How to solve a modified wave equation using pde toolbox?

Dear all,

I tried to solve the following 1D - pde using the pde-app:

u_tt - (x*u_x)_x= d*u_xxt

where d is a positive constant and the indices indicate the corresponding derivatives. As you can see, the first derivative with respect to time is needed. The hyperbolic mode of pde-tool only accepts the second derivative with respect to time. So I tried to disassemble the problem into a system of two equations:

u_t = v

v_t = (x*u_x)_x + d*v_xx

The tool accepts it as a generic system of parabolic equations but the solution diverges after the first steps of time ( timestep = 0.001).

So my questions are:

• Is it possible to disassemble the hyperbolic equation this way?
• How can I get the first derivative of time in the hyperbolic mode of pde-tool?

Thanks for your answers!

Some additional informations:

• Shape of the computational domain: rectangle (origin (0, 0), height = 0.1 , width = 10)
• Boundary conditions: left side (x = 0), bottom and top: Neumann => 0
• Right side (x = 10): time dependent Dirichlet condition.
• Initial condition: 0

Answer 1

[Ludwig]

I have similar problem and would be interested in the answer.