Lecture 16 Solving the Laplace equation in 2-D Remember Phils Problems and your notes = everything Only 6 lectures

Lecture 16Lecture 16

Solving the Laplace equation in 2-DSolving the Laplace equation in 2-D

http://www.hep.shef.ac.uk/Phil/PHY226.htmRemember Phils Problems and your notes = everything

Only 6 lectures left

• Come to see me before the end of term• I’ve put more sample questions and answers in Phils Problems• I’ve also added all the answers to the tutorial questions in your notes• Have a look at homework 2 (due in on 12/12/08)

02 u

Poisson’s equation

t

uiVuu

m

22

2

2

2

22 1

t

u

cu

Introduction to PDEsIntroduction to PDEs

In many physical situations we encounter quantities which depend on two or more variables, for example the displacement of a string varies with space and time: y(x, t). Handing such functions mathematically involves partial differentiation and partial differential equations (PDEs).

t

u

hu

22 1

02 u

0

2

uAs (4) in regions

containing mass, charge, sources of heat, etc.

Electromagnetism, gravitation,

hydrodynamics, heat flow.

Laplace’s equation

Heat flow, chemical diffusion, etc.

Diffusion equation

Quantum mechanicsSchrödinger’s

equation

Elastic waves, sound waves, electromagnetic

waves, etc.Wave equation

2 2

2 2 2

( , ) 1 ( , )y x t y x t

x c t

)()(),( tTxXtxy

Ndt

tTd

tTcdx

xXd

xX

2

2

22

2 )(

)(

1)(

)(

1

SUMMARY of the procedure used to solve PDEs

9. The Fourier series can be used to find the particular solution at all times.

1. We have an equation with supplied boundary conditions

2. We look for a solution of the form

3. We find that the variables ‘separate’

4. We use the boundary conditions to deduce the polarity of N. e.g.

5. We use the boundary conditions further to find allowed values of k and hence X(x).

6. We find the corresponding solution of the equation for T(t).

7. We hence write down the special solutions.

8. By the principle of superposition, the general solution is the sum of all special solutions..

2kN

L

xnBxX nn

sin)( kxBkxAxX sincos)( so

kctDkctCtT sincos)(

nn L

ctnEtT cos)(

nnn L

ctn

L

xnBtxY

cossin),(

1

cossin),(n

nn L

ctn

L

xnBtxy

L

ct

L

x

L

ct

L

x

L

ct

L

x

L

ct

L

xdtxy

7cos

7sin

49

15cos

5sin

25

13cos

3sin9

1cossin

8),(

2

www.falstad.com/mathphysics.html

http://www.falstad.com/mathphysics.html

The Laplace equation in 2D can be applied to any system

in which the value u does not change with distance.

In ‘steady state’ problems where nothing is changing with time, the equation

simplifies to which is the Laplace equation.

Solving the Laplace equation in 2 dimensionsSolving the Laplace equation in 2 dimensions

t

T

hdx

Td

22

2 1

02 T

Heat flow is governed by the diffusion equation,

We will look at this equation in 2D by considering the following exercise from the notes

Exercise Consider a rectangular metal plate 10 cm wide and very long. The two long sides and the far end are held at 0ºC and the base at 100ºC.

Find the steady state temperature distribution in x and y inside the plate using the boundary conditions.

02

2

2

22

dy

ud

dx

udu

Solving the Laplace equation in 2 dimensionsSolving the Laplace equation in 2 dimensionsExercise Consider a rectangular metal plate 10 cm wide and very long. The two long sides and the far end are held at 0ºC and the base at 100ºC.


02

2

2

22

y

T

x

TT

Step 1: Separation of the VariablesSubstitute this into the Laplace equation:

Separating variables: Step 2: Rearrange the equation

Laplace equation in 2-D is

0)(

)()(

)(2

2

2

2

y

yYxX

x

xXyY

2

2

2

2 )(

)(

1)(

)(

1

y

yY

yYx

xX

xX

Our boundary conditions are true at special values of x and y, so we look for solutions of the form T (x, y) = X(x)Y(y).


Step 3: Equate to a constant

Now we have separated the variables. The above equation can only be true for all x, y if both sides are equal to a constant.

So we have which rearranges to (i)

which rearranges to (ii)

We know that X(0) = X(L) = 0 and we know that in the Y direction up the page we expect an exponential drop or something similar from T(x,0) = 100 to T(x,∞) = 0. It is clear therefore that for a solution in x such that X(x) = 0 more than once, the constant must be negative (like a LHO). For convenience we choose the constant as -k2 so….

)()(

2

2

xNXx

xX

)()(

2

2

yNYy

yY

Nx

xX

xX

2

2 )(

)(

1

Ny

yY

yY

2

2 )(

)(

1

2

2

2

2 )(

)(

1)(

)(

1

y

yY

yYx

xX

xX

Step 4: Decide based on situation if N is positive or negative

Also since then

Solution to (i) is We know that X(0) = X(L) = 0 so A = 0.

Step 4 continued: Decide based on situation if N is positive or negative

Solving the Laplace equation in 2 dimensionsSolving the Laplace equation in 2 dimensionsExerciseConsider a rectangular metal plate 10 cm wide and very long. The two long sides and the far end are held at 0ºC and the base at 100ºC.


So we have (i) (ii))()( 2

2

2

xXkx

xX

)()( 2

2

2

yYky

yY

kxBkxAxX sincos)(

and

Step 5: Solve for the boundary conditions for X(x)

We know that X(0) = X(L) = 0 and we know that for Y we expect an exponential drop or something similar from T(x,0) = 100 to T(x,∞) = 0.

nkL L

xnBxX

sin)( so we can say kLBLX sin0)(

L

xnBCeyxT ky sin),( But and L = 10 so this can be written …..

Solution to (ii) is since we know that T (x, y) → 0 as y→∞

then we can state that D = 0 and


kyky DeCeyY )(

Step 6: Solve for the boundary conditions for Y(y)

We know that for Y we expect an exponential drop or something similar from T(x,0) = 100 to T(x,∞) = 0.

kyCeyY )(

Step 7: Write down the special solution for (x, y) i.e. T(x,y) =X(x)Y(y)

10sinsin),( 10 xn

PeL

xnCBeyxT

yn

L

yn

nkL

where P = CB

)()( 2

2

2

yYky

yY



Step 8: Constructing the general solution for (x, y)

The general solution of our equation is the sum of all special solutions:

1

10

10sin),(

n

yn

n

xnePyxT

This already satisfies the boundary conditions for x, namely that T(0,y) = T(L,y) = 0. All that remains is calculate the required values of Pn such that the T(x,0) =100 is satisfied

Step 9: Use Fourier series to find values of Pn

Since the temperature at y = 0 is 100 then

1 1

10

0

10010

sin10

sin)0,(n n

n

n

n

xnP

xnePxT



Step 9 continued: Use Fourier series to find values of Pn

So if temperature at y = 0 is 100 so general solution here is

1

10010

sin)0,(n

n

xnPxT

Here is a lateral jump that isn’t obvious!!!!

1

sin)(n

n L

xnbxf

L

n dxL

xnxf

Lb

0sin)(

2

As you know the Half-range sine series is given by:

where

Look how similar this is to the expression for T(x,0) if we set f(x)=100.

So all we have to do now is calculate the half-range sine series in the usual way in order to find the values of Pn.



Step 9 continued: Use Fourier series to find values of Pn

In effect we’re finding the half range sine series for a pulse as defined below.

L

n dxL

xnxf

Lb

0sin)(

2

1cos200

0cos10

10cos

200

10cos

1020

10

0

n

n

n

n

xn

nPn

Half-range sine series coefficient

1

10010

sin)0,(n

n

xnPxT

dxxn

dxL

xnxf

LP

L

n 10

00 10

sin10010

2sin)(

2

400

11200

0)11(2

200

3

400)11(

3

200

3

2

1

Pnn

So in order for the boundary conditions for T(x,0) = 100 to be satisfied, we must take the following values of Pn in the sum.

oddn

nnn

xn

n

xnPxT

11 10sin

400

10sin)0,(



Step 10: Finding the full solution for all x and y

oddn

n

yn

n

yn

n

xne

n

xnePyxT

1

10

1

10

10sin

400

10sin),(

oddn

n

xn

nxT

1 10sin

400)0,(

So in order for the boundary conditions for T(x,0) = 100 to be satisfied:

Remember from step 8 the general solution of our equation is the sum of all special solutions:

1

10

10sin),(

n

yn

n

xnePyxT

So the full solution can be written as:

Let’s check that this fulfils all boundary conditions



oddn

n

yn xne

nyxT

1

10

10sin

400),(

T(x, 0) = 100 :

T(x, ∞) = 0 :

T(0, y) = 0 :

T(L, y) = 0 :

oddn

n

xne

nxT

1

0

10sin

400)0,(

This is Fourier series for f(x) = 100 we just found

010

sin400

),(1

oddn

n

xne

nxT

00sin400

),0(1

10

oddn

n

yn

en

yT

010

10sin

400),(

1

10

oddn

n

yn ne

nyLT

Just think about how T(x, y) would look if you were to plot it on a graph

Revision for the examRevision for the exam

http://www.shef.ac.uk/physics/exampapers/2007-08/phy226-07-08.pdf

Above is a sample exam paper for this course

There are 5 questions. You have to answer Q1 but then choose any 2 others

I’ll put previous years maths question papers up on Phils Problems very soon

Q1: Basic questions to test elementary concepts. Looking at previous years you can expect complex number manipulation, basic integration, solving ODEs, applying boundary conditions, plotting functions. Easy stuff.

Q2-5: More detailed questions usually centred about specific topics: InhomoODE, damped SHM equation, Fourier series, Half range Fourier series, Fourier transforms, convolution, partial differential equation solving (including applying an initial condition to general solution for a specific case), Cartesian 3D systems, Spherical polar 3D systems, Spherical harmonics

The notes are the source of examinable material – NOT the lecture presentations

I wont be asking specific questions about Quantum mechanics outside of the notes

http://www.shef.ac.uk/physics/exampapers/2007-08/phy226-07-08.pdf

Revision for the examRevision for the examThe notes are the source of examinable material – NOT the lecture presentations

Things to do now

Read through the notes using the lecture presentations to help where required.

At the end of each section in the notes try Phils problem questions, then try the tutorial questions, then look at your problem and homework questions.

If you can do these questions (they’re fun) then you’re in excellent shape for getting over 80% in the exam.

Any problems – see me in my office or email me

Same applies over holidays. I’ll be in the department most days but email a question or tell me you want to meet up and I’ll make sure I’m in.

Look at the past exam papers for the style of questions and the depth to which you need to know stuff.

You’ll have the standard maths formulae and physical constants sheets (I’ll put a copy of it up on Phils Problems so you are sure what’s on it). At present you don’t need to know any equations e.g. Fourier series or transforms, wave equation, polars.

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Lecture 16 Solving the Laplace equation in 2-D Remember Phils Problems and your notes = everything Only 6 lectures