Model for soil temperature underground using the Heat Equation Assuming that the temperature in the ground is a function of time $t$ and depth $x$ only and assuming that at x=0, ground level, the approximate temperature at the surface is 
$$u(0,t) = T_{0}+A_{0}\cos(\omega t),$$
where $T_{0}$ is the average temperature, $A_{0}$ is the amplitude of the seasonal temperature variation, and $\omega$ the frequency such that 
$$\omega = \frac{2\pi}{year}=\frac{2\pi}{31557341}s^{-1}\approx1.991 \times 10^{-7}s^{-1}.$$
Also assuming that the temperature satisfied the heat equation
$$\frac{\partial{U}}{\partial{t}}={\kappa}^2 \frac{\partial^2{U}}{\partial{x^2}}, $$
For convenience we introduce a temperature deviation $U(x,t)=u(x,t)-T_{0}$. With the one boundary that $$U(0,t)=A_{0}\cos(\omega t)$$
We are only interested in periodic solutions of the form:
$$U(x,t)=V(x)\cos(\omega t) + W(x)\sin(\omega t).$$
How may I show that we may regard $U$ to be the real part of the complex function $\tilde{U},$ $$\tilde{U}(x,t)=X_{(x)}e^{i\omega t},$$ where $X$ may be complex, and show that $\tilde{U}$ obeys the heat equation 
$$\frac{\partial{\tilde{U}}}{\partial{t}}={\kappa}^2 \frac{\partial^2{\tilde{U}}}{\partial{x^2}}$$
with boundary condition given by $$\tilde{U}(0,t)=A_{0}e^{i \omega t}.$$
Thank you for any guidance you can give me.
 A: If you view this question by power series method approach, you will find that it is very easy to solve.
Similar to Free Schroedinger equation:
Let $u(x,t)=\sum\limits_{n=0}^\infty\dfrac{x^n}{n!}\dfrac{\partial^nu(0,t)}{\partial x^n}$ ,
Then $u(x,t)=\sum\limits_{n=0}^\infty\dfrac{x^{2n}}{(2n)!}\dfrac{\partial^{2n}u(0,t)}{\partial x^{2n}}+\sum\limits_{n=0}^\infty\dfrac{x^{2n+1}}{(2n+1)!}\dfrac{\partial^{2n+1}u(0,t)}{\partial x^{2n+1}}$
$=\sum\limits_{n=0}^\infty\dfrac{x^{2n}}{\kappa^{2n}(2n)!}\dfrac{\partial^nu(0,t)}{\partial t^n}+\sum\limits_{n=0}^\infty\dfrac{x^{2n+1}}{\kappa^{2n}(2n+1)!}\dfrac{\partial^{n+1}u_x(0,t)}{\partial t^n}$
$=\sum\limits_{n=0}^\infty\dfrac{x^{2n}}{\kappa^{2n}(2n)!}\dfrac{\partial^n(T_0+A_0\cos\omega t)}{\partial t^n}+\sum\limits_{n=0}^\infty\dfrac{x^{2n+1}}{\kappa^{2n}(2n+1)!}\dfrac{\partial^{n+1}u_x(0,t)}{\partial t^n}$
$=T_0+\sum\limits_{n=0}^\infty\dfrac{(-1)^nA_0\omega^{2n}x^{4n}\cos\omega t}{\kappa^{4n}(4n)!}-\sum\limits_{n=0}^\infty\dfrac{(-1)^nA_0\omega^{2n+1}x^{4n+2}\sin\omega t}{\kappa^{4n+2}(4n+2)!}+\sum\limits_{n=0}^\infty\dfrac{x^{2n+1}}{\kappa^{2n}(2n+1)!}\dfrac{\partial^{n+1}u_x(0,t)}{\partial t^n}$
A: Hey how are u doing with your coursework?. Remember that the deadline is on Friday and it's not good to cheat.
Cheers
James V.
