I'm trying to solve heat equation on a half-plane using fourier transform and I don't understand why it doesn't work (yes, I know we can use method of images or sine-transform, but I want to understand why I cannot use this way). We've got following problem $$\left\{ \begin{array}{l}\frac{{\partial \psi}}{{\partial t}} = \frac{{{\partial ^2}\psi}}{{\partial {x^2}}}\\\psi (0,t) = 0\\\psi (x,0) = \delta (a - x)\\x \ge 0,t > 0\end{array} \right. $$ Extend the task to whole axis and use fourier transform respect to x $$\hat \psi (k,t) = \int\limits_{ - \infty }^{ + \infty } {\psi (x,t){e^{ikx}}dx} $$ we get $$\left\{ \begin{array}{l}\frac{{\partial \hat \psi }}{{\partial t}} = - {k^2}\hat \psi \\\int\limits_{ - \infty }^{ + \infty } {\hat \psi (k,t)dk} = 0\\\hat \psi (k,0) = {e^{ika}}\end{array} \right. $$ The solution of ODE (1) is $$\hat \psi = C(k){e^{ - {k^2}t}} $$ and according to (2) it is any odd function $$C(k) = \sum\limits_{n = 1}^{ + \infty } {{b_n}\sin (nk)} $$ but from (3) we've got $C(k) = {e^{ika}}$ and it's contradiction. I know the right solution and it's $$f(x,t) = \frac{1}{{2\sqrt {\pi t} }}[{e^{ - \frac{{{{(x - a)}^2}}}{{4t}}}} + {e^{ - \frac{{{{(x + a)}^2}}}{{4t}}}}] $$ so the right function is $$C(k) = 2i\sin (ka) $$ and it seems i need to use $$f(x,0) = \delta (a - x) - \delta (a + x) $$ but it's reflection principle again, and I can't understand why the ordinary method doesn't work (The actual problem why I don't want to use reflection is that I need to solve more difficult task, where we can't use it).

Actually, I heard about Wiener-Hopf method for a half-plane problem, but I cant' understand how to use it for this task, maybe there is some not-so difficult litherature of applying Wiener-Hopf methot to heat equation?


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