Mathematical Methods in Science and Engineering, 2nd Edition

Chapter 13Fractional Calculus

The integral form of the diffusion equation is written as

13.1

where is the concentration of particles and is the current density. The left-hand side of this equation gives the rate of change of the number of particles in volume and the right-hand side gives the number of particles flowing past the boundary, of volume per unit time. In the absence of sources or sinks, these terms are naturally equal. Using the Gauss theorem we can write the above equation as

13.2

13.3

For an arbitrary volume element, we can set the expression inside the square brackets to zero, thus obtaining the partial differential equation to be solved for concentration:

13.4

With a ...