How to solve a modified wave equation using pde toolbox?

Question

Thomas Mueller 2015-2-18

2
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https://ww2.mathworks.cn/matlabcentral/answers/179036-how-to-solve-a-modified-wave-equation-using-pde-toolbox

回答： Ludwig 2015-2-19

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