November 18, 2014
One of the fundamental concepts when solving partial differential equations is the Green’s function. This function allows you to represent the exact solution as an integral. It may be hard to find, and if found, may be hard to evaluate (needing many terms to converge). But, there are problems for which one is known. Problem 10 from my website (www.ChemEComp.com) shows how to use the Green’s function to provide error bounds for a numerical solution. In this case the problem is simple: flow in a square or rectangular duct. An example of the Green’s function is shown in the figure as a function of x and y, for a particular value of zeta and eta, (0.7, 0.3). The figure becomes sharper as more and more terms are kept in the infinite series, but the world does march on! The problem statement gives some mathematical background for deriving the error bounds, and the solutions (obtained through SIAM) gives the answers.