u1-Ta)

du2/dt=-Cs*u2*exp((-b)/(u1-Ta))

How can i solve it ?

Answer 1

[Prateekshya]

Hello Kostas,

To solve a system of nonlinear parabolic partial differential equations (PDEs) like the one you described, you can use numerical methods, as analytical solutions are typically not feasible for such complex systems. MATLAB provides tools like the Partial Differential Equation Toolbox, but for custom PDEs, you might need to implement a finite difference method or use MATLAB's pdepe function for simpler cases.

However, pdepe is more suited for 1D problems, so for a 2D problem, you will need to use a more general approach. Here is a basic outline of how you might proceed using finite difference methods:

• Discretize the Domain: Divide the spatial domain into a grid with points . Choose a time step Δt for temporal discretization.
• Initialize the Variables: Set initial conditions for u1 and u2 over the spatial domain.
• Apply Finite Difference Method: Use finite difference approximations for the spatial derivatives. For example, use central differences for second derivatives:
• Time Integration: Use an explicit or implicit time-stepping scheme (e.g., Forward Euler, Backward Euler, or Crank-Nicolson). For stability, implicit methods are often preferred for parabolic PDEs.
• Iterate Over Time: Update u1 and u2 at each time step using the discretized equations.
• Boundary Conditions: Implement appropriate boundary conditions (e.g., Dirichlet or Neumann) for u1 and u2.

You can write a basic MATLAB code according to this and modify later according to your actual requirements.

I hope this helps!