Another useful method for finding analytical solutions to stable diffusion problems is eigenfunction expansion. This technique involves expanding the initial condition or boundary condition, which describes the system's behavior at a specific time or location, into a series of eigenfunctions. Each eigenfunction corresponds to a specific mode of the system's behavior, and the resulting series allows us to represent the complete solution.

For example, let's consider a scenario where a rod with an initial temperature distribution is subjected to stable diffusion. By expanding the initial temperature distribution as a series of eigenfunctions, we can express it as a linear combination of the eigenfunctions multiplied by time-dependent coefficients. By obtaining the eigenfunctions and coefficients, we can determine the temperature distribution at any point and time, providing an analytical solution to the diffusion problem.