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

Volterra Equations Integral Equations that have global memory Basically, 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 tools Another 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 method Multiply 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 f y 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 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.

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

An Example (Linear) Equation: y 0 (t)=cos(t) f(t,s,y(s))=sin(T+S)*y(s) t 0 =0 t f =2*pi (t stays constant within the integral)

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 Rectangular Time (s) Mixed Time (s) Partial Derivative Time (s)

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

Mixed

Partial Derivative

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

Linear: Rectangular vs. Mixed

Nonlinear: Rectangular vs. Partial Derivative Approximation 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 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.

Volterra Equations in Population Dynamics A two-species model of population can take the form: dx/dt=f 1 (t,y)-f 2 (t,x,y) dy/dt=g 1 (t,y)-g 2 (t,x,y) where f 1 denotes the net birth rate and f 2 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 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

Questions?