 Before we jump into numerical methods in heat transfer, what I want to do is answer a couple of questions and these are sometimes questions that help students understand a little bit better what numerical methods actually are. So the first question is what are numerical methods? I took a course in numerical methods when I was an undergrad and I'll be honest I went through the entire course and didn't really know what was going on. It was able to get a very good grade, do the programs, write programs, but it was a little bit on the confusing and abstract side mainly because there weren't really any tangible applications that could provide an understanding of how you would apply numerical methods. So I think it can be beneficial for students to answer these questions before we jump in. So to begin with what are numerical methods? So from the perspective of engineering, engineering education, engineering programs, we can say numerical methods are the solution to mathematical physics problems and so we create mathematical physics problems in almost every engineering course. Numerical methods provide a method by which we can solve for certain aspects of these problems. Another thing that we can say about numerical methods, numerical methods provide us with approximate solutions to these mathematical physics problems. So that's another point that is important to realize and this is opposed or different than if you are able to come up with an exact analytic solution. So that might be something like f of x is equal to x squared plus 2y. With an analytic solution you come up with, actually that should have been f of x y because we have the two variables there, but you'll come up with an exact analytic relationship with a numerical method. You have things on a grid at discrete points and so you're just coming up with a solution at those discrete points and that's where you know your answer and what goes on between those points you don't really know. You could guess in order to obtain that you would have to have a finer resolution grid and then you would be able to figure out what is going on in the intermediate points. So that is another thing that is unique about numerical methods. They are approximate solutions to the problems that we're solving and so the next question is what types of problems can we solve with numerical methods? So let's take a look at that and I'll also talk about what types of methods we might use. So if you take a course in numerical methods quite often it will begin with very simple types of problems and usually it'll begin with interpolation of a function. And so if you're doing interpolation of a function the simplest is linear interpolation. So you know two data points and you're trying to find out some value in the middle you draw a linear line between the two and that enables you to get the midpoint. So that's linear interpolation and then as we get more complex you can have cubic spline interpolation and that involves matching conditions that go higher than just the slope. It might be second-order derivatives at the boundary points when we're doing our interpolation. So that's one type of problem that you can solve with numerical methods. Another one is linear algebra, a system of equations. And with a system of equations quite often what we end up doing is inverting a matrix. And so in numerical methods there are ways that you can do matrix inversion. Now I realize that we today have very modern tools such as Matlab and Mathematica, things like that that enable us to do this quite quickly. But with numerical methods you can get programs that you can develop methods of inverting a matrix using these programs. Other things you might be doing in a course in numerical methods and another topic is integration of a function. And example applications might be something like area under a curve or volume of a shape. And when you're doing these you might use techniques like the trapezoid method or simpsons method. So if you've taken a course in numerical methods you'll recognize a lot of these. Another very common application in engineering is integration of ordinary differential equations. And there are different techniques that we can use but a very common one used in engineering is the I'll probably say it incorrectly. I've heard Rungikata and Rungikata. I'm not sure how to actually pronounce the first name there so I apologize if I am saying it incorrectly. So that's a common technique. And then the next in this escalating scale of complexity of numerical methods applications we get to what we're going to be looking at in this course and that is solutions of partial differential equations which we know the heat diffusion equation is a partial differential equation. And so we can look at heat conduction as we're going to in this course. You can look at fluid flow external aerodynamics so the flow overbodies. And these latter two here are quite often referred to as being CFD or computational fluid dynamics. And in solving PDEs or partial differential equations we have a couple of techniques. The one that we're going to look at is the finite difference technique. And the other one that I mentioned is the finite element method. So now this is not exclusive but quite often finite difference is used for heat or for CFD fluid flow problems. And finite element quite often is used in structural analysis or in solid mechanics. But what we're going to be covering in this course we're going to be looking at the finite difference method. And we will in the next segment go through the different steps of applying the finite difference method which is a common technique within numerical methods to the heat diffusion equation. And so we'll be going through all the steps that you go through in applying numerical methods to solve heat transfer problems. So I just wanted to give you that segment to kind of go through what numerical methods are and what types of problems you can solve with numerical methods. So from here on we're going to dive in and start applying this to heat transfer.