MATLAB by rudra pratap

5.5 Ordinary Differential Equations
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5.5.6 Event location
In solving initial value problems, usually the termination condition is specified in
terms of the independent variable. As shown in the previous Lwo examples, the
solution stops when the independent variable t reaches a prescribed value tfinal
(specified as the second element of tspan). Thus, we obtain the solution for a
certain time span. Sometimes, however, we need to stop the solution at a specified
value of the dependent variable and we do not know when (at what value of t) the
solution will reach there. For example, say we want to solve a projectile problem.
We write the equations of motion (two ODEs for x and y), and we would like
to solve the equations to find when the projectile hits the target. If we do not
know tfinal apriori, what value of tfinal should we specify in tspan? One thing
we know is that the solution should stop when the target (some specified location
Xtarget or Ytarget) is hit, that is, when x or y reach a particular value. When the
dependent variables (here x(t) or y(t)) reach some specified value, we call that an
event, and the problem of finding the time of the event is called event location. Event
location problems are solved by following the solution until the solution crosses the
event, then backtracking to the time just before the event, and then taking smaller
and suitably computed time steps to land exactly at the event. There are several
strategies and algorithms for event location. In the literature, the event location
problem is also referred to as integmting across discontinuities.
Fortunately, the ODE solvers in MATLAB have built-in ability to solve event
location problems. Let us take a simple example-a projectile is thrown with initial
speed Vo at an angle e. We want to find out (i) the instant when the projectile
hits the ground, (ii) the range, and (iii) the trajectory of the projectile. The ODEs
governing the motion of the projectile are
x 0
ii -g
The initial conditions are x(O) = y(O) = 0, x(O) = v0 cos (}, and y(O) = v0 sin 8.
The equations of motion are to be integrated until y = 0 (the projectile hits the
ground).
Before we write a function to code our equations, we need to convert them into
a set of first-order equations:
Let x = x1,x = x2,Y = X;>, and y= X4, then x1 = x2,x2(= i) = O,::i:3 = X1 , and
x4(= jj) = -g. In vector form:
(5.5)
Now, eqn. (5.5) is ready for coding in a function.