I to do a Fourier Sine Transform

[j3n5k1]

Homework Statement

A function u(x, t) satisfies the heat equation
$$K$$$$\frac{\delta^{2}u}{\delta x^{2}}$$ = $$\frac{\delta u}{\delta t}$$
on the half line x $$\geq$$ 0 for t > 0, where $$K$$ is a positive constant. The initial
condition is
u(x, 0) = cxe$$^{\frac{-x^{2}}{4a^{2}}}$$
with c and a being constants, and the boundary conditions are
u(0, t) = 0
u(x, t) $$\rightarrow$$ 0 as x $$\rightarrow$$ $$\infty$$
Prove that the Fourier sine transform of u with respect to x is given by
$$\hat{u}$$(s,t) = $$\hat{u}$$(s, 0)e$$^{-Kts^{2}}$$
Hence find the solution of the heat equation.

I have absolutely no idea what to do, and my books aren't helping at all. Anyone able to help me?

 

[lippyka]

Homework Equations K\frac{\delta^{2}u}{\delta x^{2}} = \frac{\delta u}{\delta t}u(x, 0) = cxe^{\frac{-x^{2}}{4a^{2}}}u(0, t) = 0u(x, t) \rightarrow 0 as x \rightarrow \inftyThe Attempt at a SolutionI have no idea what to do.