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For an initial heat distribution localized at a point, use a Green's function approach, where the solution is built up from the fundamental solution corresponding to that point heat source.

Solving the heat equation with one end held at a constant temperature (Dirichlet) and the other insulated (Neumann), leads to a mixed series solution, often involving sine and cosine functions or modified sine series.

Robin boundary conditions which involve a linear combination of temperature and flux can lead to solutions that use eigenfunctions satisfying associated Sturm-Liouville problems.

The initial condition must be expanded as a Fourier sine series to match the eigenfunctions derived from the homogeneous boundary conditions.

If Neumann boundary conditions are used, then the initial condition can often be represented by a Fourier cosine series consistent with these boundary conditions.

Periodic conditions assume the ends of the domain are connected, leading to a continuous and differentiable temperature profile that calls for a Fourier series solution involving both sines and cosines.

Solution involves cosine series as the eigenfunctions, reflecting no heat flux at the ends, implying $\frac{\partial u}{\partial x}(0,t) = \frac{\partial u}{\partial x}(L,t) = 0$.

A non-trivial flux at the boundaries complicates the problem. Just like with Dirichlet conditions, seek a steady solution plus a transient solution with homogeneous Neumann conditions.

Non-zero boundary conditions mean we may need to split the solution into a steady part and a transient part, where the steady part satisfies the static heat equation.