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Numerical Solutions of Integral Equations and Associated Control and Estimation Problems

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Title: Numerical Solutions of Integral Equations and Associated Control and Estimation Problems


1
Numerical Solutions of Integral Equations and
Associated Control and Estimation Problems
  • Jeffrey Carroll, Sophomore
  • Dr. S. A. Belbas

2
Volterra Equations
  • Integral Equations that have global memory
  • Basically, each successive value depends on all
    values before it.

3
System 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

4
What 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

5
Rectangular
  • This is the most basic method
  • Multiply the previous value of the function by
    the step size to approximate the integral

6
Trapezoidal 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.

7
Trapezoidal 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

8
User 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.

9
Output
  • 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

10
An Example (Linear) Equation
  • y0(t)cos(t) f(t,s,y(s))sin(TS)y(s)
  • t00 tf2pi
  • (t stays constant within the integral)

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15
Tests
  • 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

16
Consistent 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
17
Partial Derivative
Mixed
Rectangular
18
Accuracy
  • 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

19
Rectangular
20
Mixed
21
Partial Derivative
22
Looks 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

23
Linear Rectangular vs. Mixed
24
Nonlinear Rectangular vs. Partial Derivative
Approximation
Partial Derivative
Rectangular
25
Basically,
  • 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.

26
Volterra 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.

27
Moving to Volterra Equations
  • These partial derivatives translate directly into
    two Volterra equations
  • So population models can be solved with the
    Volterra approximation methods

28
In 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

29
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