Am I missing something, or does this simple PDE lack an explicit solution due to the nature of its (also simple) boundary conditions? Short version:
Below, I present a simple PDE with simple boundary conditions (BCs), which has a simple solution. I then modify one of the BCs, and end up with a transcendental equation for the wave number parameter, preventing me from going further. This surprises me, since the resulting problem doesn't seem too complicated. My main question: Am I right to conclude that an explicit solution is impossible to find in this case?
Long version:
Consider the equation (EQ1) (Note: I'm starting from the heat conduction equation here but omitting some constants for clarity, consider all variables and constants dimensionless)
$$ \theta_{xx} = \theta_t, \qquad 0 < x < a,$$
where $\theta = \theta(x,t)$ and the subscripts denote partial derivatives. The boundary conditions are (BC1 and BC2, resp.)
$$ \theta_x = 0, \qquad x = 0, $$
$$ \theta_x = 0, \qquad x = a. $$
The initial condition IC is
$$ \theta = \theta_0(x), \qquad t = 0. $$
The solution is obtained by a straightforward application of separation of variables: $\theta(x,t) = X(x)T(t) \Rightarrow$
$$ \frac{X''(x)}{X(x)} = \frac{T'(t)}{T(t)} = -\lambda^2 \Rightarrow $$
$$ T(t) = C_1\exp({-\lambda^2t}), $$
$$ X(t) = C_2\sin(\lambda x) + C_3\cos(\lambda x). $$
Now, BC1 $\Rightarrow C2 = 0$ and BC2 $\Rightarrow$
$$ -C_3\lambda\sin(\lambda a) = 0 \Rightarrow \sin(\lambda a) = 0 \Rightarrow $$
$$ \lambda_n = \frac{n\pi}{a}. $$
It follows that
$$ X_n(x) = C_3\cos(\frac{n\pi x}{a}) $$
and finally
$$ \theta(x,t) = \sum_{n=0}^{\infty}c_n\cos(\frac{n\pi x}{a})\exp({-\frac{n^2\pi^2 t}{a^2}}). $$
Now, the coefficients $c_n$ can be obtained by applying the initial condition and using the trigonometric orthogonality property. The latter requires knowledge of the spatial period of the solution.
Now, consider replacing BC2 with
$$ \theta_x = \theta, \qquad x = a. $$
Then, instead of the $\lambda_n$ given above, we would obtain
$$ -C_3\lambda\sin(\lambda a) = C_3\cos(\lambda a) \Rightarrow $$
$$ -\lambda\tan(\lambda a) = 1. $$
This is a transcendental equation in $\lambda$, so no explicit expression for $\lambda$ can be obtained.
Does this mean that we can't go any further? That more or less complicated numerical methods are the only way to proceed? Or am I missing something here?
 A: No, you do not miss anything: no explicit $\lambda$s can be found. However, the general theory tells us: 


*

*there are infinitely many lambda's

*there is minimal $\lambda$ and there is no maximal one

*the eigenfunctions corresponding to lambdas are orthogonal on the interval


Hence we can always write the solution as an infinite series. And if you want to actually calculate the function, you will need a numerical routine to find your eigenvalues (this is really simple in your case)
Added: About the point, raised by @Lost1 in comments. Note that your boundary conditions imply (physically) that the left side of the bar is isolated, and at the right side of the bar the flux of temperature is equal to the temperature itself. This means that there will be always incoming flux of the energy in your bar, which means obviously that the temperature has to approach infinity, which implies that you have to assume that you constant of separation is positive. Take not $-\lambda^2$ but $\mu^2$ and see the implications. And this also means that $\cos$ are not the only eigenfunctions in your problem. 
