Second Home work

1. Solve analytically the following Initial Value Problem:
   $$u_{tt}(x,y,t) + u_{xxxx}(x,y,t)+ u_{yyyy}(x,y,t)=0, \ u(x,y,t=0)=... ...2)),\ u_t(x,y,t=0)=0, u(x=-\pi,y,t)=u(x=\pi,t) = u(x,y=-\pi,t)=u(x,y=\pi,t)=0.$$
   Produce an animation of $u(x,y,t)$ as a function of time. If you use Mathematica, the “Plot3D” and “Animate” commands may do the trick.

   If you can, add the $2 u_{xx} u_{yy}$ term to the equation, it will make the equation more physically realistic.

2. In class we have solved by using perturbation method and multiple time scale method the Lienard type equastion
   $$\ddot x(t) + x(t) + \epsilon x^2(t) \dot x(t) = 0, x(0)=0, \dot x(t)=1.$$
   We have done the first order perturbation theory to find the slow time evolution of the Amplitude.
3. Please consider the Duffing oscillator:
   $$\ddot x(t) + x(t) + \epsilon x^3(t) = 0, x(0)=0, \dot x(t)=1,$$
   and use use perturbation theory and multiple time scale to find the slow evolution of the amplitude and nonlinear frequency shift.
4. Proove that

   $$\sum_{n=0}^{N-1} e^{\,2\pi i (k-\ell)n/N}$$

   $$= N\,\delta_{k\ell}.$$

   (3)

   Consider the Discrete Fourier Tranform.

   	$$X_k$$	$$= \sum_{n=0}^{N-1} x_n \, e^{-\,2\pi i\, kn/N}, \qquad k=0,1,\dots,N-1 ,$$	(1)
   	$$x_n$$	$$= \frac{1}{N} \sum_{k=0}^{N-1} X_k \, e^{\,2\pi i\, kn/N}, \qquad n=0,1,\dots,N-1 ,$$	(2)

   Proove that the inverse Fourier transform of a Fourier transform is identity transform.