Question
Solve the following heat equation on the semi-infinite rod by Fourier Transform
$$u_t=ku_{xx}$$ where $x,t>0$ and
$u_x(0,t) =0$ and $u(x,0)=\begin{cases}
1, & 0 < x <2 \\
0, & 2\leq x
\end{cases}
$
My attempt
If we apply Fourier Trans. to both sides, we get
$ \frac{\partial \mathcal{F} \{U(w,t)\}}{\partial t} = -kw^{2} \mathcal{F} \{U(w,t)\}$
Solving the ODE, we have
$U(w,t)=C(w)e^{-kw^2t}$
to find $C(w)$ we use the initial condt. $u(x,0)$
$C(w)=\mathcal{F} \{u(x,0)\}=\frac{1}{(2\pi)^{1/2}}\int_{-\infty}^{\infty} u(x,0) e^{-iwx} \ dx $
Could you help me to solve the rest? What is the solution $$u\left(x,t\right)=\mathrm{TF}^{-1}\{U(w,t)\}=\frac{1}{(2\pi)^{1/2}}\intop_{x\in\mathbb{R}}C(w)e^{-kw^2t}e^{iwx}dw$$?
(I can't calculate $C(w)$ since $u(x,0)$ is undefined on $-\infty<x<0$ )
Best Answer
Let $u_0(x) =u(x,0)$ then $$C(w)=\mathcal{F} \{u_0\}(w)=\frac{1}{2\pi}\int_{-\infty}^{\infty} u(x,0) e^{-iwx} \ dx$$ We also know the following $$ e^{-kw^2t} = \mathcal{F}\left(\frac{1}{\sqrt{2kt}} \displaystyle{e^{-\frac{x^2}{4kt}}}\right)(w)$$
Whence it follows that, $$U(w,t)= C(w)e^{-kw^2t}=\mathcal{F} \{u_0\}(w)\cdot\mathcal{F}\left(\frac{1}{\sqrt{2kt}} \displaystyle{e^{-\frac{x^2}{4kt}}}\right)(w)$$
Using the product rule for Fourier we have $$u\left(x,t\right)=\mathcal{F}^{-1}\{U(w,t)\} =\mathcal{F}^{-1}\left\{\mathcal{F} \{u_0\}\cdot\mathcal{F}\left(\frac{1}{\sqrt{2kt}} \displaystyle{e^{-\frac{x^2}{4kt}}}\right)\right\}\\= u_0\star\frac{1}{\sqrt{2kt}} \displaystyle{e^{-\frac{x^2}{4kt}}}~~~\text{convolution }\\= \frac{1}{\sqrt{2kt}}\int_\Bbb R u_0(y)e^{-\frac{(x-y)^2}{4kt}}dy \\=\frac{1}{\sqrt{2kt}}\int_{2}^{\infty} \underbrace{u_0(y)}_{0}e^{-\frac{(x-y)^2}{4kt}}dy +\frac{1}{\sqrt{2kt}}\int_{0}^{2} \underbrace{u_0(y)}_{1}e^{-\frac{(x-y)^2}{4kt}}dy \\+\frac{1}{\sqrt{2kt}}\int_{-\infty}^{2}\underbrace{u_0(y)}_{?}e^{-\frac{(x-y)^0}{4kt}}dy\\=\frac{1}{\sqrt{2kt}}\int_{0}^{2} e^{-\frac{(x-y)^2}{4kt}}dy +\frac{1}{\sqrt{2kt}}\int_{-\infty}^{2}\underbrace{u_0(y)}_{?-u_0~~is ~~not~~define}e^{-\frac{(x-y)^0}{4kt}}dy $$
YOU HAVE TO DEIFNE $u_0$ FOR $x<0$