CHAPTER 05.02: NM SHOOTING METHOD: THE METHOD 

Let's go and see how the Shooting Method works. So. let's suppose if we have a differential equation like this: d2y/dx2=f(x, y, dy/dx), let's go ahead and do the general case. So here you are given the condition at y(a) is ya and at y(b) it is yb, so what you are finding out here is that you are given the two boundary conditions, what is happening, what is happening to y at x equal to a and what is happening to y at x equal to b. Now what Shooting Method does is that it basically uses the initial value, it uses the initial value of problem methods. So you do have a boundary value problem, but it's going to use initial value problem methods to solve the problem. So what that means is that if you had a second order differential equation which was of this particular form and you were solving the initial value method, then you would need the condition at y(a) which is given as ya and then you would need the cindition of y'(a) equal to, let's suppose, let's call it Ya. Then I will see that, let's suppose somebody was saying that hey, I'm going to use initial value problem method such as Euler's Method, Runge Kutta Second Order Method, Runge Kutta Fourth Order Method and I want to solve this boundary value problem with these boundary condtitions, but I want to use initial value problem methods to be able to do that. Now you can't simply say okay hey, we can go ahead and use the initial value problem methods like Euler's Method and solve the problem because initial value problems for a second order differential equation such as this one will require you to have the intial value of y at x equal to a and also the initial slope at x equal to a. That's the only way you can solve the problem. So what Shooting Method is all about is that what you are going to do is you are going to say hey I know this initial, I know this condition at x equal to a because that is given to me, however; what I am doing is I am replacing this condition by this and I cannot just replace it because I like it, what I am doing is I am just replacing this condition, by this condition right here because that's what initial value problem methods are going to require me to do. So what I'm going to do is I'm going to use this as my initial value so what that means is that I'll have to guess, I'll have to guess this one. I'll have to guess this one and what I am going to is I'm going to go from a to b so let's suppose if I am at x equal to a and I want to, and the conditions are given at x equal to b, I am going to use my initial condition which is y(a) is ya and y'(a) is Ya, let's suppose, and I am going to use that to find the value at some point which is h away, so whether you are using Euler's Method, Runge Kutta Methods you are going to find the value of y at this particular point and similarly, you are going to find the value of y at this particular point, y at this particular point by following these numerical methods and eventually you are going to end up right here and what's going to happen is that you are going to get the value of y(b) by this using some numerical method with the initial value problem. But that y(b) is not going to be the same as this number right here. Why? because there is no way of knowing that hey whether this guess which you are used was appropriate or not. So, what you are going to do is you are going to find some value y(b) so that's why it is called Shooting Method because you are shooting for, hoping that the answer that y(b) will turn out to be very close or exactly to be this number right here. But that's not going to happen in almost every case that, once you have chosen this initial guess, no matter how good you are at guessing and understanding the physics of the problem, once you have chosen this guess right here, this y(b) is not going to equal to this y(b). So what happens is that once you have chosen y(b), once you have obtained y(b) by using the initial value problem method, such as Euler's Method and Runge Kutta Second Order Method, you are going to choose another, you are going to choose another value for y'(a) and let's call it Za, so you are going to chose another value, another initial guess and you are going to get a different y(b). So let's suppose here you will getting, let's suppose, p, now based on this value here, we are getting p, now on this value here you are going to follow the same procedure, starting from here with this as the initial value of y, this as the initial value of y'(a) and take the steps and go all the way up to here and you are going to get y(b), let's suppose equal to q. Now you have some idea of knowing that hey, if I get by using, by using this value, Ya, I'm getting the value of p at b, and using this value I get the value of q. Now you can do some kind of interpolation method to see that hey, what should I have chosen, what should I have chosen as my initial slope here based on trying to get the value of y(b) to be this yb right here. So that's how the Shooting Method works and that's why it's called the Shooting Method becaue you are simply trying to guess the initial slope and hoping that the value of y which will turn out, which is given at the other boundary will turn out to be close to the numbe which is given to you, but you can follow a scientific method by choosing another guess let's suppose and what happens is that when you choose antoher guess you are going to get another value of y(b) and based on some interpolation technique you can say okay hey, I want y(b) not to be p, not ot be q, but I want it to be yb. What should I choose as my initial values of y'(a)? All of these things will be very clear once we do an example but you do need to understand the theoretical basis of the Shooting Method. And that's the end of this segment.