Title: Numerical Solutions of Integral Equations and Associated Control and Estimation Problems
1Numerical Solutions of Integral Equations and
Associated Control and Estimation Problems
- Jeffrey Carroll, Sophomore
- Dr. S. A. Belbas
2Volterra Equations
- Integral Equations that have global memory
- Basically, each successive value depends on all
values before it.
3System Used
- The fastest possible system that runs MATLAB is
ideal. - MATLAB Ideal for working math problems has its
own programming language and nice graphing tools - Another feature of MATLAB is modularity it can
have two functions that reference each other
4What I Did
- Basically, I estimated the solutions to Volterra
equations using a number of different methods. - Three main approximations (some with slight
variations) were used - Rectangular
- Mixed Trapezoidal
- Trapezoidal using fy, the partial derivative of
the integrated function with respect to y
5Rectangular
- This is the most basic method
- Multiply the previous value of the function by
the step size to approximate the integral
6Trapezoidal Mixed
- Since the regular trapezoidal approximation
requires the value of the function at two points,
a rectangular approximation is used for the last
value and the trapezoidal is used for that value
in the next calculation set.
7Trapezoidal Using fy
- Basically the same as the last trapezoidal,
except a different method is used to find the
last value. - This method involves the Partial Derivative with
respect to y. - For most cases this is relatively easy to
calculate, because all terms not involving f(s)
are treated as constants
8User Interface
- The interface is a typical MATLAB interface
command-line prompts - The user has to input the following
- If the user does not wish to use a previously
used equation, the new equation (initial value
equation and integrand) must be put in in a text
format such as sin(T-S)y(s).2 - Follows basic MATLAB rules in input
- For the partial derivative approximation the user
is prompted for the partial derivative - This is then changed to a string that MATLAB can
recognize to perform calculations - The user then inputs the final time to which he
wishes to approximate and the number of
subintervals used.
9Output
- First the program outputs the approximation used
for the integral equation - After calculations finish, the program displays
the step size and the final value of the function - A graph of the value at every step is then
displayed - The user can simply type y to see all numerical
values of the function at each step
10An Example (Linear) Equation
- y0(t)cos(t) f(t,s,y(s))sin(TS)y(s)
- t00 tf2pi
- (t stays constant within the integral)
11(No Transcript)
12(No Transcript)
13(No Transcript)
14(No Transcript)
15Tests
- The three methods were compared to find
- If they had consistent results
- Whether one method was more accurate or not
- If relative calculation time was an issue among
the three methods
16Consistent Results?
- Three methods were tested using different values
for the number of increments to see if they
approached a certain value. - As the next page shows, they all seemed to
approach a similar value, though the partial
derivative approximation seems to get closer
faster. - The data
Intervals 10 20 30 50 75 100 150 200 300
Rectangular Time (s) 8.4008 8.7886 9.0676 9.3559 9.5245 9.6155 9.7113 9.7610 9.8120
Mixed Time (s) 7.9274 8.7970 9.0923 9.3727 9.5342 9.6217 9.7144 9.7628 9.8129
Partial Derivative Time (s) 12.5942 10.5632 10.2008 10.0197 9.9634 9.9436 9.9294 9.9244 9.9208
17Partial Derivative
Mixed
Rectangular
18Accuracy
- The best way to determine accuracy is using each
methods relative accuracy to the other. - The following graphs show small-step values for
the functions (5 steps) - This reveals that at least with small numbers of
steps, the methods are extremely different
19Rectangular
20Mixed
21Partial Derivative
22Looks Good, But How Much Time Does It Take?
- It depends on how accurate you wish to be a
greater number of steps means greater accuracy as
well as greater time - The partial derivative method in general takes
slightly more time, but is much more accurate
than rectangular. It is ideal for nonlinear
methods (mixed will not work properly and
rectangular is less accurate.) - Mixed is most likely the best choice for Linear
functions
23Linear Rectangular vs. Mixed
24Nonlinear Rectangular vs. Partial Derivative
Approximation
Partial Derivative
Rectangular
25Basically,
- As the number of intervals increases the time
increases more (because it has to do all the same
calculations again, plus new ones) - Mixed and Rectangular take almost equal times, so
mixed is better due to accuracy - The Partial Derivative approximation is
increasingly slower than Rectangular as the
number of intervals increases, but it is more
accurate. - For extreme accuracy, a large amount of time is
required.
26Volterra Equations in Population Dynamics
- A two-species model of population can take the
form - dx/dtf1(t,y)-f2(t,x,y)
- dy/dtg1(t,y)-g2(t,x,y)
- where f1 denotes the net birth rate and f2
denotes how much y eats x. - In the simplest case, these functions can be
reduced to different variables.
27Moving to Volterra Equations
- These partial derivatives translate directly into
two Volterra equations - So population models can be solved with the
Volterra approximation methods
28In Conclusion
- Three different methods of approximating Volterra
Integral Equations were developed in MATLAB - These equations were then tested to determine
which methods were better depending on the
situations. - The results are summed as follows
- Use Trapezoidal Mixed Approximation for Linear
functions - Use Trapezoidal with Partial Derivatives
approximation for nonlinear functions - If speed is all that matters, use Rectangular
29Questions?