2.3.7 In a square

problem number 89

Added December 20, 2018.

Example 28, taken from https://www.mapleprimes.com/posts/209970-Exact-Solutions-For-PDE-And-Boundary--Initial-Conditions-2018

In 2 space dimensions Solve for \(f(x,y,t)\) \[ I \hslash f_t = - \frac {\hslash ^2}{2 m} \nabla ^2 f \] With initial conditions \(f(x,y,0) = \sqrt {2} \left ( \sin (2 \pi x) \sin (\pi y)+ \sin (\pi x) \sin (3 \pi y) \right )\) and boundary conditions \begin {align*} f(0,y,t) &= 0 \\ f(1,y,t) &= 0 \\ f(x,1,t) &= 0 \\ f(x,0,t) &= 0 \end {align*}

pict
Figure 2.22:PDE specification

Mathematica


\[\left \{\left \{f(x,y,t)\to \sqrt {2} e^{-\frac {5 i \pi ^2 h t}{m}} \left (\sin (\pi x) \sin (3 \pi y)+\sin (2 \pi x) \sin (\pi y) e^{\frac {5 i \pi ^2 h t}{2 m}}\right )\right \}\right \}\]

Maple


\[f \left (x , y , t\right ) = \sqrt {2}\, \left (2 \cos \left (\pi x \right ) {\mathrm e}^{-\frac {5 i \pi ^{2} \mathit {hbar} t}{2 m}} \sin \left (\pi y \right )+{\mathrm e}^{-\frac {5 i \pi ^{2} \mathit {hbar} t}{m}} \sin \left (3 \pi y \right )\right ) \sin \left (\pi x \right )\]