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Chapter 8

Partial Differential equationsPDEs

MA1506

Mathematics II

Chew T S MA1506-14 Chapter 8 1

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In this Chapter, we will study

(1) Method of separation of variables (MSV)

(2) Solution of wave equation (Two versions)

(b) dAlemberts method

1

( , ) sin cosnn

n n y x t A x c t

L L

=

=

0

2( )sin

L

n

n A f x x dx L L

= where

(a)By MSV

[ ]1( , ) ( ) ( )2

y x t f x ct f x ct = + +

(3) Solution of Heat equation

22

1( , ) sin

nc t

Ln

n

nu x t A x e L

=

=

0

2( )sin

L

n

n A f x x dx

L L = where

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Notation

( , )u x y x xudenoted by

( , )u x y y

yudenoted by2

2

( , )u x y

x

xxudenoted by

2 ( , )u x y y x

xyudenoted by


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Partial Differential Equations

0 xxu u = 0 x yu xu+ =

y xxu u= yy xxu u=

0 xx yyu u+ =Equations above involving partial derivativesare called partial differential equations


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By method of separation of variables, weassume that solution ( , )u x y can be written as

( , ) ( ) ( )u x y X x Y y=Hence' ( ) ( ) xu X x Y y=

'( ) ( ) yu X x Y y=

' '( ) ( ) xyu X x Y y=

'' ( ) ( ) xxu X x Y y= ''( ) ( ) yyu X x Y y=

8.1 SV for PDE

8.1 Separation of variables for PDE N L pp 7-12

In this section, , we study one simple case

by method of separation of variables .


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Example

solve 0 x yu xu+ =SolutionLet ( , ) ( ) ( )u x y X x Y y=

Then ' '( ) ( ) ( ) ( ) 0 X x Y y xX x Y y+ =Hence

' '1 ( ) ( )( ) ( )

X x Y y x X x Y y

= holds for any x and y

Use method of S V to8.1 SV for PDE


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Therefore'' '

00

( )1 ( ) ( )( ) ( ) ( )

Y y X x Y y x X x Y y Y y= = for any fixed 0 yThus

' '

1 ( ) ( ) constant( ) ( )

X x Y y k x X x Y y

= = =

holds for any x and y

So ' X kxX = 'Y kY =

8.1 SV for PDE

for any fixed k


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Solve the above two 1 st order ODE, get

2 /2( ) kx X x Ae= ( ) kyY y Be=Hence

( , ) ( ) ( )u x y X x Y y=2 2/2 /2kx ky kx ky

ABe e Ce e

= =

8.1 SV for PDE


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USEFUL FACTS for studying wave and heat equationsUSEFUL FACTS

Sturm-Liouville equationconsider ( ) + ( ) = 0 with initial conditions (0) = 0, (0) = 0 Then we know that zero is a solution.Zero is the only solution .

Now, if we replace the above initial conditions byboundary conditions

Then we have Sturm-Liouville equation

( ) + ( ) = 0 (0) = 0,

( ) = 0

(0) = 0, ( ) = 0 Chew T S MA1506-14 Chapter 8 9

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We shall write down solutions ofSturm-Liouville equation without proofs.

Case 1 < 0 Zero is the only solution

Case 2 = 0

Zero is the only solution

Case 3 > 0

There are two subcasesSubcase 1 When 2, = 1,2, . . .

Zero is the only solution

USEFUL FACTS


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Subcase 2 = 2, = 1,2, . . .

= 2, = 1,2, . . . called eigenvalueFor eachthere exists nonzero solution

called eigenfunction

USEFUL FACTS

( ) = sin


This case is very important

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1. if

n m

n m=1

sin sin L

L

n m x x dx

L L L

=

0

1

if

where m, n are positive integers

USEFUL FACTSSome special integrals

1cos cos

L

L

n m x x dx

L L L

= 2.0

1

if

if

n mn m=

1sin cos 0

L

L

n m x x dx

L L L

= 3.for any positive

integers m, n

where m, n are positive integers


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You dont have to spend time on working out

Examples 1, 2,3.Just need to know the results.

Use 1, 2 and 3,

21 (sin 4sin 2 23cos3 ) x x x dx

+ +

2 2 21 4 23= + +

we get

expand

USEFUL FACTS


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1. Odd function f(x) : symmetric w.r.t. origin

2. Even function f(x): symmetric w.r.t. y-axis

USEFUL FACTSOdd and even functions

( ) = ( )

( ) = ( ) Chew T S MA1506-14 Chapter 8 14

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3. sinx, sin x are odd, where is any real number

4. cosx, cos x are even, where is any real number

USEFUL FACTS


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5. oddeven =odd functionoddodd =even functioneveneven =even function

6.

( ) 0

L

L odd dx =sin( ) 0

L

L x dx

=

sin cos 0 L L

n m x x dx L L

= for any integers n, m

odd even

means

multiplication

USEFUL FACTS


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7.0

( ) 2 ( ) L L

L

even dx even dx

=

8. Let f be odd.

( )sin = 2 ( )sin 0 odd odd

USEFUL FACTS


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sin( ) =sin( )

2 cos( )

2sin( ) = 2 sin( )2 + 2 22)cos(3

9.

11.

sin( ) = 1 cos 10.

USEFUL FACTS


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2 1

cos (1 cos 2 )2 x x= +

2 1sin (1 cos 2 )

2 x x=

1sin cos sin 2

2

x x x=

1sin sin [cos( ) cos( )]

2 x y x y x y= +

1cos cos [cos( ) cos( )]

2 x y x y x y= + +

1sin cos [sin( ) sin( )]2

x y x y x y= + +

12.


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14.

1sin(2 1) ( 1)2

nn + =

sin 0n =1,2,3,...n =

1,2,3,...n =

USEFUL FACTS

sin(2

) =

sin(

2

) = Chew T S MA1506-14 Chapter 8 21

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8.2 Wave Equation8.2 wave equation

Elastic string of length L

tightly stretched

The string is set in motion.It vibrates in vertical plane.

Animation slide


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Let be vertical displacement at pt , at time( , ) y x t x t Let c be constant velocity of propagation

of wave along string

8.2 wave equation

Animation slide

x


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Now we shall use the method of S Vto solve the above wave equation

Solution

By MSV, let ( , ) ( ) ( ) y x t u x v t =

Then ( , ) ( ) ( ) xx y x t u x v t =( , ) ( ) ( )tt y x t u x v t =

Hence2

( , ) ( , ) xx tt c y x t y x t =implies 2 ( ) ( ) ( ) ( )c u x v t u x v t =

8.2 wave equation


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Therefore2

( ) 1 ( )( ) ( )

u x v t u x c v t

= holds for

ALL x, and t

So 2( ) 1 ( )( ) ( )

u x v t u x c v t

= = for any fixed

Hence ( ) ( ) 0u x u x + =2( ) ( ) 0v t c v t + =

For convenience, we use -

8.2 wave equation


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Now we shall solve

( ) ( ) 0u x u x + =with boundary conditions:

(0, ) 0, ( , ) 0, for all 0 y t y L t t = = >i.e., (0) ( ) 0, ( ) ( ) 0, for all 0u v t u L v t t = = >

i.e., (0) 0, ( ) 0u u L= =

8.2 wave equation


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So now we solve

( ) ( ) 0u x u x + =with boundary conditions: (0) 0, ( ) 0u u L= =Recall the above equation is

Sturm-Liouville equation.

= 2, = 1,2, . . . Whenthere exists nonzero solution

( ) = sin

8.2 wave equation


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now look at corresponding

For

2( ) ( ) 0v t c v t + =

The general solution is

( ) cos sinn nn n

v t A c t B c t L L = +

However, from one of the initial conditions( ,0) 0t y x = i.e., (0) 0v = we have 0n B =

= 2, = 1,2, . . . 8.2 wave equation


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So for each n,

sin cosnn n

A x c t L L

is a solution of the given PDE

with boundary conditions and initial condition( ,0) 0t y x =

1sin cos

m

nn

n n A x c t L L

=

Hence

is again a solution

8.2 wave equation


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1

( , ) sin cosnn

n n y x t A x c t

L L

=

=

In fact

is the general solution of the given PDE

Now we shall use the 2nd

initial condition

to find the value ofn

A( ,0) ( ) y x f x=

8.2 wave equation


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1 1

( ) ( ,0) sin cos 0 sinn nn n

n n n f x y x A x c A x

L L L

= =

= = =

1

( ) sinnn

n f x A x

L

=

=

How to find n A

where 0 x L<

( ,0) , ( ,0) 0, where 0t y x x y x x = = <

(0, ) 0, ( , ) 0, for all 0 y t y t t = = >( ,0) , ( ,0) 0, where 0t y x x y x x = = <

(0, ) 0, ( , ) 0, for all 0 y t y t t = = >

( ,0) sin(10 ), ( ,0) 0, where 0t y x x y x x = = <

(0, ) 0, ( , ) 0, for all 0 y t y L t t = = >

( ,0) ( ), ( ,0) 0, where 0t y x f x y x x L= = <


8 3 Heat Equation

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with boundary conditions:

(0, ) 0, ( , ) 0, for any 0u t u L t t = = >

0 0Temperatures are always ZERO at both end points

with initial condition:

( ,0) ( ), where 0u x f x x L=

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ChewMA1506-14 Ch8