 I'm now going to go through a relatively quick overview of how to apply the finite difference method to heat transfer and then in the next lecture we'll go into more detail about how the equations come about and how you set up the method. So the first step in applying the finite difference method to the heat equation or the heat diffusion equation is we do what is referred to as being grid generation. So let's imagine we have a rectangular plate which was the problem that we looked at when we were looking at solutions to the heat diffusion equation and any finite difference method that you apply to the heat diffusion equation, remember we're looking at in two dimensions study, we're looking for the temperature distribution within that plate and what I'm going to do I'm going to prescribe even more complex boundary conditions than what we were able to do using the separation of variable techniques. So there we have a plate with four different boundary conditions, one on each side and this is the problem of a square plate and then what we want to do we want to convert that object into a grid of finite nodes so we would recast that. So what we've done is we have represented our square plate through a series of grid points and then what we would do is we specify the grid spacing in this case it's delta x delta y and then we go through and we assign node numbers to each of the individual nodes. So what I've done here is I've expressed this node as being some generic node m, and we'll talk a little bit more about how the nodes relate to the equation in a moment and so this process in any finite difference technique is referred to as being grid generation. The next step in the finite difference method applied to the heat diffusion equation is to rewrite the mathematical physics equation in finite difference form. So if we have a case of the steady heat transfer in two dimensions with internal generation the heat diffusion equation and partial differential equation would look like this. Now what we want to do is we want to be able to express that in finite difference form and so going through a process that I will get into in the next lecture the following would result. So that's what the finite difference equation or version of the heat diffusion equation would be transformed into and the TMN in these equations here so there and there that refers to the temperature at nodal location m, n and the MN subscripts are basically just a bookkeeping approach that we use in order to keep track of the temperatures at different points within our grid that we generate in the first step. So that is the second step you recast your mathematical physics equation. The third step is to figure out how to handle your boundary conditions. So looking back at the plate that we're trying to solve our boundary conditions are here and we can see what is specified for this particular problem is a specific temperature along each of the walls of the plate. So an example for one of the boundary conditions could be the temperature at nodal location 1, 0 equals 2, 0 equals 3, 0 equals 70 degrees C and so that would be an example of how you would handle the boundary conditions on one of the sides. And the last thing that we do in this process is we need a way to be able to solve for the temperature distribution and the way that we do that is the following. So what you do is you apply the finite difference equation the one that we came up with here so this is the finite difference equation the finite difference form of our mathematical physics equation you apply that to every interior node so looking back at our grid that would be all of the interior node locations you apply the finite difference the equation the external surfaces have all had the boundary conditions applied there now the corners you got to be a little careful with because the boundary conditions are going to be essentially an average of what's going on on each of the walls but those ones have been handled by the boundary conditions it's the interior nodes that we're going to apply the mathematical physics equation to and that is going to result in a linear a series of linear algebraic equations that you need to solve with some technique and essentially you're using a matrix inversion technique and if you look in numerical methods textbooks there are different methods of doing matrix inversion for linear algebraic equations and one is the gauss-jordan elimination technique and this is a direct method another one is the gauss-seidel technique and this one requires iteration a final one and and then this is a code that hopefully i will be able to show you in two lectures from now but it is using excel and that involves a circular reference and iteration so those are different techniques that we can use to solve our set of linear algebraic equations that result from each of the interior nodes and applying the finite difference the equation to them and once we've done that what will come out of that process we will then obtain t m comma n for all the interior nodes and that essentially is then giving us t x y where x and y are at each of the discrete node locations that we established in the grid generation step at the beginning so if you want finer resolution make delta x and delta y smaller and you'll get finer resolution in your solution but of course there'd be more points and it would take a little bit longer in terms of the numerical analysis so that in a nutshell is kind of what the finite difference method applied to heat transfer looks like and what we'll be doing in this course we're only going to be looking at 2d solutions that doesn't mean that we can't solve 3d problems 3d problems we can solve using the same technique but essentially what happens is they require a little bit more bookkeeping because then you have a third index on your index notation for the temperatures and and it gets a little bit more complex but nonetheless would be the same technique that you'd be applying so bookkeeping not necessarily a technical term but it just means that you got a little more things you got to keep track of because then you're going to have t m and let's say oh not the best thing to put for an index and numerical methods how you could do ijk that would probably be another way it doesn't really matter what you put there it could be anything okay so that is a quick summary of the finite difference method applied to the heat diffusion equation what we're going to do in the next lecture we're going to go into more detail and i will go through a little bit more of an extensive derivation coming up with the translation of the heat diffusion equation at the finite difference form and then we'll take a look at a lot of the different boundary conditions that we can have in heat transfer and with that we'll be ready to solve problems provided that we have it all set up which hopefully if all goes well i will do in the following lecture show you an excel solution for that so that is the end of this lecture the next one we go into more details of this technique