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Classification of PDEs

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					   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
Advection-diffusion equation (linear)

           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
           Steady-state : parabolic  elliptic
           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)

                          
 Advection equation
                     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

    Steady, compressible potential flow

           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
Usually “steady-state”
Diffusion / dissipation, smooth gradient

                                                       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
 steady-state configuration
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)
Unsteady, transient, steady shock, boundary-layer (space-
 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
                                   Ady  Bdx  Cdx
         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
 changing what already happened (history!)
                                                          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
Steady 2D incompressible Navier-Stokes equations
  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
 Steady 2D incompressible Navier-Stokes equations
                                          
                              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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