# Eigenfunction Expansion and Fourier Series

What is the difference of Eigenfunction Expansion solution and Fourier Series solution in solving Partial Differential Equation? Are they the same? Please to tell me about what it is the eigen value and eigen function and their expansion please. I don't really understand about this. Thanks.

• If you truly know nothing about this topic, asking here won't help you much. You need to pick up a PDE textbook or google this topic and come back with concrete questions, if you don't understand some of the definitions or examples. – Yuriy S May 21 at 11:28
• As for your particular question, Fourier expansion is a particular kind of eigenfunction expansion. Trigonometric functions are eigenfunctions of the following operator $L=\frac{\partial^2}{\partial x^2}$ with zero boundary conditions or periodic boundary conditions on some finite interval $[-a,a]$. There can be many other kinds of eigenfunctions depending on the operator and the boundary conditions. – Yuriy S May 21 at 11:31
• More generally the $e^{2 i\pi nx}$ are the $1$-periodic eigenfunctions of the convolution operators, and the derivatives are convolution operators (that is linear and $(T f)(x+a) = (T f(.+a))(x)$) – reuns May 21 at 11:58
• The Fourier Series is an eigenfunction expansion of the second derivative operator with periodic conditions. So the Fourier Series is a special case of an eigenfunction expansion. – DisintegratingByParts May 23 at 18:06