I am solving a PDE numerical problem. And I have already had a algorithm. However, it seems to be hard to find a specific example to test my solution. Could you please give me one?
The equation should be like that: \begin{equation} u_{t} = a^{2}u_{xx} + f \end{equation} \begin{equation} u(x,0) = u_{0}(x) \end{equation} \begin{equation} u(x,t) = u(x + 1,t) \end{equation} Please tell me the $f(x)$ , $u_{0}(x)$ , $a$ and of course the real solution $u(x,t)$.
My try: Use the Poisson Formula, I can get: \begin{equation} u(x,t) = \int_{-\infty}^{\infty} K(x-\xi,t)u_{0}(\xi)\mathbb{d}\xi + \int_{0}^{t}\mathbb{d}\tau \int_{-\infty}^{\infty} K(x-\xi,t-\tau)f(\xi,\tau)\mathbb{d}\xi \end{equation} where$$ K = \left\{ \begin{array}{ll} \frac{1}{2a\sqrt{\pi t}}\mathbb{e}^{-x^{2} / (4a^{2}t)}, & t > 0, \\ 0, & t\leq 0 \end{array} \right.$$ To simplify it, I assume $f\equiv0$ and $u_{0}(x) = x$ in $ [0,1)$. Then substitute it: \begin{equation} \int_{-\infty}^{\infty} \frac{1}{2a\sqrt{\pi t}}\mathbb{e}^{-(x-\xi)^{2} / (4a^{2}t)}u_{0}(\xi) \mathbb{d}\xi = \int_{-\infty}^{\infty}\frac{1}{2a\sqrt{\pi t}}\mathbb{e}^{-(x+1-\xi)^{2} / (4a^{2}t)} u_{0}(\xi) \mathbb{d}\xi. \end{equation} Because of the periodic condition, the integration can become: \begin{equation} \sum_{j=-\infty}^{+\infty} \int_{j}^{j+1} \frac{1}{2a\sqrt{\pi t}}\mathbb{e}^{-(x-\xi)^{2} / (4a^{2}t)}(\xi - j) \mathbb{d}\xi = \sum_{j=-\infty}^{+\infty} \int_{j}^{j+1} \frac{1}{2a\sqrt{\pi t}}\mathbb{e}^{-(x+1-\xi)^{2} / (4a^{2}t)}(\xi - j) \mathbb{d}\xi. \end{equation} But the integration is too tedious, maybe $u_{0}(x)$ can be more simple?
Apparently, if $u$ is $1-$periodic, so is $f$. Thus, writing $$ u(x,t)=\sum_{k\in\mathbb Z}\hat u_k(t)\,\mathrm{e}^{2\pi ikx}, \quad $f(x)=\sum_{k\in\mathbb Z}\hat f_k\,\mathrm{e}^{2\pi ikx}, $$ the PDE becomes: $$ 0=u_t-a^2u_{xx}-f=\sum_{k\in\mathbb Z}\Big(\frac{d}{dt}\hat u_k(t)+(2\pi ka)^2\hat u_k(t)-\hat f_k\Big)\,\mathrm{e}^{2\pi ikx}, $$ which implies that $$ \frac{d}{dt}\hat u_k(t)+(2\pi ka)^2\hat u_k(t)=\hat f_{\!\!k}, \quad\text{for all}\,\,\,k\in\mathbb Z, $$ etc...