# Classification of PDEs

Document Sample

```					   Chapter 2

Partial Differential
Equations (PDEs)

1
Classification of PDEs
Elliptic Type
Parabolic Type
Hyperbolic Type
different mathematical and physical behaviors

Fluid flow equations
Time : first-derivative (second-derivative for wave eqn)
Space: first- and second-derivatives
System of coupled equations for several variables
2
First-Order PDEs
First-order linear wave equation (advection eq.)
u     u
c    0
t    x
Propagation of wave with speed c
Advection of passive scalar with speed c
First-order nonlinear wave equation (inviscid
Burgers’s equation)
u    u
u    0
t    x                    3
Second-Order PDEs

T    T     2T
u    
t    x    x 2

Burger’s equation (nonlinear)

u    u    u  2
u     2
t    x   x
4
Other Common PDEs
Korteweg-de Vries (KdV) equation

u    u  u 3             Nonlinear
u    3 0             dispersive wave
t    x x
Laplace and Poisson’s equations

 2u  2u             f  0 : Laplace

 2 u  2  2  f ( x, y)   
x   y               f  0 : Poisson

5
Other Common PDEs
Helmholtz equation
 u  u
2       2             Time-dependent harmonic waves
 2 k u0
2

x 2
y               Propagation of acoustic waves

Tricomi equation

 u  u
2       2    y  0 : elliptic

y 2  2 0                                Mixed-type
x  y            y  0 : hyperbolic

6
Other Common PDEs
Wave equation

 u
2
2  u
2
c        0
t 2
x 2

Fourier equation (Heat equation)

T    T   2
 2
t   x
7
Navier-Stokes Equations
Navier-Stokes equation
Vorticity / stream function formulation

 2  


                
2  2
    u     v     2  2 
 x    
 t
        x    y       y 

8
Navier-Stokes Equations
Navier-Stokes equation
Primitive variables
  u v
       0
  x y

 u   u    u    1 p     2u 2u 
 u      v            2  2 
 x
 t   x    y     x          y   

 v   v    v    1 p      2v  2v 
 u      v            2  2 
 x
 t   x    y     y          y      9
RANS Equations: Turbulent Flows
Reynolds-Averaged Navier-Stokes equation
 U V
         0
  x y
                                  2 U  2 U   uu  uv
 U     U        U    1 P
    U      V               2 
 x          
2 

 t     x        y     x           y  x       y
                                  2V  2V     uv  vv
 V  U  V  V   V

1 P
  2  2       
 t     x        y     y     x   y      x  y
                                             
 k     k     k               2k  2k 
 U        V      (   t ) 2  2   G  
 x
 t     x     y                   y   

   U   V   (   )         (C G  C  )
2    2

t 
 2          
2 
 t     x     y               x   y  k
1    2
                                                            10
2.1 Background
Linear second-order PDE in two independent
variables (x,y), (x,t), etc.
2u    2u   2u    u    u
A 2 B      C 2  D    E     Fu  G  0
x    xy   y     x    y

A, B, C, …, G are constant coefficients (may be
generalized)
 B 2  4 AC  0 : elliptic
 2
Classification  B  4 AC  0 : parabolic
(discriminant)  B 2  4 AC  0 : hyperbolic

11
Classification of PDEs
Au xx  Bu xy  Cu yy  Du x  Eu y  Fu  G  0
 The classification depends only on the highest-order
derivatives (independent of D, E, F, G)
 For nonlinear problems [A,B,C = f(x,y,u)], the discriminant
can still be used for classification of local flows
 The equation types are coordinate invariant, i.e., coordinate
transformation will not change the type of equations
 Physical processes are independent of coordinates
 Introduction of simpler flow categories (approximations)
may change the equation type
Boundary-layer : elliptic  parabolic            12
Classification of PDEs
 General form of second-order PDEs (2 variables)

Au xx  Bu xy  Cu yy  Du x  Eu y  Fu  G  0

(1) Hyperbolic PDEs (Propagation)

      
 t  u x  0

(first - order)
 2
    c 2    0 (second  order)
2
Wave equation
 t 2
           x 2

13
Classification of PDEs
 General form of second-order PDEs (2 variables)

Au xx  Bu xy  Cu yy  Du x  Eu y  Fu  G  0

(2) Parabolic PDEs (Time- or space-marching)

        2
Burger’s equation   u     ν 2
 t  x    x             Diffusion /
                          dispersion

Fourier equation      2
 t
     x 2
14
Classification of PDEs
 General form of second-order PDEs (2 variables)

Au xx  Bu xy  Cu yy  Du x  Eu y  Fu  G  0

(3) Elliptic PDEs (Diffusion, equilibrium problems)
  2  2
 2  2 0
 x   y
Laplace equation
  2  2

Possion’s equation  2  2  f ( x , y )
 x   y
  2  2
Helmholtz equation       2  c 2  0
 x 2 y
                              15
Classification of PDEs
 General form of second-order PDEs (2 variables)

Au xx  Bu xy  Cu yy  Du x  Eu y  Fu  G  0

(4) Mixed-type PDEs

 2  2   M  1 : subsonic
(1  M ) 2  2  0 
2

s   n     M  1 : supersonic

16
Classification of PDEs
 General form of second-order PDEs (2 variables)

Au xx  Bu xy  Cu yy  Du x  Eu y  Fu  G  0

(5) System of Coupled PDEs
  u v
       0
Navier-Stokes  x y
 u
      u    u    1 p      2u  2u 
 u      v            2  2 
 x
Equations      t   x    y     x          y   
 v   v    v    1 p      2v  2v 
 u      v            2  2 
 x
 t
      x    y     y          y  
17
General Curvilinear Coordinates
Divergent Channel
Trapezoidal section

Curved Channel

General Curvilinear Coordinates
Coordinate Transformation
Physical plane  Transformed plane


Physical Plane
y


x

Transformed Plane
   ( x , y )    x  x ( , )
   ( x , y )  
                   y  y( , )

20
Coordinate Transformation
Au xx  Bu xy  Cu yy  Du x  Eu y  Fu  G  0
Physical plane Transformed plane
   ( x , y )
u( x , y )  u( , ) 
   ( x , y )
Chain rule of transformation (first-derivatives)
u x  u  x  u x u x   x  x  u 
u  u   u   u                  u 
 y      y       y   y   y  y    
 
 
Jacobian of
22
transformation
Coordinate Transformation
Chain rule of transformation (second-derivatives)
u xx  ( u x ) x  ( u  x  u x ) x
 ( u ) x  x  u  xx  ( u ) x  x  u xx
 [(u )  x  ( u )  x ] x  u  xx
 [(u )  x  ( u )  x ] x  u xx
 u   2 u  x x  u   u  xx  u xx
2
x
2
x

Similarly
u yy  u  y  2u  y y  u  y  u  yy  u yy
2                       2
23
Coordinate Transformation
Chain rule of transformation (mixed derivatives)
u xy  ( u x ) y  ( u ) y  x  u  xy  ( u ) y  x  u xy
 u  x y  u ( x y   y x )  u  x y  u  xy  u xy
Therefore,
Au xx  Bu xy  Cu yy  H  0
 ( A x  B x y  C y )u  ( A x  B x y  C y )u 
2                2             2                2

 [ 2 A x x  B( x y   y x )  2C y y ]u
 [( A xx  B xy  C yy )u  ( A xx  B xy  C yy )u  H ]
 Au  B u  C u   H   0                               24
Classification of PDEs
Discriminant in the transformed plane
( B ) 2  4 AC   [ 2 A x x  B( x y   y x )  2C y y ]2
 4( A x  B x y  C y )( A x  B x y  C y )
2                2       2                2

 ( x y   y x ) 2 ( B 2  4 AC )
 J 2 ( B 2  4 AC )
 J 0: one-to-one mapping
 Same classification in the transformed plane
 The type of PDE determines the nature of physical problems
 The nature of physical process is independent of the coordinate
system we choose to represent it
 Coordinate invariant – invariant under coordinate transformation
25
Equilibrium Problems
Boundary Value Problems
Elliptic PDE
“Jury” Problem (every juror must agree on the same
verdict)
The entire solution is passed on a jury requiring
satisfaction of all boundary conditions and all internal
requirements

26
Physical Diffusion
Irreversible processes
Move towards the most probable state

High concentration 
Fluid A    Fluid B
low concentration

The probability that a fluid particle A stays in
compartment A is ½. If there are 100 fluid particles,
then the probability that the two fluids remain
separate is (½)100 = 0.79*10-30
Thermal diffusion: high T  low T
27
Eigenvalue Problems
Extension of equilibrium problems
Critical values of certain parameters need to be
determined in addition to the corresponding
Buckling and stability of structures, natural
frequency in vibration, resonance in acoustics

28
Propagation Problems
Initial Value Problems
Hyperbolic or Parabolic
“Marching” problems (inverse problem unstable)
marching), …
The solution marched out from the initial state guided and
modified in transient by the side boundary conditions
Parabolic – marching in certain direction, equilibrium in
the other directions

29
2.1.1 Nature of a Well-Posed Problem

(A) Mathematically well-posed
Governing equations - infinite many solutions
Auxiliary (initial and boundary) conditions

 (1) the solution exists (existence)
 (2) the solution is unique (uniqueness)
 (3) the solution depends continuously on the auxiliary data

Under-prescription of BCS: non-unique solution
Over-prescription of BCS: unphysical solution
30
Well-Posed Problem
(B) Numerically well-posed
Discretization equations
Auxiliary conditions (discretized, approximated)

 (1) the computational solution exists (existence)
 (2) the computational solution is unique (uniqueness)
 (3) the computational solution depends continuously on the
approximate auxiliary data
 (4) the algorithm should be well-posed (stable) also

31
Well-Posed Problems
Computational procedures

Specified       Computational          Computational
data            algorithm              solution
a                u = f(a)                 u

Auxiliary   Discretization equations   Discrete nodal
data          (must be stable)           values

32
2.1.2 Boundary and Initial Conditions

 Initial conditions: starting point for propagation problems
 Boundary conditions: specified on domain boundaries to
provide the interior solution in computational domain
R

R

(i) Dirichletcondition : u  f on R

                          u        u              s
(ii) Neumann condition :      f or      g on R
                          n         s
n
                               u
(iii) Robin (mixed)condition : n  ku  f on R

33
2.1.3 Classification by Characteristics

Consider PDEs of two independent variables
Seek “characteristic directions” along which
the equations involve only total differentials
Coordinate transformation to Canonical form
in characteristic directions
The equation type can be determined from the
number of “real” characteristics

34
First-order PDE
First-order PDE in (x,t)
Aut  Bux  C
homogeneou solution: uh  f ( Ax  Bt )  f ( )
s

                           C      C
particular solution: u p  A t or B x


Characteristic lines:  = Ax  Bt = const
dx B
Characteristic direction: d = 0             
dt A      35
Characteristics of
first-order PDEs
u = f( ) along the characteristic direction  = constant
Characteristic line
t          = 1  = 2     = 3
t = t3

t = t2

t = t1
d = 0
t = t0
x
dt A
slope    
dx B                                                  36
Characteristics of
first-order PDEs
 Along the characteristic direction     Non-characteristic
d = 0,  = constant               t         line       Characteristic line

u = f( ) = constant                                                     t = t3
 The solution remains the same
along the characteristic direction                             t = t2
 An observer moving with  =
constant sees no changes                              t = t1
(stationary) in wave form u
 The profile will change if the               t = t0
observer moves faster or slower                                               x
than the characteristic line
37
Characteristics of
first-order PDEs
 Along the characteristic direction d = 0,  = constant

  Ax  Bt  const ,            d  0  Adx  Bdt
C         C                                      dx B
u  f ( )  r ( t )  s( x )        along   Ax  Bt  const,   
A         B                                      dt A
 du df     d      C      C          du C
 dt  d       r  r , if d  0
           dt      A      A          dt  A
               involves only
                                    
 du  df   d      C     C                        total derivatives
 s  s , if d  0    du  C
 dx d
           dx      B      B          dx B


 Hyperbolic PDE - involves only total differentials along the
characteristic directions
38
Characteristics of
first-order PDEs
Alternative derivation
 Aut  Bu x  C            du      dx              du      dt
                     ut      ux    ;       ux      ut
ut dt  u x dx  du        dt      dt              dx      dx
Aut dt  Bu x dt  Cdt
A(du  u x dx )  Bu x dt  Adu  ( Bdt  Adx )u x  Cdt

 If we choose Bdt - Adx = 0 (characteristics), then du/dt =
C/A (or du/dx = C/B) involves only total differential du

dx B   du C                        du C
       &                        
dt A   dt A                        dx B
39
First-Order Hyperbolic PDEs
Characteristic line

d  Adx  Bdt  0

d   x dx   t dt  0
dx B          t
     
dt A         x
 Characteristic direction (x ,t )   = const line
 v = dx/dt = B/A is the propagation velocity of the
characteristic line
 If the observer moves faster or slower than v = B/A, the flow
will still change in space and time (with relative velocity)
40
First-Order Hyperbolic PDEs
Consider a transport quantity 

                v : convective velocity of 
v    0
t    x
 Along  = x  vt = const (v = dx/dt), the property  remains
the same (i.e., d /dt = 0)
 Consider an airplane (or a train) moving at a velocity v, the
passengers inside the airplane (or train) see everything
remains stationary while a ground observer sees partial
derivatives (d /dt = 0, but  /t  0)
Lagrangian description along the characteristic line
41
Second-Order PDEs
Second-order PDE in two variables
Au xx  Bu xy  Cu yy  H  0
 P  ux         Px  uxx (  R), Q y  u yy (  T )

let           then 
Q  u y         Py  Q x  uxy (  S )

Express every derivative in terms of uxy

dP  Px dx  Py dy  u xx dx  uxy dy


dQ  Q x dx  Q y dy  uxy dx  u yydy

dP         dy          dQ         dx
 u xx       uxy     ; u yy        uxy
dx         dx          dy         dy
42
Second-Order PDEs
Au xx  Bu xy  Cu yy  H
 dP      dQ               dy             dx 
 A          dy   H  u xy   A dx   B  C dy   0
  C    
 dx                                         

Eliminate the dependence on partial derivatives
2
 dy       dx         dy   dy 
Choose  A   B  C     0  A   B   C  0
 dx       dy         dx   dx 
 dP      dQ                involves only
then    A          dy   H  0
  C                 total differentials
 dx         
43
Characteristic Equation
Characteristic equation for second-order PDE
2
 dy   dy         dy B  B 2  4 AC
A   B   C  0     
 dx   dx         dx     2A

Classification of second-order PDEs
Hyperbolic : B 2  4 AC  0, two real roots (character istics)

Parabolic : B  4 AC  0, one real root (character
2
istics)

Elliptic :   B 2  4 AC  0, two complex roots (cannot identify

                              the propagation directions)
44
2.1.4 System of Equations
Consider two coupled first-order PDEs

 A11 u x  B11 u y  A12 v x  B12 v y  E 1


 A21 u x  B21 u y  A22 v x  B22 v y  E 2

In matrix form
 A11     A12  u x   B11   B12  u y   E 1 
A             v    B          v    E 
 21      A22   x   21     B22   y   2 
             u
or      A qx  B q y  E, q   
v                     45
System of Equations
 Questions: Is the behavior of the solution just above P uniquely
determined by the information below and on the curve ?
 Are the data sufficient to determine the directional derivatives
at P in directions that lie above the curve ?

solution to be                 du  u x dx  u y dy

calculated          D       
dv  v x dx  v y dy

C       P
solution           Find dy/dx (characteristic
domain             direction) along which only
A                  B       the total differentials du and
I.C.s              dv appear                    46
System of Equations
 Under what conditions are the derivatives (ux, uy, vx, vy)
uniquely determined at P by values of (u,v) on ?

 A11 u x  B11 u y  A12 v x  B12 v y  E 1

 A21 u x  B 21 u y  A22 v x  B 22 v y  E 2

                       u x dx  u y dy  du
                        v x dx  v y dy  dv

 Determinant  0: unique solution (linearly independent)
 Determinant = 0: multiple or no solutions

47
System of Equations
 Determinant  0: unique solution
 A11   A12     B11     B12  u x   E 1 
A                           v   E       A
 21    A22     B21     B22   x   2             B 
    C              
 dx     0      dy       0  u y   du     I dx I dy 
           
                              
 0      dx      0      dy   v y   dv 
 Characteristic equation det [C] = 0


det C  det A dy  B dx      A11 dy  B11 dx
A21 dy  B 21 dx
A12 dy  B12 dx
A22 dy  B 22 dx
 ( A11 A22  A12 A21 )dy 2  ( A12 B 21  A21 B12  A11 B 22  A22 B11 )dxdy
 ( B11 B 22  B12 B 21 )dx 2
 Ady 2  B dxdy  C dx 2  0                                     48
Characteristic Equation
 Characteristic directions along which (ux,uy,vx,vy) are not
defined uniquely
 Multiple solutions possible, discontinuity may occur
2
 dy     dy 
A   B    C   0 Characteristic equation
 dx     dx 
dy  B   B  2  4 AC 
                           Characteristic directions
dx         2 A

 Classification of equation types -- depends on the
discriminant of the characteristic equation
49
Characteristic Equation
 Discriminant DIS
DIS  B  2  4 AC    A12 B21  A21 B12  A11 B22  A22 B11 
 4( A11 A22  A12 A21 )(B11 B22  B12 B21 )
 0 , 2 real roots, 2 characteristics directions; Hyperbolic

B 2  4 AC  0 , 1 real root, 1 characteristics direction; Parabolic
 0 , 2 com plex roots, no real characteristics; Elliptic


 DIS < 0 (elliptic), cannot identify characteristic directions
along which discontinuity may occur across 
Elliptic equation - continuous (smooth) solution

50
Second-Order PDEs
 Transformation of higher-order PDE to first-order PDEs
A xx  B xy  C yy  H  0
u   x                    u x   xx , u y   xy

let         ( i .e .,V   )  
v   y                     v x   xy , v y   yy

 Convert to first-order PDEs

 Au x  Bu y  Cv y  H  0



               uy  vx  0
 A 0  u x   B C  u y   H                  
 0 1  v     1 0   v    0     Aqx  Bq y  E
       x            y       
51
Second-Order PDEs
 Transformation of second-order PDE to two first-order PDEs
          
A qx  B qy  E
det C  det A dy  B dx 
dx       dy
 Ady 2  Bdxdy  Cdx 2  0
 Characteristics equation and characteristic directions
2
 dy      dy               dy B  B 2  4 AC
A   B   C  0               
 dx      dx               dx      2A
 Hpberbolic : B 2  4 AC  0;   2 characteristics

 Parabolic : B  4 AC  0;
2
1 characteristics
 Elliptic :   B 2  4 AC  0;   no real characteristics

52
Hyperbolic PDEs
Two real roots, two characteristic directions
Two propagation (marching) directions
Domain of dependence
Domain of influence
(ux,uy,vx,vy) are not uniquely defined along the
characteristic lines, discontinuity may occur
Boundary conditions must be specified
according to the characteristics

53
Parabolic PDEs
One real (double) root, one characteristic direction
(typically t = const)
The solution is marching in time (or spatially) with
given initial conditions
The solution will be modified by the boundary conditions
(time-dependent, in general) during the propagation
Any change in boundary conditions at t1 will not affect
solution at t < t1, but will change the solution after t = t1
Irreversible: You can control your future, but not
54
Elliptic PDEs
det [C]  0 in every direction
The derivatives (ux,uy,vx,vy) can always be uniquely
determined at every point in the solution domain
No marching or propagation direction !
Boundary conditions needed on all boundaries
The solution will be continuous (smooth) in the entire
solution domain
Jury problem - all boundary conditions must be
satisfied simultaneously

55
System of Equations
One variable, n = 1
         
Aqx  Bqy  E     Au x  Bu y  C

 A B  u x   C 
dx dy  u   du
       y   

          
det A dy  B dx  Ady  Bdx  0 
dy
dx

λx B
λy

A

One real root  hyperbolic
d   x dx   y dy  Ady  Bdx  0

  Ay  Bx  const (or   Ay  Bx )           56
System of Equations
Two variables, n = 2
          
A qx  B qy  E
           
det A dy  B dx  Ady 2  B dxdy  C dx 2  0
dy    λ x  B   B  2  4 AC   dy  dx  0
                            
dx    λy          2 A             dy  dx  0
Two characteristic lines
d   x dx   y dy  0
 1    y  x  const

 2    y  x  const                   57
System of N first-order PDEs
• Two-dimensional (two variables x, y)
 u1 
                                                  
A qx  B qy  E
                     A       B  q x   E         u2 
                                    ;   q 
dxq x  dyq y  dq
                     I dx
        I dy  q y  dq 
                    
u n 
 

Characteristic equation                   
det C  det A dy  B dx  0   
Characteristic lines        const, dλ  λ x dx  λ y dy  0

Characteristic directions
dy
dx
λx

  , det A λ x  B λ y  0
λy

58
Example
u x  v y  0
                                               3 coupled PDEs
uu x  vu y  p x  ( u xx  u yy ) / Re  0   for (u, v, p)

uv x  vv y  p y  (v xx  v yy ) / Re  0
let R  v x , S  v y   u x , T  u y , then
 R y  S x  v xy  v xy  0
                                               Convert to 4
 S y  T x   u xy  u xy  0                 first-order PDEs

 uS  vT  p x  (  S x  T y ) / Re  0     for 4 unknowns
uR  vS  p  ( R  S ) / Re  0               (R, S, T, p)
               y       x     y                            59
Example
          
A qx  B qy  E
 0        1  0    0   R x   1   0      0     0Ry          0    
 0                                                   
          0  1    0 S x   0
            1      0     0 S y  
          0    

         
 0      1/Re 0    1  T x   0     0    1/Re   0   T y   uS  vT 
                                                  
  1/Re   0  0    0   p x   0  1/Re    0     1  p y   uR  vS 

 Characteristic equation
 y       x        0     0

           
det A  x  B  y 
0
0
y       x     0
    x   y   0
1 2
 x /Re   y /Re  x Re
2 2

  x /Re   y /Re    0     y
Elliptic system (all complex roots, no real roots)           60
System of Equations
• Three variables (Fourier analysis)                         (x, y, z)
 u1 
              
 A q x  B q y  C qz  E          
                                 u2 
                        ;   q             = constant
dxq x  dyq y  dzq z  dq
                                                            d=0
un 
 

x
Characteristic equation det C  det A λ  B λ  C   0
y     z           
Characteristic surface      dλ  λx dx  λ y dy  λz dz  0 (  const )
λx , λy , λz 
Characteristic directions                   Normal to characteristic
surface  = constant
61
System of N Equations
Classification for first-order equations in n
variables
Hyperbolic: n real roots
Parabolic: m real roots, 1 m < n, and no
complex roots
Elliptic: no real roots
Mixed: some real and some complex roots,
assumed to be elliptic if any complex roots
occur
62
Canonical Forms of PDEs
Any PDE can also be transformed into
Canonical form (x,y)  (,)
 A xx  B xy  C yy  D x  E y  F  G


 A  B  C    D  E   F   G 

Choose (,) to be characteristic directions
along which A = C = 0
 A  A x  B x y  C y  0

2                2


C
    A x  B x y  C y  0
2                 2
63
Canonical Forms of PDEs
 Characteristic equations: A = C = 0
   2      x 
 A x 
B    C  0  x   B     B 2  4 AC
 
  y                   y            2A
              y
        2
 x         
   B  x   C  0   x   B 
 A 
B 2  4 AC
 y                   y            2A
              y
 Characteristic directions:  = const,  = const (d ,d = 0)
  c1 , d   x dx   y dy  0 
                                     dy    x     x B  B 2  4 AC
                                               
  c 2 , d   x dx   y dy  0  dx
                                           y    y       2a
2
 dy   dy 
A  C   0  A   B   C  0         Characteristic equation
 dx   dx                                     64
Canonical Forms
Hyperbolic PDE: B2  4AC > 0, 2 real characteristics
  h1 ( , ,  ,  ,  ),
                                        A  C   0 , B  1        
                                                                 
     h1 ( , ,  ,  ,  ), A  1, B  0 , C   1     
              

Parabolic PDE: B2  4AC= 0, 1 real characteristics

  h2 ( , ,  ,  ,  ), A  1, B  C   0
Elliptic PDE: B2  4AC < 0, no real characteristics

     h3 ( , ,  ,  ,  ), A  1, B  0 , C   1
65
2.1.5 Classification by Fourier Analysis

Characteristic polynomial
The root determine the characteristic surfaces ( = const)
or directions (x, y, z, …)

Fourier Analysis
The roots have a different physical interpretation
Produces the same characteristic polynomial from the
“principal part” of the governing equation. However,
(x, y, z, …) also determine the solution of PDE, e.g.,
oscillatory, exponential growth, wavelike, …
Avoids the construction of intermediate first-order PDEs66
Classification by Fourier Analysis
Consider second-order PDE
Au xx  Bu xy  Cu yy  0
General solution in Fourier series
                            
      
1
u( x , y ) 
4   2    u
ˆ
j   k  
jk   exp i ( x ) j x exp i ( y )k y

ˆ
u jk      amplitude                        Determined by boundary conditions

 x , y  exponents
                                            Determine the nature of solutions
67
Classification by Fourier Analysis
Take derivatives of the general solution

                              
      
1
ux  4 2     i(         x   ) j u jk exp i ( x ) j x exp i ( y )k y
ˆ
            j   k  

                              
      
u  1
 y 4 2       i(         y k ) u jk exp i ( x ) j x exp i ( y )k y
ˆ
            j   k  

If A, B, C are not function of u (linear), then the relation
between x and y is the same for all modes
“Superposition” gives all possible functional forms for
linear equation
68
Classification by Fourier Analysis
Consider one single model only (x )j=x , (y )k=y
      
expi x x expi y y 
1
u( x , y ) 
4   2    u
ˆ
j   k  
jk

ux  i x u, uxx  ( i x )2 u   x u, uxy   x y u

2



 u y  i y u, u yy  ( i y )2 u   y u
2

 Au xx  Bu xy  Cu yy  0   A x  B x y  C y  0
2              2

2
Characteristic polynomial                         x   x 
A   B   C  0
     
  y   y       69
Characteristic Equation
Fourier analysis (discrete mode)
2
 x   x          x  B  B 2  4 AC
A   B   C  0     
               y        2A
  y   y
Discriminant B2  4AC
            x        Propagation/marching
 0 , real  
          problems (sin/cos functions)
            y
B  4 AC 
2

                  x  Growth/decay/propagation
 
 0 , complex   
                 y  Exponential Growth/decay
– pure imaginary (x /y )70
Characteristic Equation
Fourier Transform (continuous spectrum)

u( x , y )      u( x , y ) exp i x x exp i y y dxdy  Fu
   
ˆ
                            u                                                   u
( x , y ) exp i x x exp i y y dxdy  F
                      
 i x u( x , y )   
ˆ
    x                                                  x

i u( , )    u ( x , y ) exp i x exp i y dxdy  F u


y
ˆ x y              y                     x             y
y

Integration by parts
u        u                   i x x  i  y y
F                ( x , y )e            e          dxdy Fourier transform of u/x
x        x

 
 

    x
ue   i x x  i  y y
e                       
dxdy    (  i x )ue
 
 i x x  i y y
e         dxdy
                             
 i x x  i  y y
   ue              e               dy  i x Fu  i x u
ˆ
                                                                          71
= 0, if u = 0 at x
Characteristic Equation
Classification by Fourier transform
             u                       2u
i x u  F
ˆ           , i x  u  F 2
ˆ
2

             x                      x
u  Fu  
ˆ
i y u  F u , i y 2 u  F  u
2
ˆ                      ˆ

             y                     y 2
 Au xx  Bu xy  Cu yy  0 
                                          
A( i x ) 2  B( i x )(i y )  C ( i y ) 2 u  0
ˆ
2
Characteristic polynomial             x   x 
A   B   C  0
     
  y   y       72
Fourier Analysis
 Particularly usseful for system of equations with higher-order
PDEs (avoid the construction of first-order PDEs)
 Example: Navier-Stokes equations

u x  v y  0

uux  vu y  p x  ( u xx  u yy ) / Re  0

uv x  vv y  p y  (v xx  v yy ) / Re  0
iσ x u  iσ y v  0
ˆ        ˆ


 iuσ x u  ivσ y u  iσ x p  ( iσ x ) 2 u  ( iσ y ) 2 u / Re  0
ˆ          ˆ       ˆ              ˆ             ˆ


 iuσ x v  ivσ y v  iσ y p  ( iσ x ) 2 v  ( iσ y ) 2 v / Re  0
ˆ         ˆ        ˆ             ˆ             ˆ
73
Fourier Analysis
 Characteristic equations
                   iσ x                                  iσ y                 0   u  0 
ˆ
                                                                                     
 i ( uσ x  vσ y )  ( x   y ) / Re
2     2
0                  iσ x   v   0 
ˆ

                    0                  i ( uσ x  vσ y )  ( x   y ) / Re iσ y   p  0 
2     2
 ˆ   

 Determinant = 0
2                       1           2 
 i ( x   y ) i ( u x  v y ) 
2
( x   y )  0
2

                     Re             
first-derivative     second-derivative
 Consider only the highest-order derivatives
1
Re

 x  y
2    2
2
0       Elliptic system (complex roots)
74

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