A TUTORIAL ON METHOD OF MOMENTS

A TUTORIAL ON METHOD OF MOMENTS

Ercumend Arvas

ISTANBUL MEDIPOIL UNIVERSITY1Method of moments is a general procedure for solving linear equations.

Many problems that cannot be solved exactly can be solved approximately by this method.

Many results are from R.F. Harrington, Field Computation by Moment MethodsAs an example, consider the Capacitance of two parallel plates (including the fringing fields):

Here L is a LINEAR operator; g(x) is a known function (usually excitation in a linear system), and f(x) is the unknown function (usually response) to be found. Consider A Linear Operator Equation4 Definition of Linear Operators5Examples of Linear OperatorsExample 1:Here L is multiplication by 5.

Note that L(2f) = 5 (2f) = 10 f = 2(5f) = 2 L(f)

and

L(3f+4g) = 5(3f+4g) = 3 (5f)+4(5g) = 3L(f)+4L(g)Examples of Linear OperatorsExample 2:Here L is derivative operator. Note thatExamples of Linear OperatorsExample 3:Example 4:Examples of Linear OperatorsExample 5:Example 6:L(x)=[A] xwhere

[A] is an nxn matrix, and x is an nx1 column vector.Examples of Linear Operator Equation

Example 1:

Examples of Linear Operator Equation

Example 2:11Approximating a function by other functionsTesting TestingTestingSummary of Fourier SeriesConvergence of Fourier SeriesExample 1:Convergence of Fourier SeriesConvergence of Fourier Series

Convergence of Fourier SeriesConvergence of Fourier SeriesExample 2:In practice, we use a truncated Fourier series to represent a periodic function approximately.f(x)x-2 1 -1f(x)x-21 -1Convergence of Fourier Series

xf(x)-11Convergence of Fourier SeriesConvergence of Fourier SeriesQuestionCan we use the Fourier series if the function f(x) is not periodic as shown?

f(x)x1AnswerYes, use even or odd periodic extension of f(x):

x1x1-1Another QuestionAssume that I do not know the function f(x), but I have some information about it. For example, I may know its derivative or I may know that it is the solution of a linear (differential or integral) equation. Can I still come up with an approximation to f(x)?

Yes MoM is one way of doing that.Answer:Back to MoMUse MoM to find f(x) satisfying

Example:Solution:Expansion FunctionsIn this example, we should choose expansion functions that are twice differentiable and satisfy the boundary conditions at x = 0 and x = 1.

Expansion FunctionsResidualPoint MatchingResults

r(x) = 0 at x=1/2xClose up

xMatching at two pointsMatching at two points

Results r(x) = 0 at x =1/2

Close upMatching at three pointsPoint Matching Delta Functions01xPoint Matching Delta FunctionsDifferent Testing FunctionsTo make average r(x) equal to zero, we can test the approximate operator equation by a pulse p(x-0.5).

01xp(x-0.5)0.51Different Testing Functions

Different Testing Functions

01xp(x-0.25)0.51p(x-0.75)Formal Summary of MoMGiven a linear operator equation

Formal Summary of MoMFormal Summary of MoMBack to our example:Back to our example:

Point matching results:Back to our example:Back to our example:Galerkins Results

An application of MoM: Optical Fiber

An application of MoM: Optical Fiber

An application of MoM: Optical Fiber

An application of MoM: Optical Fiber

An application of MoM: Optical Fiber

An application of MoM: Optical Fiber

An application of MoM: Optical Fiber

An application of MoM: Optical Fiber

An application of MoM: Optical Fiber

An application of MoM: Optical Fiber

An application of MoM: Optical FiberThank you

Any questions?