ABSTRACT
A brief overview of the historical developments in the field of mathematical physics for modeling diffusion since the inception of fractional calculus in 1695 up to fPDEs widely studied today is the point of focus of this chapter. It is interesting to note that these developments were made in parallel routes.
Fourier’s heat equation forms the foundation of a mathematical description of physical processes in different media and has been extended to many other fields, with the fractional diffusion-wave equation as its latest form for different flow and transport processes.
The CTRW theory, which connects mathematical models with physical processes, using stochastic parameters, has also been included in this chapter. Zaslavsky’s transport exponent (Zaslavsky 2002) links the scaling parameters for fractals and the orders of fractional derivatives, hence uniting fractional PDEs with fractal geometry.
Another important piece of work presented here is the ‘the generalized self-diffusion’ proposed by Zwanzig in 1961, but no analytical results in the time domain have been given since. The results presented using the Wright function in section 4.2 have been shown to be connected to the findings by Philip in 1986 (reached at using a different concept). This is a new finding following the historical review.
The rest of the chapter presents the key concepts and the fPDEs in different forms which have been expanded from Chapters 5 onwards.
