How do I undo a Fourier cosine transform to solve a heat equation problem?

[member 428835]

Homework Statement

Find the solution $u(x, t)$ to the semi-infinite interval problem

$$ u_t = u_{xx} - 4u, \hspace{2 mm} 0 < x < \infty, \hspace{2 mm} t>0\\
u_x(0,t) = -1, \hspace{2 mm} t>0\\
\lim_{x \to \infty}u(x,t) = 0, \hspace{2 mm} t>0\\
u(x,0) = e^{-x}, \hspace{2 mm} 0 \leq x < \infty.$$

Homework Equations

fourier cosine series, which I shall denote $C(u)$ when in variable $x$ operating over some function $u$.

The Attempt at a Solution

after taking Fourier cosine series of the governing PDE i arrive at $$
\frac{\partial}{\partial t}C(u) = \frac{2}{\pi} - \omega^2 C(u) -4C(u) \implies\\
C(u) = \frac{2}{\pi(\omega^2 + 4)} + C_1e^{-(\omega^2 +4)t}$$
Notice $C(u(x,0)) = C(e^{-x}) = \frac{2}{\pi(1+\omega^2)}$. This implies $C_1 = \frac{2}{\pi(1+\omega^2)} - \frac{2}{\pi(\omega^2 + 4)}$. Thus we arrive at $$
C(u) = \frac{2}{\pi(\omega^2 + 4)} + \frac{2}{\pi}\left[ \frac{1}{(1+\omega^2)} - \frac{1}{(\omega^2 + 4)} \right] e^{-(\omega^2 +4)t}$$
I have skipped some in-between steps, but I think the idea is here. My question is, how do I undo this Fourier cosine transform, if I have arrived to a correct solution?

Thanks so much!

Josh

 

[Brian T]

Use the inverse Fourier cosine transform

 

[member 428835]

Cool, that's what I did! Just wanted to make sure my work was correct. Thanks!