partial differential equations & wavesjmb/lectures/pdelecture1.pdf · 2006. 1. 2. · partial...
TRANSCRIPT
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Partial Differential Equations
& waves
Professor Sir Michael Brady FRS FREng
Michaelmas 2005
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Analysing physical systemsFormulate the most appropriate mathematical model for the
system of interest – this is very often a PDE
This is what a large part of Engineering science & practice is about.
• Diffusion of charge, flow of heat, absorption of a drug• Propagation of waves across water, electrical networks, with/without loss of energy• “steady state” – no further change – in stress analysis, heat or fluid flow, …
– We will recall from ODEs: a single equation can have lots of very different solutions, the boundary conditions determine which
Figure out the appropriate boundary conditions, apply them
In this course, solutions will be analytic = algebra & calculusReal life is not like that!! Numerical solutions include finite difference and finite element techniques
Solve the PDE
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…but why partial differential equations
A physical system is characterised by its state at any point in space and time
now here,in re temperatu,),,,( tzyxu
tu
∂∂State varies over time:
yxu∂∂
∂2
like thingsState also varies over space:
Surely, we need to relate these variations to each other…e.g.
2
2
xuk
tu
∂∂
=∂∂
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How do we relate spatial variations to temporal variations?
• Constituent equations which you met in vector calculus embody physical constraints such as– “conservation of mass”, – “conservation of enthalpy”
Don’t panic! We’ll work mostly in one spatial dimension
In the case of an insulated, diffusing distribution of heat, the equation (which we will derive later) is:
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂
=∂∂
2
2
2
2
2
2
zu
yu
xuk
tu
That is, the spatial change is directly proportional k to the temporal change
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An example of a PDE: the one-dimensional heat equation
2
22
xuc
tu
∂∂
=∂∂
material the ofdensity heatspecific
tyconductivi thermal
:case this In
===
=
ρσ
σρK
Kc2
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Another example:the one-dimensional wave
equation
2
22
2
2
xuc
tu
∂∂
=∂∂
string the of length mass/unit string the in tension
:case this In
==
=
ρ
ρT
Tc2
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Background to this course
Partial Differential Equations
Partial differentiation
Ordinary Differential Equations
Fourier series
Numerical methods
Vector calculus
Electrical engineering
Mechanical engineering
Civil engineering
Biomedical engineeringWe now give brief reminders of partial differentiation,
ODEs, and Fourier series. Please re-read the relevant parts of Kreysig if you are shaky on some particular part
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Partial derivatives*
cyaxxux +=
∂∂ 2: to respect withderivative Partial
From which:
cuau
xy
xx
== 2
dcxyyxayxu ++−= )(),( 22 :function the Consider
* Please refer to Kreysig, 8th Edition, pages A57 – A60 for a refresher
xuxu
≡∂∂ :Notation
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Partial differentiation with respect to y
cu
au
cxayu
yx
yy
y
=
−=
+−=
2
2
Evidently, changing the order of differentiation makes no difference:
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂∂
=⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
yu
xxu
y
dcxyyxayxu ++−= )(),( 22
This is the case whenever u varies “smoothly” with respect to x and y. This is almost always so.
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Chain rule*
),( vuC),( and ),( yxvvyxuu ==
Suppose that we are given a function
where
dvvCdu
uCdC
∂∂
+∂∂
=The total variation in C is
yvyuy
xvxux
vCuCCvCuCC
+=+=
From which we find
*Kreysig, 8th Edition, page 444
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Ordinary differential equations
First order axAeyaydxdy −=⇒=+ 0
Kreysig, 8th Edn, pp 19-21
)sincos()(
, 2121
xBxAeyjaeBAxy
BeAey
ax
x
xx
ωωω
λ
λλλ
λλ
+=⇒±−
+=⇒
+=⇒
− rootscomplex :3 Case roots equal real, :2 Case roots unequal real, 1: Case
Second order* 0 0 22
2
=++⇒=++ cbmamcdxdyb
dxyda
Auxiliary equation
*Kreysig, 8th Edn, Chapter 2
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Homogeneous equations: superposition of solutions
shomogeneou called then is ODE the , Ifs.derivative its of one or function unknown the contain
: of side left the on terms The
0)()(
)()(')(''
=
=++
xrxy
xryxqyxpy
Fundamental theorem* about homogeneous ODEs:
solution. a is
solutions of ionsuperposit linearany generally, More
constants. are wheresolution, a also is then ODE, given a to solutions are and if
∑
+
iii
i
xyc
cxycxycxyxy
)(
)()()()(
2211
21
*Kreysig, 8th Edn, p66
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How we use superposition of solutions
Consider: 02
2
=− kydx
ydwhere k can be +ve, -ve, 0
xFxEykkDCxyk
BeAeykk xx
βββ
α αα
sincos say ,0 0 say ,0
2
2
+=⇒−=<
+=⇒=+=⇒=> −
We “superpose” these solutions, and leave it to analysis of the boundary conditions to help us figure out which bits are relevant in any given case
)sincos()()( xFxEDCxBeAey xx ββαα +++++= −
For example, if we are told: xBeyxy α−=∞→→ then , as ,0
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Partial Differential Equations generally have many different solutions
axu 22
2
=∂∂
and ayu 22
2
−=∂∂
Evidently, the sum of these two is zero, and so the function u(x,y) is a solution of the partial differential equation:
0yu
xu
2
2
2
2
=∂∂
+∂∂
Laplace’s Equation
Recall the function we used in our reminder of partial derivatives:
dcxyyxayxu ++−= )(),( 22
This choice was not random! Recall that we showed:
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A completely different solution to Laplace’s Equation
( ) xeyxv y cos, −=Consider the entirely different function:
xexv y cos2
2−−=
∂∂We find
xeyv y cos2
2−=
∂∂ and
02
2
2
2
=∂∂
+∂∂
yv
xv
So that the function v(x,y) also satisfies
Boundary conditions determine the solution in any particular case
solution a also is :ionsuperpositby Evidently, ),(),( yxvyxu +
Showing that particular functions satisfy particular PDEs is the subject of Q1, Q2 on the first tutorial sheet
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An example of applying specific boundary conditions
Consider the superposition of the two solutions u(x,y)+v(x,y) suppressing constants, which would make no difference:
( ) (1) DCxy)yx(BxcosAey,xu 22y ++−+= −
And, suspending reality for a moment, suppose this represents the stress in an infinite plate with a circular hole:
F F
By considering x and y at infinity, it is clear that for (1) to be a physically plausible solution, then because the stress must remain finite, we conclude that B = C = 0.
x
y
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Applying the problem-specific boundary condition that the end of the bar (x=L) is maintained at zero temperature, we have
LnqnqL
qLAqLAππ ==
===
:is that , so these, of first the in interested not are We
orthat so .0sin0,0sin
Every value of n corresponds to a solution, so we use superposition to find the general solution:
∑∞
=
⎟⎟⎠
⎞⎜⎜⎝
⎛−
=0
sin),(2
22
n
Ltkn
n LxneAtxT π
π
How Fourier series enter the game
qxAetxT tkq sin),(2−=
Anticipating lecture 2, suppose we are solving a specific case of the Heat Equation, to find the temperature of a bar of length L. We will find that the solution is given (in that case) by the temperature
We then apply Fourier series to solve for the nA
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Fourier series* in 3 steps1. Fourier theory asserts that for any periodic function, f(θ), with period
2π, coefficients an and bn can be found such that
( ) θθθ nsinbncosaf n1n
n0n
∑∑∞
=
∞
=
+=
*Kreysig, 8th Edn, Sections 10.1-10.4, p526
2. Many functions of interest are not specified as periodic; but they can be made so by judicious choices
Lx
T
To
πθ
=Lx
T
T0
- T0
θ
3. To find the constants an and bn, we proceed in one of two ways:
a. Look up the solution in HLT
b. Figure them out from first principles using “orthogonality relations”
We’ll do both in the next lecture
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A page scanned from HLT
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( ) θθθ nbnaf nn
nn
sincos10
∑∑∞
=
∞
=
+= : that told are weSuppose
The orthogonality relationships massively simplify finding the coefficients. We first multiply the function f(θ), by cosmθ and integrate between 0 and 2π
( ) =∫ θθθπ
π
dmcosf21 2
0
θθθπ
θθθπ
ππ
dmcosnsinb21dmcosncosa
21
n1n
2
0n
0n
2
0∑∫∑∫
∞
=
∞
=
+
Reversing the orders of summation and integration on the right hand side gives
( ) θθθπ
θθθπ
θθθπ
πππ
dmcosnsinb21dmcosncosa
21dmcosf
21
n
2
01nn
2
00n
2
0∫∑∫∑∫
∞
=
∞
=
+=
How to apply orthogonality relationships
This is always zeroThis is zero unless m=n
( )2
admcosmcosa21dmcosf
21 m
m
2
0
2
0
== ∫∫ θθθπ
θθθπ
ππ
giving
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( )2
admcosmcosa21dmcosf
21 m
m
2
0
2
0
== ∫∫ θθθπ
θθθπ
ππ
giving
so
( ) θθθπ
π
dmcosf1a2
0m ∫=
Similarly,
( ) θθθπ
π
dmsinfbm ∫=2
0
1
These are the coefficients for the full-range series, ie those for which 0 < θ < 2π. Orthogonality relationships also hold for half-range series (ie those for which 0 < θ < π) which are also useful. They are
( ) θθθπ
π
dncosf2a0
n ∫= ( ) θθθπ
π
dnsinf2b0
n ∫=
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Three equations dominate
• Diffusion (or heat) equation
• Laplace’s (or potential) equation
• Wave Equation
02
2
2
2
=∂∂
+∂∂
yu
xu
2
21xu
tu
∂∂
=∂∂
κ
2
22
2
2
xuc
tu
∂∂
=∂∂
Diffusion problems, transient heat transfer, concentration in fluids, transient electric potential
Steady state problems in stress analysis, heat transfer, electrostatics, fluid flow…..
Wave phenomena in mechanical systems (vibrations), fluids, electricity…..
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The general second order PDE
),(),(),(),(
),(),(),(
yxGuyxFuyxEuyxD
uyxCuyxBuyxA
yx
yyxyxx
=++
+++
042 <− ACB042 =− ACB042 >− ACB
Elliptic, if
Parabolic, if
Hyperbolic, if
LaplaceDiffusionWave
The three PDEs arise most frequently in practice, and they cover the most interesting basic PDEs
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Overview of the Course1. General introduction, revision of partial differentiation, ODEs, and
Fourier series2. Wave equation in 1D part 1: separation of variables, travelling
waves, d’Alembert’s solution3. Heat equation in 1D: separation of variables, applications4. limitation of separation of variables technique. Sometimes, one
way to proceed is to use the Laplace transform5. Laplace’s equation: first, separation of variables (again), Laplace’s
equation in polar coordinates, application to image analysis 6. Wave equation in 1D part 2: phase and transverse velocity,
characteristic impedance, wave number, circular fequency, standing waves; impedance boundaries, lossy (dispersive) waves, amplitude modulation
7. Water waves8. Another look at separation of variables: Sturm-Liouville Equations
and orthogonal functions. Legendre and Bessel functions.
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Books
1. Kreyszig Advanced Engineering Mathematics 8th Edition. Very big, impresses fellow students, mostly unread but can support a stereo or three pints. Most of the course “follows” the treatment in this book.
2. James: Advanced Modern Engineering Mathematics. Again comprehensive, perhaps a bit easier than Kreyszig. Somewhat duller and less impressive for your tutor.
3. Main Vibrations and Waves in Physics. Used, with James for waves section. Unlikely that your tutor will believe that you bought it, or even read it.
4. Pain The Physics of Vibrations and Waves, ditto Main.
5. Pearson Partial Differential Equations. Wonderful book, if you are a mathematician at a US ivy league university. No pictures. Generally dull.
Moral of the tale: read the notes AND Kreysig. K has more detail, fewer jokes
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Reminder of the orthogonality relations*
The orhogonality relations exploit values of integrals like:
θθθπ
π
dmcosncos21 2
0∫
θ is periodic with period 2π, and n and m are integer.
First take the case m ≠ n.
( ) ( )[ ]
( ) ( ) 0nmsinnm
1nmsinnm
141
dnmcosnmcos41dmcosncos
21
2
0
2
0
2
0
=⎥⎦
⎤⎢⎣
⎡−
−++
+=
−++= ∫∫π
ππ
θθπ
θθθπ
θθθπ
*Kreysig 8th Edn, page 530 & A3
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Now take the case m = n.
( )212
21
4121
41
21 2
0
2
0
22
0
=⎥⎦⎤
⎢⎣⎡ +=+= ∫∫
πππ
θθπ
θθπ
θθπ
nsinn
dncosdncos
We can do similar things for sinnθ sinmθ and sinnθ cosmθand so obtain the orthogonality relationships:
( )nm0
0nm1nm5.0dmcosncos
21 2
0 ≠=====
∫ forwhenfor
θθθπ
π
( )nm
0nm0nm5.0dmsinnsin
21 2
0 ≠=====
∫ for 0whenfor
θθθπ
π
n,m0dmcosnsin21 2
0
allfor=∫ θθθπ
π