The Infinite Slot

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The Infinite Slot
Two infinite, grounded, metal plates lie parallel
to the x-z plane, one at y0, and the other at
ya. The left end, at x0, is insulated from the
two plates and held at a potential V0(y).
Find the potential inside the slot!
Slot extends infinitely in the z-direction. --gt
potential does not depend on z

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The Infinite Slot
Laplace
Assume that we can separate the potential

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The Infinite Slot
We can try to determine the four constants A-D
from our four boundary conditions
1. V ? 0 for x ? ?
2. V 0 at y 0
3. V 0 at y a

with n 0,1,2,3,
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The Infinite Slot
So that the complete solution is
Example with constant potential V0(y) 40 V.

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Numerical Integration
In many physics problems we need to calculate the
value of a definite integral over an interval
a,b
Numerical integration provides general methods
how to efficiently calculate this integral.
--gt Numerical Quadrature

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Numerical Quadrature
Numerical Quadrature refers to a basic method
involved in approximating a definite integral
Quadrature rule is a weighted sum of a finite
number of sample values of the integrant function.
To obtain the desired level of accuracy at low
computational cost,
How should the sample points be chosen?
How should their contributions be weighted?

Computational cost required number of
evaluations of f(x)
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Numerical Quadrature
An n-point quadrature rule has the form
Points xi are called the nodes or the abscissas
Multipliers wi are called the weights
Distinguish between two cases
Closed Quadrature x0 a and xn-1 b
Open Quadrature a lt x0 and xn-1 lt b

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Numerical Quadrature
Quadrature rules are based on polynominal
interpolation
(1) Integrant function f is sampled at a finite
set of points
(2) Polynominal interpolating those points is
determined
(3) Interpolating polynominal is integrated
In practice, the interpolating polynominal is not
determined explicitely, but used to determine the
weights.

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Newton-Cotes Formulas
In the general quadrature rule
the nodes dont need to be spaced evenly.
The Newton-Cotes Formulas relate to evenly spaced
nodes.
The Newton-Cotes Formulas can be obtained for the
closed and open cases.
Lets first look at the closed cases.

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Closed Newton-Cotes
The simplest and for many cases most practical
quadrature scheme is the trapezoidal rule

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Closed Newton-Cotes
Error estimation for the Trapezoidal Rule
Error in the 1st order interpolation is given by

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Closed Newton-Cotes
Simpsons Rule
Lets use 3 points x0 a, x1 h
a(b-a)/2, x2 b, to specify an
interpolating quadratic polynominal

Simpsons 3 point rule is exact for polynominals
up to degree 3
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Closed Newton-Cotes
Simpsons Rule
Check Simpsons rule is exact for a cubic
Use Simpsons rule x00, x1 0.5, x21

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Closed Newton-Cotes
Simpsons 3/8 Rule
Lets use 4 points x0a, x1, x2 , x3b,
to specify an interpolating 3rd order
polynominal
Simpsons 3/8-Rule has the same degree of
accuracy as Simpsons 3-Point Rule.
This phenomena is true for all even-order
Newton-Cotes formulas

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Another Example
Example
Lets use Simpsons rule
-- gt Closed formula does not work

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Open Newton-Cotes

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Open Newton-Cotes
Example
Closed formula does not work
MidpointRule (n0)
n1 Open Rule
n2 Open Rule

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Drawbacks of Newton-Cotes
The Newton-Cotes rules are simple and often
effective, but they have drawbacks
Using a large number of equally spaced nodes may
incur erratic behavior associated with
high-degree polynominal interpolation