On a generalized basis for solving the one dimensional transport equation: Theory

The most general basis for approximating the transport equation has been studied. The application of the Gram-Schmidt procedure has been shown to unify the complete class of functions (polynomial type or non-polynomial type), applicable to this equation. The completeness of the series of functions is proved. A generalized version of the Fick's law is introduced. It is shown that the spectrum of the transport equation obtained by this method agrees with the conventional methods of obtaining the spectrum.

► The essence of the previous approaches to approximating the transport equation is captured. ► The class of functions, providing a recursive formula for the streaming term, are recognized. ► This class of functions are orthogonalized according to the Gram-Schmidt procedure. ► A generalized version of Fick's law is obtained. ► The spectrum of the transport equation is calculated.