Mat-F February 21, 2005 Separation of variables
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Mat-FFebruary 21, 2005
Separation of variables
Åke Nordlund
Niels Obers, Sigfus Johnsen / Anders SvenssonKristoffer Hauskov Andersen
Peter Browne Rønne
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Overview
Follow-up Computer problems Maple TA problems Maple TA changes New web pages
Chapter 19 Separation of variables Exercises today
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Follow-up:Computer problems
Client-server problems Over-loaded servers Over-loaded network??
Browser problems Mozilla / Maple / JAVA Netscape plugin missing
Don’t panic! “… it said in warm, red letters …”
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Follow-up:Computer problems
Some problems will be fixed Browser: Netscape with plugin
Some hardware constraints need work-arounds Network limitations??
Possibly use D315 & A122 CPU limitations
Make sure logins and windows are on localhost
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Follow-up:Maple TA
Maple TA improvements No longer accepts “0” or “whatever” as solutions
Remaining limitations Still accepts less than general PDE solutions
Use human judgment! I will !
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Follow-up:Maple TA
Maple TA changes
For those who could not come Wednesday For continued exercises (computer problems) For additional training
Maple TA exercises will be (are) available in an open version from the day (Thursday) after the exercises.
Maple TA exercises will be (are) available in an open version from the day (Thursday) after the exercises.
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New web pages
Maple Examples Examples / templates for turn-in assignments
Exercise Groups Time & room schedule added
Rooms could possibly change next week But not this week!
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19. Partial Differential Equations:Separation of variables …
Main principles Why?
Techniques How?
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Main principles
Why? Because many systems in physics are separable In particular systems with symmetries Such as spherically symmetric problems
Example: Spherical harmonics
u(r,Θ,φ,t) = Ylmn(r,Θ,φ) eiωt
Example: Spherical harmonics
u(r,Θ,φ,t) = Ylmn(r,Θ,φ) eiωt
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19. Partial Differential Equations:Separation of Variables …
Main principles Why?
Common in physics! Greatly simplifies things!
Techniques How?
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19.1 Separation of variables:the general method
Suppose we seek a PDE solution u(x,y,z,t)
Ansatz:
u(x,y,z,t) = X(x) Y(y) Z(z) T(t)
Ansatz:
u(x,y,z,t) = X(x) Y(y) Z(z) T(t)
This ansatz may be correct or incorrect!
If correct we have
This ansatz may be correct or incorrect!
If correct we have
u/x = X’(x) Y(y) Z(z) T(t)
2u/x2 = X”(x) Y(y) Z(z) T(t)
u/x = X’(x) Y(y) Z(z) T(t)
2u/x2 = X”(x) Y(y) Z(z) T(t)
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19.1 Separation of variables:the general method
Suppose we seek a PDE solution u(x,y,z,t):
Ansatz:
u(x,y,z,t) = X(x) Y(y) Z(z) T(t)
Ansatz:
u(x,y,z,t) = X(x) Y(y) Z(z) T(t)
This ansatz may be correct or incorrect!
If correct we have
This ansatz may be correct or incorrect!
If correct we have
u/y = X(x) Y’(y) Z(z) T(t)
2u/y2 = X(x) Y”(y) Z(z) T(t)
u/y = X(x) Y’(y) Z(z) T(t)
2u/y2 = X(x) Y”(y) Z(z) T(t) u/y = X(x) Y’(y) Z(z) T(t)
2u/y2 = X(x) Y”(y) Z(z) T(t)
u/y = X(x) Y’(y) Z(z) T(t)
2u/y2 = X(x) Y”(y) Z(z) T(t)
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19.1 Separation of variables:the general method
Suppose we seek a PDE solution u(x,y,z,t):
Ansatz:
u(x,y,z,t) = X(x) Y(y) Z(z) T(t)
Ansatz:
u(x,y,z,t) = X(x) Y(y) Z(z) T(t)
This ansatz may be correct or incorrect!
If correct we have
This ansatz may be correct or incorrect!
If correct we have
u/y = X(x) Y’(y) Z(z) T(t)
2u/y2 = X(x) Y”(y) Z(z) T(t)
u/y = X(x) Y’(y) Z(z) T(t)
2u/y2 = X(x) Y”(y) Z(z) T(t) u/z = X(x) Y(y) Z’(z) T(t)
2u/z2 = X(x) Y(y) Z”(z) T(t)
u/z = X(x) Y(y) Z’(z) T(t)
2u/z2 = X(x) Y(y) Z”(z) T(t)
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19.1 Separation of variables:the general method
Suppose we seek a PDE solution u(x,y,z,t):
Ansatz:
u(x,y,z,t) = X(x) Y(y) Z(z) T(t)
Ansatz:
u(x,y,z,t) = X(x) Y(y) Z(z) T(t)
This ansatz may be correct or incorrect!
If correct we have
This ansatz may be correct or incorrect!
If correct we have
u/y = X(x) Y’(y) Z(z) T(t)
2u/y2 = X(x) Y”(y) Z(z) T(t)
u/y = X(x) Y’(y) Z(z) T(t)
2u/y2 = X(x) Y”(y) Z(z) T(t) u/t = X(x) Y(y) Z(z) T’(t)
2u/t2 = X(x) Y(y) Z(z) T”(t)
u/t = X(x) Y(y) Z(z) T’(t)
2u/t2 = X(x) Y(y) Z(z) T”(t)
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Examples
u(x,y,z,t) = xyz2sin(bt)u(x,y,z,t) = xyz2sin(bt)
u(x,y,z,t) = xy + ztu(x,y,z,t) = xy + zt
u(x,y,z,t) = (x2+y2) z cos(ωt)u(x,y,z,t) = (x2+y2) z cos(ωt)
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Examples
u(x,y,z,t) = xyz2sin(bt)u(x,y,z,t) = xyz2sin(bt)
u(x,y,z,t) = xy + ztu(x,y,z,t) = xy + zt
u(x,y,z,t) = (x2+y2) z cos(ωt)u(x,y,z,t) = (x2+y2) z cos(ωt)
Separable?
No!
Yes!
Hm…?
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Examples
u(x,y,z,t) = xyz2sin(bt)u(x,y,z,t) = xyz2sin(bt)
u(x,y,z,t) = xy + ztu(x,y,z,t) = xy + zt
u(x,y,z,t) = (x2+y2) z cos(ωt)u(x,y,z,t) = (x2+y2) z cos(ωt)
Separable?
No!
Yes!
Hm…?
(x2+y2) p2(x2+y2) p2
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Examples
u(x,y,z,t) = xyz2sin(bt)u(x,y,z,t) = xyz2sin(bt)
u(x,y,z,t) = xy + ztu(x,y,z,t) = xy + zt
u(p,z,t) = p2 z cos(ωt)u(p,z,t) = p2 z cos(ωt)
Separable?
No!
Yes!
Yes!
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Example PDEs
The wave equation
where
c22u = 2u/t2 c22u = 2u/t2
2 2/x2 + 2/y2 + 2/z2 2 2/x2 + 2/y2 + 2/z2
Ansatz:
u(x,y,z,t) = X(x) Y(y) Z(z) T(t)
Ansatz:
u(x,y,z,t) = X(x) Y(y) Z(z) T(t)
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Separation of variables in the wave equation
For the ansatz to work we must have (lets set c=1 for now to get rid of a triviality!)
Divide though with XYZT, for:
X”YZT + XY”ZT + XYZ”T = XYZT”X”YZT + XY”ZT + XYZ”T = XYZT”
X”/X + Y”/Y + Z”/Z = T”/TX”/X + Y”/Y + Z”/Z = T”/T
This can only work if all of these are constants!This can only work if all of these are constants!
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Separation of variables in the wave equation
Let’s take all the constants to be negative Negative bounded behavior at large x,y,z,t
X”/X + Y”/Y + Z”/Z = T”/TX”/X + Y”/Y + Z”/Z = T”/T
- 2 - 2 - 2 = - µ2 - 2 - 2 - 2 = - µ2
This is typical: If / when a PDE allows separation of variables, the partial derivatives are replaced with ordinary derivatives, and all that remains of the PDE is an algebraic equation and a set of ODEs – much easier to solve!
This is typical: If / when a PDE allows separation of variables, the partial derivatives are replaced with ordinary derivatives, and all that remains of the PDE is an algebraic equation and a set of ODEs – much easier to solve!
These are called separation constantsThese are called separation constants
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Separation of variables in the wave equation
Solutions of the ODEs:
Example: x-direction
X”/X = - 2
Example: x-direction
X”/X = - 2
General solution:
X(x) = A exp(i x) + B exp(-i x)
or
X(x) = A’ cos( x) + B’ sin( x)
General solution:
X(x) = A exp(i x) + B exp(-i x)
or
X(x) = A’ cos( x) + B’ sin( x)
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Separation of variables in the wave equation
Combining all directions (and time)
Example:
u = exp(i x) exp(i y) exp(i x) exp(-i cµt)
or
u = exp(i ( x + y + x - ω t))
Example:
u = exp(i x) exp(i y) exp(i x) exp(-i cµt)
or
u = exp(i ( x + y + x - ω t))
This is a traveling wave, with wave vector {,,} and frequency ω. A general solution of the wave equation is a super-position of such waves.
This is a traveling wave, with wave vector {,,} and frequency ω. A general solution of the wave equation is a super-position of such waves.
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Example PDEs
The diffusion equation
where
2u = u/t 2u = u/t
2 2/x2 + 2/y2 + 2/z2 2 2/x2 + 2/y2 + 2/z2
Ansatz:
u(x,y,z,t) = X(x) Y(y) Z(z) T(t)
Ansatz:
u(x,y,z,t) = X(x) Y(y) Z(z) T(t)
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Separation of variables in the diffusion equation
For the ansatz to work we must have (lets set =1 for now to again get rid of that triviality)
Divide though with XYZT, for:
X”YZT + XY”ZT + XYZ”T = XYZT’X”YZT + XY”ZT + XYZ”T = XYZT’
X”/X + Y”/Y + Z”/Z = T’/TX”/X + Y”/Y + Z”/Z = T’/T
Again, this can only work if all of these are constants!Again, this can only work if all of these are constants!
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Separation of variables in the diffusion equation
Again, we take all the constants to be negative Negative bounded behavior at large x,y,z,t
X”/X + Y”/Y + Z”/Z = T’/TX”/X + Y”/Y + Z”/Z = T’/T
- 2 - 2 - 2 = - µ - 2 - 2 - 2 = - µ
We see the typical behavior again: All that remains of the PDE is an algebraic equation and a set of ODEs – much easier to solve!
We see the typical behavior again: All that remains of the PDE is an algebraic equation and a set of ODEs – much easier to solve!
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Separation of variables in the diffusion equation
Solutions of the ODEs:
Example: x-direction
X”/X = - 2
Example: x-direction
X”/X = - 2
General solution:
X(x) = A exp(i x) + B exp(-i x)
or
X(x) = A’ cos( x) + B’ sin( x)
General solution:
X(x) = A exp(i x) + B exp(-i x)
or
X(x) = A’ cos( x) + B’ sin( x)
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Time-direction:
T’/T = - µ
Time-direction:
T’/T = - µ
Separation of variables in the diffusion equation
General solution:
T(t) = A exp(- µ t) + B exp(µ t)
General solution:
T(t) = A exp(- µ t) + B exp(µ t)
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Separation of variables in the diffusion equation
Combining all directions (and time)
Example:
u = exp(i x) exp(i y) exp(i x) exp(-µ t)
or
u = exp(i ( x + y + x )) exp(- µ t)
Example:
u = exp(i x) exp(i y) exp(i x) exp(-µ t)
or
u = exp(i ( x + y + x )) exp(- µ t)
This is a damped wave, with wave vector {,,} and damping constant µ. A general solution of the diffusion equation is a super-position of such waves.
This is a damped wave, with wave vector {,,} and damping constant µ. A general solution of the diffusion equation is a super-position of such waves.
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Example PDEs
The Laplace equation
where
2u = 0 2u = 0
2 2/x2 + 2/y2 + 2/z2 2 2/x2 + 2/y2 + 2/z2
Ansatz:
u(x,y,z,t) = X(x) Y(y) Z(z)
Ansatz:
u(x,y,z,t) = X(x) Y(y) Z(z)
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Separation of variables in the Laplace equation
For the ansatz to work we must have
Divide though with XYZT, for:
X”YZT + XY”ZT + XYZ”T = 0X”YZT + XY”ZT + XYZ”T = 0
X”/X + Y”/Y + Z”/Z = 0X”/X + Y”/Y + Z”/Z = 0
Again, this can only work if all of these are constants!Again, this can only work if all of these are constants!
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Separation of variables in the Laplace equation
This time we cannot take all constants to be negative!
X”/X + Y”/Y + Z”/Z = 0X”/X + Y”/Y + Z”/Z = 0
- 2 - 2 - 2 = 0 - 2 - 2 - 2 = 0
At least one of the terms must be positive imaginary wave number; i.e., exponential rather than sinusoidal behavior!
At least one of the terms must be positive imaginary wave number; i.e., exponential rather than sinusoidal behavior!
Let’s assume for simplicity (as in the example in the book) that there is only x & y dependence!
Let’s assume for simplicity (as in the example in the book) that there is only x & y dependence!
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Separation of variables in the Laplace equation
Solutions of the ODEs:
Example: x-direction
X”/X = - 2 < 0 assumed
Example: x-direction
X”/X = - 2 < 0 assumed
General solution:
X(x) = A exp(i x) + B exp(-i x)
or
X(x) = A’ cos( x) + B’ sin( x)
General solution:
X(x) = A exp(i x) + B exp(-i x)
or
X(x) = A’ cos( x) + B’ sin( x)
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Separation of variables in the Laplace equation
Solutions of the ODEs:
Example: y-direction
Y”/Y = + 2 > 0 follows!
Example: y-direction
Y”/Y = + 2 > 0 follows!
General solution:
Y(y) = C exp( y) + D exp(- y)
or
Y(y) = C’ cosh( y) + B’ sinh( y)
General solution:
Y(y) = C exp( y) + D exp(- y)
or
Y(y) = C’ cosh( y) + B’ sinh( y)
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Separation of variables in the Laplace equation
Combining x & y directions
Example:u = [A exp(i x) + B exp(-i x)] [C exp( y) + D exp(- y)]
oru = [A’ cos( x) + B’ sin( x)] [C’ cosh( y) + D’ sinh(- y)]
Example:u = [A exp(i x) + B exp(-i x)] [C exp( y) + D exp(- y)]
oru = [A’ cos( x) + B’ sin( x)] [C’ cosh( y) + D’ sinh(- y)]
A general solution of the Laplace equation is a superposition of such functions (which everywhere have a ’saddle-like’ behavior – net curvature = 0)
A general solution of the Laplace equation is a superposition of such functions (which everywhere have a ’saddle-like’ behavior – net curvature = 0)
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19.2 Superposition of separated solutions
If the PDE with separable solutions is linear … Wave equation Diffusion equation Laplace & Poisson eq. Schrødinger eq.
… then any linear combination (superposition) is also a solution Boundary conditions will make a limited selection!
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Exercises today:
Exercise 19.1 Separation of variables
Exercise 19.2 Diffusion of heat
Exercise 19.3 Wave equation on a stretched membrane
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Examples in the text of the book
Heat diffusion Stationary (u/t = 0)
equivalent to Laplace equation problem!
Various boundary conditions selects particular solutions NOTE: a diffusion (heat flow) equation may well have
a mixed sinusoidal & exponential behavior in space! consider the limit when the time dependence is weak!
2u = 0 2u = 0
2u = u/t 2u = u/t
2u = (1/)u/t → 0 2u = (1/)u/t → 0
- 2 - 2 - 2 = - µ → 0 - 2 - 2 - 2 = - µ → 0
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Enough for today!
Good luck with the exercises 10:15-12:00Good luck with the exercises 10:15-12:00