Solving second-order non-linear PDE

I am trying to solve this second order differential equation

Where

θ is a function of space (x) and time (t),

κ is a function of space. This is a known ramp function that starts at 0 and increases to a fixed value.

v is constant and is

A is a constant.

With initial conditions at of ,

I have tried using pdepe but I am struggling to get it into a form that is acceptable. I have also attempted reformating it as an ODE but wasn't able to get any resonable solutions.

Is this a feasible equation that can be solved with Matlabs solvers?

Thanks

[Aditya Patil]

Can you verify the following? If v is constant and v = x/t, then theta is function of only t(or x), as x = vt. Similarly k is also function of t.

[Felix]

Yes, with the chain rule we can make it into solely a function of x with , here v is constant so (and the dash is derivative wrt x). This gives .

But i can't solve this one either.

Answer 1

[Aditya Patil]

As per my understanding, the core issue here is with the variable k which needs to be saturated. In other words,

k = min(0, max(C, x))

For some constant C.

This is currently not supported by the ODE solvers. More about this in this answer.

As a workaround, you can set the above condition in the odefun parameter of the solver, say ode45.

On a side note, you can also use Simulink. See the attached file for example.

t = [1:0.1:20];
x = sin(t);
input = [t;x]';
sim("differentialExample");