Solutions to the heat equation, spatial or time decay?

Question

In reading standard texts there seems to be two standard solution to the heat equation:

$$\frac{\partial T}{\partial t}=D\frac{\partial^2T}{\partial x^2}$$

The first is obtained when assuming a real separation constant (if solving by separation of variables) this gives you a wave of the form:

$$e^{-t}\sin(x)$$

The second is obtained when you allow for a complex separation constant and this gives you a solution of the form:

$$e^{-x}\sin(t-x)$$

I'm becoming very confused as to which solution to use in which scenario, I have answered two separate questions which seem to be the same but each one is looking for a different solution.

Could anybody help me work out which one is appropriate and when?

Answer 1

There are not two solutions, but two class of solutions. Which class to use depends on the boundary and initial conditions.

Answer 2

Given $$u_t=a u_{xx}$$ coupled with initial condition and homogeneous (spatial) boundary conditions. assume $u= X(x)T(t)$ then $$XT'=a X''T$$ divide by $a u$ to get $$\frac1a\frac{T'}T = \frac{X''}X$$ since a function of $t$ equals to a function of $x$ for any $x,t$ can only be when both are constant functions one write $$\frac1a\frac{T'}T = \frac{X''}X=\lambda$$ next you solve $$X''-\lambda X=0$$ together with homogeneous boundary conditions to determine (countable set of) eigenvalues $\lambda_n$ and eigenfunctions $X_{\lambda_n}\equiv X_n$ . Then for each $\lambda_n$ one solves $$T'-a\lambda_n T=0$$ to for a general solutions $T_n$.

Finally one writes $$u=\sum_n A_n X_n T_n$$ and use initial condition to determine $A_n$, some time the solution has closed form.

The function $e^{-x}\sin(t-x)$ doesn't solve any problem with homogeneous boundary condition. In order to solve such problem one solve auxiliary inhomogeneous problem ($v_t=a v_{xx}+f(x,t)$) with homogeneous boundary conditions.