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Introduction to Partial Introduction to Partial Differential EquationsDifferential Equations

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What is a Partial What is a Partial Differential Equation ?Differential Equation ? Ordinary Differential Equations have only one independent

variable

Partial Differential Equations have more than one independent variable

subject to certain conditions: where u is the dependent variable, and x and y are the independent variables.

 

5)0(,353 2 yeydx

dy x

222

2

2

2

3 yxy

u

x

u

Example of an Ordinary Example of an Ordinary Differential EquationDifferential Equation

Assumption: Ball is a lumped system.Number of Independent variables:

One (t)

Hot Water

Spherical Ball

dt

dmChA a

Example of an Partial Example of an Partial Differential EquationDifferential Equation

Assumption: Ball is not a lumped system.Number of Independent variables: Four

(r,θ,φ,t)

Hot Water

Spherical Ball

aTrTtt

TC

T

r

kT

r

k

r

Tr

rr

k

)0,,,(,0,sin

sinsin 2

2

2222

2

Classification of 2Classification of 2ndnd Order Order Linear PDE’sLinear PDE’s

where are functions of ,and is a function of

02

22

2

2

Dy

uC

yx

uB

x

uA

CBA and,,yx and D

, , and , .u u

x y ux y

Classification of 2Classification of 2ndnd Order Order Linear PDE’sLinear PDE’s

can be: Elliptic Parabolic Hyperbolic

02

22

2

2

Dy

uC

yx

uB

x

uA

Classification of 2Classification of 2ndnd Order Order Linear PDE’s: Linear PDE’s: EllipticElliptic

02

22

2

2

Dy

uC

yx

uB

x

uA

042 ACBIf ,then equation is elliptic.

Classification of 2Classification of 2ndnd Order Order Linear PDE’s: Linear PDE’s: EllipticElliptic

02

22

2

2

Dy

uC

yx

uB

x

uA

Example:

where, giving

therefore the equation is elliptic.

02

2

2

2

y

T

x

T

1,0,1 CBA

04)1)(1(4042 ACB

Classification of 2Classification of 2ndnd Order Order Linear PDE’s: Linear PDE’s: ParabolicParabolic

02

22

2

2

Dy

uC

yx

uB

x

uA

2 4 0B AC If ,then the equation is parabolic.

Classification of 2Classification of 2ndnd Order Order Linear PDE’s: Linear PDE’s: ParabolicParabolic

02

22

2

2

Dy

uC

yx

uB

x

uA

Example:

where, giving

therefore the equation is parabolic.

2

2

x

Tk

t

T

0,0, CBkA

ACB 42 ))(0(40 k 0

Classification of 2Classification of 2ndnd Order Order Linear PDE’s: Linear PDE’s: HyperbolicHyperbolic

02

22

2

2

Dy

uC

yx

uB

x

uA

2 4 0B AC If ,then the equation is hyperbolic.

Classification of 2Classification of 2ndnd Order Order Linear PDE’s: Linear PDE’s: HyperbolicHyperbolic

02

22

2

2

Dy

uC

yx

uB

x

uA

Example:

where, giving

therefore the equation is hyperbolic.

2

2

22

2 1

t

y

cx

y

2

1,0,1

cCBA

)1

)(1(4042

2

cACB

0

42

c

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