CHAPTER 08.01: FINITE DIFF METHOD FOR ODEs: Background Part 1 of 2 

In this segment we are going to look at Finite Difference Method to solve ODEs, to solve, I should say boundary value ODEs So, Finite Difference Method is one of the methods used to solve boundary value ordinary differential equations. There is another method called Shooting Method which is in a seperate segment and in this particular segment I'm going to just give you the background of it. And the best way to show you how this Finite Difference Method works for ODEs is basically to check, to go through examples. So let me go through a simple example here. Let's suppose somebody gives you a differential equation like this: d2u/dr2 + (1/r)(du/dr) - u/r2 = 0 d2u/dr2 + (1/r)(du/dr) - u/r2 = 0 d2u/dr2 + (1/r)(du/dr) - u/r2 = 0 So here is a differential equation which is given to you. It's a second order ordinary differential equation and the dependent variable is u, independent variable is r, and two boundary conditions are given. You are told that hey, u(5)=0.008 and u(9)=0.007, so let's suppose that these are the two boundary conditions which are given to you. So now this different from an initial value problem where you are given the values of the conditions at an initial point, but now the value is given at 5 and the value given at 9, so what this bascially means is that you have to find out to be able to solve this differential equation, we are basically looking for, trying to find out the value of u between the values of 5 and 9. In fact, this is a differential equation which governs a pressure vessel. So if you have a thick pressure vessel, the radial displacement, so if this is r, this is the radial location, the radial displacement is given by this ordinary differential equation. So how much is it really expanding or contracting when, let's suppose, you are putting pressure inside it? We are taking some simpler boundary conditions here where we are given what the, so if this is, if this is 5 and this is 9 units let's suppose, you are given the boundary condition right here on this surface and you are given the boundary condition on this surface that it is displacing 0.008 units at 5 and 0.007 units at r=9. So what you want to be basically able to do it that since you are given the value of the displacement at this point and the displacement at this point, you want to be able to find out the displacement at some other points. That's what you want to be able to do. So if you are able to find out displacements at some of these points then you can develop a profile for the dependent variable u and that's what you are basically looking for. You are trying to find out u as a function of r and as we know that numerically we can just find it at some points. Now, so how do we solve this particular ordinary differential equation by using, by using Finite Difference Method is to be able to rewrite these equations in terms of the finite difference approximations for the derivatives.