# S11.

Solution:

The partial differential equation can be written by expressing the Laplace operator as follow:

Let’s apply the Fourier transformation first by x and then y.

It is known from the Fourier transformation rules:

Let’s execute Fourier transformation on both side of partial differential equation:

The second ordered partial differential equation has been transformed to second order linear differential equation with constant coefficient.

It is known, the solution of that differential equation is:
like harmonic oscillator.

Let’s use the available initial condition in Fourier frequency space:

Consequently:

Consequently is expressed by the initial condition in frequency space (i.e. by Fourier transformed expression):

Then can be obtained by the p and q variables inverse Fourier transformation
.