Eigenvalue of the operator L is the value of the parameter λ (complex or real) for which the equation Lu = λu has a nonzero solution. The appropriate nonzero solutions are called eigenfunctions of the operator L, corresponding to eigenvalue λ. L nonlinear operators are usually considered as differential, integral, etc. A set of eigenvalues is a discrete spectrum of the operator L. Eigenfunctions belonging to different eigenvalues are linearly independent.
The Hermitian (conjugate) linear operators [for instance, the differential operator involved in the stationary equations of heat transfer and diffusion L = div(k · gradT) play an important part in solving problems of heat transfer. If the operator L is self-conjugate, then all its eigenvalues are real. Eigenfunctions corresponding to different eigenvalues are mutually orthogonal. If a self-conjugate operator L has a purely discrete spectrum, then it has a complete orthonormal sequence of eigenvalues.
The expansion of functions into a series in terms of the orthonormal sequence of eigenfunctions (the Fourier Series) is of paramount importance in solving problems of hydrodynamics and heat transfer (in analyzing computational algorithms, in particular).
Heat & Mass Transfer, and Fluids Engineering