Numerical Solutions of Integral Equations and Associated Control and Estimation Problems

Jeffrey Carroll, SophomoreDr. S. A. Belbas

Volterra Equations

Integral Equations that have global memoryBasically, each successive value depends on all values before it.

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 toolsAnother feature of MATLAB is modularity: it can have two functions that reference each other

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

Rectangular

This is the most basic methodMultiply the previous value of the function by the step size to approximate the integral

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.

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

User InterfaceThe interface is a typical MATLAB interface: command-line promptsThe 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.

Output

First the program outputs the approximation used for the integral equationAfter calculations finish, the program displays the step size and the final value of the functionA graph of the value at every step is then displayedThe user can simply type ‘y’ to see all numerical values of the function at each step

An Example (Linear) Equation:

y0(t)=cos(t) f(t,s,y(s))=sin(T+S)*y(s)

t0=0 tf=2*pi

(t stays constant within the integral)

0 1 2 3 4 5 6 7-8

-6

-4

-2

0

2

4

6

8

10Approximated Y value vs Time using 10 subintervals

Time

Y

0 1 2 3 4 5 6 7-8

-6

-4

-2

0

2

4

6

8

10Approximated Y value vs Time using 20 subintervals

Time

Y

0 1 2 3 4 5 6 7-10

-8

-6

-4

-2

0

2

4

6

8

10Approximated Y value vs Time using 50 subintervals

Time

Y

0 1 2 3 4 5 6 7-10

-8

-6

-4

-2

0

2

4

6

8

10Approximated Y value vs Time using 200 subintervals

Time

Y

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

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

Partial Derivative

Mixed

Rectangular

Accuracy

The best way to determine accuracy is using each method’s 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

Rectangular

0 1 2 3 4 5 6 7-6

-4

-2

0

2

4

6

8

10

12Approximated Y value vs Time using 5 subintervals

Time

Y

Mixed

0 1 2 3 4 5 6 7-15

-10

-5

0

5

10

15

20Approximated Y value vs Time using 5 subintervals

Time

Y

Partial Derivative

0 1 2 3 4 5 6 7-5

0

5

10

15

20

25Approximated Y value vs Time using 5 subintervals

Time

Y

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 timeThe 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

Linear: Rectangular vs. Mixed

200 250 300 350 400 450 50010

20

30

40

50

60

70

80

90

100

110Times using a Linear model with variable Interval numbers

Intervals

Tim

e (s

)

Nonlinear: Rectangular vs. Partial Derivative Approximation

200 250 300 350 400 450 5000

20

40

60

80

100

120

Intervals

Tim

e (s

)Time vs Intervals, Nonlinear functions Rectangular and Partial Derivative Approximations

Partial Derivative

Rectangular

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 accuracyThe 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.

Volterra Equations in Population Dynamics

A two-species model of population can take the form:

dx/dt=f1(t,y)-f2(t,x,y)

dy/dt=g1(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.

Moving to Volterra Equations…

These partial derivatives translate directly into two Volterra equations:

So population models can be solved with the Volterra approximation methods

In Conclusion

Three different methods of approximating Volterra Integral Equations were developed in MATLABThese 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

Questions?