1. EEE 431 Computational Methods in Electrodynamics
Lecture 15 By
Dr. Rasime Uyguroglu

2. Integral Equations and The Moment Method

3. Integral Equation Method/ Electrostatic Charge Distribution
Finite Straight Wire (Charged) at a Constant Potential
Formulation of the Problem (In terms of the integral eqn.)
For a given charge distribution, the potential is:

4. Electrostatic Charge Distribution
Now consider a wire of length along the y direction. The wire has radius , and connected to a battery of 1 Volts.
To have 1Volts everywhere on the surface (actually inside too), a charge distribution is set up. Let this charge be

5. Electrostatic Charge Distribution
Then,
Position vector of any point in space.
Position vector of any point on the surface of the wire.
Surface charge density.

6. Electrostatic Charge Distribution
Simplifying assumptions:
Assume Also assume the wire is a solid conductor. Then:
And:

7. Electrostatic Charge Distribution
Integral Equation:
If the observation point is brought onto the surface (or into the wire) the potential integral must reduce to 1 volt for all on S or in S.
Choose along the wire axis.

8. Electrostatic Charge Distribution
Then: or

9. Electrostatic Charge Distribution
And
This is the integral equation. Solve the integral equation for

10. Electrostatic Charge Distribution
Numerical Solution: Transforming the integral equation into a matrix equation:
The inverse of the integral equation for
will be achieved numerically by discretizing the integral equation.

11. Electrostatic Charge Distribution
Let us divide the wire uniformly into N segment each of width
If is sufficiently small we may assume that is not varying appreciably over the extent , and we can take it as a constant at its value at the center of the segment.

12. Electrostatic Charge Distribution
Now take a particular and utilize the property of the segmentation.

13. Electrostatic Charge Distribution
There are N unknowns above, namely:
We need N linearly independent equations. Take k=1,2,3,…,N.

14. Electrostatic Charge Distribution
Then:

15. Electrostatic Charge Distribution
Or:
Where: (T: transpose)

16. Electrostatic Charge Distribution
Where:
(NXN) matrix to be generated.
(NX1) excitation column vector (known).
(NX1) unknown response column vector to be found.
Then the solution is:

17. Electrostatic Charge Distribution
Evaluation of the Matrix Elements:

18. Electrostatic Charge Distribution
Where,
is the distance between the m th matching point and the center of the n th source point.

19. Electrostatic Charge Distribution
Exercise: Consider a wire with , a=0.001m, V=1 Volt. Determine the charge distribution for N=5.

20. Moment Methods (Method of Moments, MoM)
The MoM is the name of the technique which solves a linear operator equation by converting it to a matrix equation.

21. Moment Methods (Method of Moments, MoM)
Consider the differential equation
Where L is a differential operator, is the unknown field and is the known given excitation.
The Method of Moments is a general procedure for solving this equation.

22. Moment Methods (Method of Moments, MoM)
The procedure for applying MoM to solve the equation above usually involves four steps:
1)Derivation of the appropriate integral equation (IE).
2)Conversion (discretization) of IE into a matrix equation using basis (or expansions) functions and weighting functions.

23. Moment Methods (Method of Moments, MoM)
3)Evaluation of the matrix elements.
4)Solving the matrix equation and obtaining the parameters of interest.
The basic tools for step 2 will be discussed.
MoM will be applied to IEs rather than PDEs.

24. Differential Equations Vs. Integral Equations
Integral equations may take several forms, e.g, Fredholm equations.

25. Moment Methods (Method of Moments, MoM)
Where is a scalar (or possibly complex) parameter. Functions K(x,t) and f(x) are known. K(x,t) is known as the kernel of the integral equation. The limits a and b are also known, while the function is unknown.

26. Moment Methods (Method of Moments, MoM)
The second class of integral equations, with a variable upper limit of integration, Volterra equations:

27. Moment Methods (Method of Moments, MoM)
If f(x)=0 the integral equations become homogeneous.
All above equations are linear.
An integral equation becomes non-linear when appears in the power of n>1 under the integral sign.

28. Differential Equations Vs. Integral Equations
Most differential equations can be expressed as integral equations, e. g.,
This can be written as the Voterra integral equation.

29. Differential Equations Vs. Integral Equations
Solve the Voterra integral equation:
In general given an integral with variable limits:
It differentiated by using the Leibniz rule:

30. Differential Equations Vs. Integral Equations
It differentiated by using the Leibnitz rule:
Differentiating
We obtain:

31. Differential Equations Vs. Integral Equations
Or:
Integrating gives:
Where is the integration constant. Or
From the given integral equation: