Chapter 5: Solutions to the Diffusion Equation

Address methods to solve the diffusion equation for a variety of initial and boundary conditions when D is constant and therefore of the simple form below

dc / dt = D grad^2 c

This equation is a second-rder linear partial-differential equation.

In a large class of initial and boundary conditions, there are theorems of and existence of solutions and theorems of maximum and minimum values

Solutions of many boundary-value problems can be adopted as solutions to corresponding diffusion problems.

It can be difficult to find a closed-form solution in problems with highly specific and complicated boundary conditions.  Numerical methods can be employed.

 The spreading Gaussian distribution describes the diffusion out into an infinite medium of from various instantaneouw localized sources

When the initial conditions can be represented by a distribution of sources, one simply superposes the solutions of individual sources by integration

Solutions can be obtained by the separation-of-variables method

Laplace transforms can be used to derive many results.

The difficult part of using the Laplace transform is back-transforming to the desired solution, which usually involves integration on the complex domain.