CHAPTER 05.01: NM SHOOTING METHOD: BACKGROUND 

In this segment we are going to talk about Shooting Method. Shooting Method is a way to solve ordinary differential equations but the boundary value problems. So, this is one of the methods to solve ordinary differential equations but which are where the conditions are given on the boundary. So that's why we call them boundary value problems. So the kind of equations which are solved by the Shooting Method are of this form: d2y/dx2 is some function of x which is the independent variable, y the dependent variable, and also the the first derivative of y with respect to x. So, and this particular differential equation will be subject to some conditions at some point x equal to a, let's suppose we call it ya and then yb which will be the other boundary there, the value will be yb. So this is the form which we can solve by using Shooting Method. To give you some idea of what I typical example might look like, so if you could have a beam, let's suppose, let's suppose you have a beam, you are simply supporting it. It's pinned on one side and supported on the other side and then what we have is some kind of load being applied. So let's suppose we have some kind of uniform load being applied. In that case what we will have is that the differential equation which will find the deflection of this particular beam is given by this. So, the deflection of this particular beam under this load q here, where x measured from here let's suppose is given by this differential equation, so you are measuring the deflection going downwards, let's suppose, q is the load which is applied, E is the Young's Modulus, I is the second moment of area, x is course a location, l is the length. So this could be the length of the beam which you are going through. So in this case this will be the differential equation and then it will be subject to of course v(0)=0 and v(l)=0 so you are finding out the conditions are given on the boundary itself. So it does fall under this category right here because the second derivative is simply a function of just x in this case. It's not, it doesn't have y terms, it doesn't have derivative of y terms in there, but they are included in a way. And then the boundary condition is at one point which is at x=0 is 0 and at the other point at x=b which is in this case l is also 0. So that's one example of what kind of problems can be solved by using the Shooting Method. Let's take a little bit more, not involved example but one which gives you things which are non-zero. If you have a pressure vessel for example, and the inner radius is a and the outer radius is b, let's suppose, and if you want to find out what, so if you are subjecting this to some kind of a pressure inside, so uniform pressure inside, what's going to happen is that the differential equation which governs the amount of radial deflection which is taking place is given by this: So r is measured as the radial location so r is the radial location. So if r is the radial location and u is the, u is the radial displacement. So because if you are going to pressurize it inside or outside, what's going to happen is that this particular pressure vessel is going to deform radially. And this is the differential equation which governs the radial displacement of this pressure vessel here. So this might be subjected to u at a equal to ua and u at b equal to ub so you might be given conditions at the, that how much it is displacing at a and how much it is displacing at b. The way you could do that is by simply putting a strain gage at r equal to a and r equal to b and then once you are measuring the normal strain, the hoop strain, that hoop strain is directly connected to how much is the deflection taking place at a and how much is the variable displacement taking place at b. But, leaving that asside, what you will have to do is, you rewrite this in the form d2u/dr2 is equal to u/r2 plus 1/r(du/dr) and you can see that this is function of the independent variable r, the function of the dependent variable u, and the function of the first derivative of u, subject to the condition that u(a) is equal to some displacement which is given at a and the displacement b is given as ub. So that's what's happening there. So this is another example which I am showing you that, where you could be asked to, or where you could apply the Shooting Method. These are the kind of differential equations which are, which you can do that to. Now the question is that what is Shooting Method in itself and that's something which we will discuss in the next part. And this is the end of this segment.