Chapter 13 Partial differential equations

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Mathematical methods in the physical sciences
3nd edition Mary L. Boas
Chapter 13 Partial differential equations
Lecture 13 Laplace, diffusion, and wave equations
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1. Introduction (partial differential equation)
ex 1) Laplace equation
gravitational potential, electrostatic
potential, steady-state temperature with no source
ex 2) Poissons equation
with sources (f(x,y,z))
ex 3) Diffusion or heat flow equation
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ex 4) Wave equation
ex 5) Helmholtz equation
space part of the solution of either the
diffusion or the wave equation
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2. Laplaces equation steady-state temperature
in a rectangular plate (2D)
In case of no heat source
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1) In the current problem, boundary conditions
are
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2) How about changing the boundary condition? Let
us consider a finite plate of height 30 cm with
the top edge at T0.
T0 at 30 cm
In this case, eky can not be discarded.
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- To be considered I
This is correct, but makes the problem more
complicated. (Please check the boundary
condition.)
- To be considered II
In case that the two adjacent sides are held at
100? (ex. CD100?), the solution can be the
combination of C100? solutions (A, B, D 0?)
and D100? (A, B, C 0?) solutions.
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-. Summary of separation of variables. 1) A
solution is a product of functions of the
independent variables. 2) Separate partial
equation into several independent ordinary
equation. 3) Solve the ordinary differential
eq. 4) Linear combination of these basic
solutions 5) Boundary condition (boundary value
problem)
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3. Diffusion or heat flow equation heat flow in
a bar or slab
cf. Why do we need to choose k2, not k2?
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Lets take a look at one example.
At t0, T0 for x0 and T100 xl. From t0 on,
T0 for xl.
For T(x0)0 and T(xl)100 at t0, the initial
steady-state temperature distribution
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- Using Boundary condition
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For some variation, when T?0,
we need to consider uf.as the final state,
maybe a linear function. In this case, we can
write down the solution simply like this.
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4. Wave equation vibrating string
node
Under the assumption that the string is not
stretched,
x0
xl
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1) case 1
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2) case 2
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3) Eigenfunctions
first harmonic, fundamental
second harmonic
third
fourth
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Mathematical methods in the physical sciences
2nd edition Mary L. Boas
Chapter 13 Partial differential equations
Lecture 14 Using Bessel equation
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5. Steady-state temperature in a cylinder
For this problem, cylindrical coordinate (r, ?,
z) is more useful.
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In order to say that a term is constant, 1)
function of only one variable 2) variable does
not elsewhere in the equation.
- 1st step
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- 2nd step
- 3rd step
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Bessels equation 1
- Bessels equation
1) Equation and solution
- named equation which have been studied
extensively. - Bessel function solution of a
special differential equation. - being
something like damped sines and cosines. - many
applications. ex) problems involving
cylindrical symmetry (cf. cylinder function)
motion of pendulum whose length increases
steadily small oscillations of a flexible chain
railway transition curves stability of a
vertical wire or beam Fresnel integral in
optics current distribution in a conductor
Fourier series for the arc of a circle.
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Bessels equation 2
- Graph
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Bessels equation 3
2) Recursion relations
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Bessels equation 4
3) Orthogonality
cf. Comparison
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Bessels equation 5
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6. Vibration of a circular membrane (just like
drum)
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cf. They are not integral multiples of the
fundamental as is true for the string
(characteristics of the bessel function). This is
why a drum is less musical than a violin.
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(1,1)
(m,n)(1,0)
(2,0)
American Journal of Physics, 35, 1029 (1967)
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Mathematical methods in the physical sciences
2nd edition Mary L. Boas
Chapter 13 Partial differential equations
Lecture 15 Using Legendre equation
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7. Steady-state temperature in a sphere
- Sphere of radius 1 where the surface of upper
half is 100, the other is 0 degree.
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Legendres equation 1
- Legendres equation
1) Equation and solution
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Legendres equation 2
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Legendres equation 3
- Legendre polynomials
- Associated Legendre polynomials
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Legendres equation 4
2) Orthogonality
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8. Poissons equation
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Example 1
grounded sphere
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Method of the images
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cf. Electric multipoles
quadrupole
octopole
monopole
dipole
2) Expansion for the potential of an arbitrary
localized charge distribution
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- n 0 monopole contribution - n 1
dipole - n 2 quadrupole - n 3 octopole