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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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