I want to solve$$\frac{\partial^2}{\partial t^2}u(z,t) + a\frac{\partial^2}{\partial z^2}u(z,t) + k\frac{\partial^4}{\partial z^4}u(z,t) = 0 $$with $u(z,0) = 1+0.1e^{-\frac{z^2}{2}}$. I'd like to study the dynamic of the wave.What is the simplest way to do it?I tried with finite difference method but my solution diverges very quickly!I use this iteration$$f_j^{n+1} = 2f_j^n - f_j^{n-1} + \frac{\Delta t^2}{\Delta z^2}a(f_{j+1}^n - 2f_j^{n} + f_{j-1}^n) +k \frac{\Delta t^2}{\Delta z^4}(f^n_{j-2}-4f^n_{j-1}+6f^n_{j}-4f^n_{j+1} + f^n_{j+2})$$where $n$ it's the time step and $j$ the space step.Improving my question after comments:I used a very small time step and this is what i obtainshock http://i62.tinypic.com/2mc7n0o.pngHow can avoid the reflection with this iteration method?
Solve a fourth order differential equation
finite difference;integral equations