The general linear 2ed order PDE in two variables x, y.

02 GFuEuDuCuBuAu yxyyxyxx

0),,,,( wwww

Chapter 2:Linear Second-Order Equations Sec 2.2,2.3,2.4:Canonical Form

hyperbolic

parabolic

elliptic

0),,,,( wwww

0),,,,( wwwbwaw

0),,,,( wwwww

1

2

3

Can

on

ical Fo

rm

The general linear 2ed order PDE in two variables x, y.

02 GFuEuDuCuBuAu yxyyxyxx

0),,,,( wwww

Chapter 2:Linear Second-Order Equations

hyperbolic1Use the transformation: η(x,y)ηyx ),,(

02 gfwewdwcwbwaw

22 2 yyxx CBAa 22 2 yyxx CBAc yyyxyxxx CBAb )(

0a 02 22 yyxx CBA 02

2

A

C

A

B

y

x

y

x

A

ACBB

y

x

2

A

ACBB

dx

dy

y

x

2

Sec 2.2,2.3,2.4:Canonical Form

Chapter 2:Linear Second-Order Equations

02 gfwewdwcwbwaw

22 2 yyxx CBAa 22 2 yyxx CBAc yyyxyxxx CBAb )(

0a 02 22 yyxx CBA 02

2

A

C

A

B

y

x

y

x

A

ACBB

y

x

2

A

ACBB

dx

dy

y

x

2

0c02

2

A

C

A

B

y

x

y

x

A

ACBB

y

x

2

A

ACBB

dx

dy

y

x

2

02 22 yyxx CBA

Sec 2.2,2.3,2.4:Canonical Form

The general linear 2ed order PDE in two variables x, y.

02 GFuEuDuCuBuAu yxyyxyxx

0),,,,( wwww

Chapter 2:Linear Second-Order Equations

hyperbolic1Step

1Form characterestic equation:

A

ACBB

dx

dy

y

x

2

A

ACBB

dx

dy

y

x

2

kyx ),( Kyx ),(

Step2

Solve these equations

:Example0)sin()(sin)cos(2 2 yyyxyxx uxuxuxu 0w

Sec 2.2,2.3,2.4:Canonical Form

The general linear 2ed order PDE in two variables x, y.

02 GFuEuDuCuBuAu yxyyxyxx

Chapter 2:Linear Second-Order Equations

2

Step1

Form characterestic equation:A

B

dx

dy

kyx ),(Step2

Solve this equation

:Example 044 yyxyxx uuu 0w

parabolic 0),,,,( wwww

0,0 CA02 ACB

A

ACBB

dx

dy

y

x

2

A

ACBB

dx

dy

y

x

2

A

B

dx

dy

Choose any cont func with cont 1st 2ed partial derivatives with 0J

Sec 2.2,2.3,2.4:Canonical Form

The general linear 2ed order PDE in two variables x, y.

02 GFuEuDuCuBuAu yxyyxyxx

Chapter 2:Linear Second-Order Equations

elliptic0),,,,( wwwbwaw

3 0),,,,( wwwww

02 gfwewdwcwbwaw

22 2 yyxx CBAa 22 2 yyxx CBAc

)()(0 yxyyxx CBBA 0b

)()( yxyyxx CBBAb

yx

yx

y

x

BA

CB

dx

dy

kyx ),(0),,,,( wwwbwaw

:Example 0432 uuuu yyxyxx 0463 www

Sec 2.2,2.3,2.4:Canonical Form

linear 2ed order PDE in two variables x, y.

Chapter 2:Linear Second-Order Equations

elliptic3 0),,,,( wwwww

02 gfwewdwcwbwaw

22 2 yyxx CBAa 22 2 yyxx CBAc

ca

)()( yxyyxx CBBAb

0)(2 GFuEuDuCuuBAu yxyyxyxx

22yx CA 22

yx CA

0b 0 yyxx CA yx CA xy AC

dyC

Adx

A

Ckyx x

yx

yx

y ),(

),( 00

),(

In which k is a constant and (x0,y0) some chosen point. Such function will satisfy (*) exactly when the line integral is independent of path.

Sec 2.2,2.3,2.4:Canonical Form

Line integrals

a

bdxxf )( Integration of a function defined over an

interval [a,b]

C xf )( Integration of a function defined along a curve C

Example 1: Evaluation of the Line Integral

Evaluate .along the the quarter-circle C

(4,0)

(0,4)

C dxxy2

Steps

1) Find the parametric equations of C :

2) let

3) Replace

bta g(t),y ),( tfxdttfdx )('

)( and )(by and tgtfyx

Under what condition the integral

is independent of the path C

QdyPdx

C

QdyPdxis independent of the path

QdyPdx is an exact differential

x

Q

y

P

linear 2ed order PDE in two variables x, y.

Chapter 2:Linear Second-Order Equations

elliptic3 0),,,,( wwwww

0)(2 GFuEuDuCuuBAu yxyyxyxx

Step1

Solve for :

y

y

x

x A

C

C

A

Step2

Write:

),( yx

dyC

Adx

A

Ckyx x

yx

yx

y ),(

),( 00

),(

:Example 0432 uuuu yyxyxx 0463 www

09

4 www

Sec 2.2,2.3,2.4:Canonical Form

Problems for section 2.1 through 2.4 (page 36-37)In each problem (a) classify (b) sketch the characteristics (c) find canonical

Chapter 2:Linear Second-Order Equations

028 )1 yxyyxyxx yuxuuuu

Sec 2.2,2.3,2.4:Canonical Form

024 )2 uxyuuuuu yxyyxyxx

023 )3 yxyyxyxx uyuuuu

>> ezplot('y-2*x+1')>> hold on>> ezplot('y-2*x+4')>> ezplot('y-2*x-6')>>

kxExample:y- 2

MATLAB