- Homogeneous & nonhomogeneous Ex.1) Important linear 2

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• An equation involving partial derivatives of an unknown function of two more independent variables

 PDE

Classification of PDES

• Linear and nonlinear PDEs

Linear PDE: There is no product of the dependent variable and/or product of its derivatives within the equation

Nonlinear PDE: The equation contains a product of the dependent variable and/or a product of the derivatives

• Classification based on characteristics (paths of propagation of physical disturbances) (I) First-order PDE: Almost all first-order PDEs have real characteristics, and therefore behave much like

hyperbolic equations of second order.

(II) Second-order PDE: A second-order PDE in two dependent variables, x and y, may be expressed in a general form as

Chap. 11. PARTIAL DIFFERENTIAL EQUATIONS

0 G y F

x E y D

y C B x

A x ₂

2 2

= + φ

∂ + φ + ∂

∂ φ + ∂

∂

∂ φ + ∂

∂ φ

∂

) nonlinear (

y x xu u x

), u nonlinear (

y x x 6 u x

u

) linear ,

order rd

3 ( y 5 u y 8

x u y x ), u linear ,

order nd

2 ( 1 y u

xy u x 2

u

2 2 2

3 3 2 2

2 2 2

3 2

2 2

2

∂ = + ∂

∂

= ∂

∂

∂ + ∂



 





∂

−

=

∂ + + ∂

∂

− ∂

=

∂ + + ∂

∂

(2)

• The equation is classified according to the expression (B²-4AC) as follows:

(B²-4AC) < 0  Elliptic equation

= 0  Parabolic equation

> 0  Hyperbolic equation (a) Elliptic equations

No real characteristic lines exist A disturbance propagates in all directions Domain of solution is a closed region

Boundary conditions must be specified on the boundaries of the domain (b) Parabolic equations

Only one characteristic line exists

A disturbance propagates along the characteristic line Domain of solution is an open region

An initial condition and two boundary conditions are required (c) Hyperbolic equations

Two characteristic lines exist

A disturbance propagates along the characteristic lines Domain of solution is an open region

Two initial conditions along with two boundary conditions are required

• Boundary conditions

(a) Dirichlet B.C. (=Essential B.C.): The value of the dependent variable along the boundary is specified (b) Neumann B.C (=Natural B.C.): The normal gradient of the dependent variable along with the

boundary is specified

(c) Mixed B.C. (Robbin B.C.): A combination of the Dirichlet and the Neumann type B.C.’s is specified

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•11.1. Basic Concepts - Linear & nonlinear

- Homogeneous & nonhomogeneous Ex.1) Important linear 2

^nd

-order PDEs

Theorem 1: Superposition or linearity principle

u

₁

, u

₂

: solutions of a linear homogeneous PDE in R, then u = c

₁

u

₁

+ c

₁

u

₂

: also solution of that equation in R

Ex. 1) A solution u(x,y) of PDE u

_xx

-u=0 u(x,y) = A(y)e

^x

+ B(y)e

^-x

Ex. 2) PDE u

_xy

= -u

_x

u

_x

= p  p

_y

=-p: p=c(x)e

^-y

 u(x,y) = f(x)e

^-y

+ g(y) where

z 0 u y

u x

u y

u x

c u t

u

) y , x ( y f

u x

0 u y

u x

u

x c u t u x

c u t

u

2 2 2

2 2

2 2 2

2

2 2 2

2 2

2

2 2 2 2

2 2 2

2

∂ = + ∂

∂ + ∂

∂

 ∂



 





∂ + ∂

∂

= ∂

∂

∂ = + ∂

∂

= ∂

∂ + ∂

∂

= ∂

∂

= ∂

∂

1D wave Eqn. 1D heat Eqn.

2D Laplace Eqn. 2D Poisson Eqn.

2D wave Eqn. 3D Laplace Eqn.



= c(x)dx )

x ( f

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11.2. Modeling: Vibrating String, Wave Equation

- Equation governing small transverse vibration of an elastic string Find the deflection u(x,t):

Assumptions: - Constant mass/unit length, perfect elastic, no resistance to bending - Negligible gravitational force

- Small transverse motion in vertical plane  vertical movement Derivation of the PDE from forces

In horizontal direction:

In vertical direction:

const T

cos T

cos

T

₁

α =

₂

β = =

2 2 1

2

t

x u sin

T sin

T ∂

Δ ∂ ρ

= α

− β

2 2

1 1 2

2

t u T

tan x cos tan

T sin T cos

T sin T

∂

∂ Δ

= ρ α

− β α =

− α β β

x

u

Δ +

 

 





∂

x

u  

 





∂

 

 





= ρ

∂

= ∂

∂

∂ T

x c c u t

u

₂

2 2 2 2

2

1D Wave Equation:

2nd-order Hyperbolic PDE

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11.3. Separation of Variables: Use of Fourier Series

- 1D Wave equation:

2 B.C.’s: u(0,t) = u(L,t) = 0 for all t 2 I.C.’s : u(x,0) = f(x),

Solving Steps: - Method of separation variables leading to two ODEs.

- Solutions of two eqns. satisfying B.C’s

- Final solution of wave eqn. satisfying I.C’s, using Fourier series First Step: Two ODEs using method of separation variables

- u(x,t) = F(x)G(t)

Second Step: Satisfying the B.C.’s

- u(0,t) = F(0)G(t) = 0 Case 1) G = 0  u = 0 ( ∴G≠0) Case 2) k=0  F=0 (∴k≠0) u(L,t) = F(L)G(t) = 0 Case 3) k= μ

²

 F=0 ∴ k = -p

²

(negative)

= 

=

= k const

F '' F G c

G

2



 

 





= ρ

∂

= ∂

∂

∂ T

x c c u t

u

₂

2 2 2 2

2

) x ( t g

u

0 t

∂ =

∂

=

Initial deflection Initial velocity

(derivative w.r.t t) (derivative w.r.t x)

0 kG c G

0 kF '' F

2

=

−

=

−

(linear system) 

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) , 2 , 1 n ( 1 B for L x

sin n )

x ( F

0 ) L ( F ) 0 ( F : s ' C . B apply px

sin B px cos A ) x ( F 0

F p '' F

n 2

=  π =

 =

=

← +

 =

= +

Solving F(x):

Solving G(t):

U

_n

: harmonic motion with frequency λ

n

/2 π=cn/2L (n

^th

normal mode) n

^th

normal mode has n-1 nodes

Tuning controlled by tension T (or c

²

=T/ ρ)

t sin B t cos B

) t ( L G

0 cn G

G

²_n

 

_n

=

_n

λ

_n

+

^*_n

λ

_n



 



 λ = π

= λ

 +

( ^B ^cos ^t ^B ^sin ^t ) ^sin ⁿ _L ^x ⁽ ⁿ ¹ ^, ² ^, ⁾

) t , x (

u

_n _n _n ^*_n _n

 = 



 



 π

λ +

λ

=

(Eigenfunctions or characteristic functions) ( λ

n

: eigenvalues or characteristic values) (p=n π/L)

Standing wave solutions

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Third Step: Solution to the Entire Problem. Fourier Series - Sum of many solutions u

_n

satisfying I.C.’s:

- Satisfying I.C.1: initial displacement (u(x,0) = f(x))

- Satisfying I.C.2: initial velocity

- Solution (I): for the simple case of g(x) = 0

 ^π

=

^L

n 0

dx

L x sin n

) x ( L f

B 2

( )



^∞

=

 

 



 π

λ +

λ

=

1 n

n

* n n

n

x

L sin n t

sin B

t cos B

) t , x ( u

(Fourier sine series)

 

 



 =

∂

=

) x ( t g

u

0 t

) x ( f L x

sin n B

) 0 , x ( u

1 n

n

π =

= 

^∞

=

) x ( g L x

sin n t B

u

1 n

n

* n 0

t

π = λ

∂ =

∂ 

^∞

= =

^λ

n

⁼ 

0^L

^π

*

n

dx

L x sin n

) x ( L g

B 2

[ ^f ⁽ ^x ^ct ⁾ ^f ⁽ ^x ^ct ⁾ ]

2 ) 1

ct x L ( sin n 2 B

) 1 ct x L ( sin n 2 B

1 L x cn

L sin n t cos B

) t , x ( u

*

* 1

n

n 1

n

n 1

n

n n

+ +

−

 =

 

 

 π +

 +

 

 

 π −

=

 

 



 λ = π λ π

=



∞

=

∞

=

∞

=

(f: odd periodic extension* of f with period 2L)

(Fourier sine series)

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Odd periodic extension of f(x) Physical Interpretation of the Solution

f*(x - ct): a wave traveling to the right as t increases constant along each line x - ct

f*(x + ct): a wave traveling to the left as t increases constant along each line x + ct

c: wave velocity

 u(x,t): superposition of above two waves

Ex. 1) Vibrating string

if the initial deflection is triangular.

See Ex. 3 in Sec. 10.4

L/3 L

0 x

t

characteristic lines

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Solution (II): for the case of f(x)=0

Solution (III): for the general case of f(x) ≠0 and g(x)≠0

Exercise: Find the solution of the wave equation with following B.C.’s & I.C.’s u

_tt

= c

²

u

_xx

B.C.’s: u

_x

(0,t) = u

_x

( π,t) = 0 for all t I.C.’s: u(x,0) = f(x), u

_t

(x,0) = g(x) (use Fourier cosine series)

 

 



 =  λ π = − 

^∞

π  =

=

∞

= n 1

* n 1

n

*

n

x G ' ( x ) g ( x )

L cos n c B )

x ( G , L x sin n B

) x ( g

[ ^G ⁽ ^x ^ct ⁾ ^G ⁽ ^x ^ct ⁾ ]

c 2 ) 1

ct x L ( cos n 2 B

) 1 ct x L ( cos n 2 B

1 L x cn

L sin n t sin B )

t , x ( u

1 n

* n 1

n

* n

n 1

n

* n

−

− +

 =

 

 

 π +

 −

 

 

 π −

=

 

 



 λ = π λ π

=



∞

=

∞

=

∞

=

[ ] [ ]

) ct x ( Q ) ct x ( P

) ct x ( G ) ct x ( c G 2 ) 1 ct x ( f ) ct x ( 2 f ) 1 t , x ( u

− +

+

=

−

− +

+

− +

+

=

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11.4. D’Alembert’s Solution of the Wave Equation - Other method to obtain the solution of the wave eqn.