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cours-maths-enseee

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28 Pages
English

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Chapter IIIClassi cation ofpartial di eren tial equations intoelliptic, parabolic and hyperbolictypesThe previous chapters have displayed examples of partial di eren tial equations in various eldsof mathematical physics. Attention has been paid to the interpretation of these equations in1the speci c contexts they were presented.In fact, we have delineated three types of eld equations, namely hyperbolic, parabolic andelliptic. The basic idea that the mathematical nature of these equations was fundamental totheir physical signi cance has been creeping throughout.Still, the formats in which these three types were presented correspond to their canonicalforms, that is, a form that one recognizes at rst glance. Such is not the general case. Forexample, it is not obvious (to this author at least!) that the following second order equation,2 2 2@ u @ u @ u @u2 4 6 + = 0;2 2@x @x@t @t @xis of hyperbolic type. In other words, it shares essential physical properties with the waveequation,2 2@ u @ u= 0:2 2@x @tIndeed, this is the aim of the present chapter to show that all equations of mathematicalphysics can be recast in these three fundamental types. By the same token, we introduce a newnotion, that of a characteristic curve. A method to solve IBVPs based on characteristics willbe exposed in the next chapter.The terminology used to coin the three types of PDEs borrows from geometry, as thecriterion will be seen to rely on the nature of the roots of ...

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Chapter III
Classi cation of
partial di eren tial equations into
elliptic, parabolic and hyperbolic
types
The previous chapters have displayed examples of partial di eren tial equations in various elds
of mathematical physics. Attention has been paid to the interpretation of these equations in
1the speci c contexts they were presented.
In fact, we have delineated three types of eld equations, namely hyperbolic, parabolic and
elliptic. The basic idea that the mathematical nature of these equations was fundamental to
their physical signi cance has been creeping throughout.
Still, the formats in which these three types were presented correspond to their canonical
forms, that is, a form that one recognizes at rst glance. Such is not the general case. For
example, it is not obvious (to this author at least!) that the following second order equation,
2 2 2@ u @ u @ u @u
2 4 6 + = 0;
2 2@x @x@t @t @x
is of hyperbolic type. In other words, it shares essential physical properties with the wave
equation,
2 2@ u @ u
= 0:
2 2@x @t
Indeed, this is the aim of the present chapter to show that all equations of mathematical
physics can be recast in these three fundamental types. By the same token, we introduce a new
notion, that of a characteristic curve. A method to solve IBVPs based on characteristics will
be exposed in the next chapter.
The terminology used to coin the three types of PDEs borrows from geometry, as the
criterion will be seen to rely on the nature of the roots of quadratic equations.
We envisage in turn rst of order equations, sets of rst order equations, and second order
equations. The use of a common terminology to class rst and second order equations is
challenged by the fact that a set of two rst order equation may be transformed into a second
1Posted, December 05, 2008; updated, December 12, 2008
5758 Classi cation of PDEs
order equation, and conversely. The point will not be developed throughout, but rather treated
via examples.
Since we are concerned in this chapter with the nature of partial di eren tial equtions, we
will not specify the domain in which they assume to hold. On the other hand, the issue surfaces
when we intend to solve IBVPs, as considered in Chapters I, II and IV.
III.1 First order partial di eren tial equations
III.1.1 A single equation
We consider rst a single rst order partial di eren tial equation for the unknown function
u =u(x;y),
u =u(x;y) unknown;
(III.1.1)
(x;y) variables;
that can be cast in the format,
@u @u
a +b +c = 0: (III.1.2)
@x @y
This equation is said to be (please think a little bit to this terminology),
- linear if a =a(x;y), b =b(x;y), and c constant;
- quasi-linear if these coe cien ts depend in addition on the unknownu;
- nonlinear if these coe cien ts depend further on the derivatives of the unknownu.
Let " #
a1ps = ; (III.1.3)
2 2a +b b
be the unit vector that makes it possible to recast the PDE (III.1.2) into the format,
sru +d = 0; (III.1.4)
p
2 2with d =c= a +b .
The curves, starting from an initial curve I , and with a slope,0
dy b
= ; (III.1.5)
dx a
are called characteristic curves. A point on these curves is reckoned by the curvilinear
abscissa ,
2 2 2(d ) = (dx) + (dy) : (III.1.6)
Typically, is set to 0 on the initial curve I .0
Then " #
dx=d
s = ; (III.1.7)
dy=d
and the partial di eren tial equation (PDE) (III.1.4) for u(x;y),
@u dx @u dy du
+ +d = +d = 0; (III.1.8)
@x d @y d d
magically becomes an ordinary di eren tial equation (ODE) foru() along a characteristic
dy=dx =b=a. Hum puzzling, how is that possible? There should be a trick here My mumBenjamin LORET 59
warned me, \my little boy, nothing comes for free in this world, except AIDS perhaps". Indeed,
there is a price to pay, and the price is to nd the characteristic curves, which are not known
beforehand.
Taking a step backward, the transformation of a PDE to an ODE is a phenomenon that
we have already encountered. Indeed, this is in fact the basic principle of Laplace or Fourier
transforms. The initial PDE is transformed into an ODE where the variable associated to the
transform is temporarily seen as a parameter. The price to pay here is the inverse transforma-
tion.
characteristic network
I2
abscissa s dy
dx
s dy
I1
dy abscissa s
ds dx
initial data
dxon I0
(s=0,s)
Figure III.1 Given data on a non characteristic initial curve I , the characteristic network and so-0
lution are built simultaneously, step by step. Each characteristic is endowed with a curvilinear
abscissa, while points on the initial curveI are reckoned by a curvilinear abscissas.0
Analytical and/or Numerical solution
The above observations provide the basics to a method for solving a partial di eren tial
equation.
If the PDE is linear, then
- the characteristics and curvilinear abscissa are obtained by (III.1.5) and (III.1.6);
- the solution u is deduced from (III.1.8).
If the PDE is quasi-linear, a numerical scheme is developed to solve simultaneously (III.1.5)
and (III.1.8):
- assume u to be known along a curve I , which is required not to be a characteristic;0
- at each point of I , one may obtain and draw the characteristic using (III.1.5), which0
provides also d by (III.1.6);
- du results from (III.1.8), whence the solution on the new curve I ;1
- the three steps above are repeated, starting from I , and so on.1
It is now clear why the initial curve I should not be a characteristic. Indeed, otherwise,0
the subsequent curves I would be I itself, so that the solution could not be obtained at1 0
points (x;y) other than on I .0
III.1.2 A system of quasi-linear equations
The concept of a characteristic curve is now extended to a quasi-linear system of rst order
0partial di eren tial equations for the n unknown functionsus,
u =u (x;t); j2 [1;n]; unknowns;j j
(III.1.9)
(x;t) variables;60 Classi cation of PDEs
that can be cast in the format,
@u @uL U = a + b + c = 0
@t @x (III.1.10)
@u @uj jL u = a +b +c = 0; i2 [1;n];ij j ij ij i
@t @x
2where the coe cien t matrices a = (a ) and b = (b ) with (i;j)2 [1;n] , and vector c = (c ),ij ij i
with i2 [1;n], may depend on the variables and unknowns, but not of their derivatives.
In order to form an ordinary di eren tial equation in terms of a (yet) unknown curvilinear
abscissa , we devise a linear combination of these n partial di eren tial equations, namely,
du
L u = p +r = 0
d (III.1.11)
dujL u = p +r = 0:i ij j j
d
The vector will appear to be a left eigenvector of the matrix adx=dt b, namely
dx
(a b) = 0;
dt (III.1.12)
dx
(a b ) = 0; j2 [1;n]:i ij ijdt
To prove this property, we pre-multiply (III.1.10) by ,
@u @u
(III.1.13) a + b + c = 0;
@t @x
which can be of the form (III.1.11) only if
a b p
= = : (III.1.14)
dt dx d
Elimination of the vector p in this relation yields the generalized eigenvalue problem (III.1.12).
For the eigenvector not to vanish, the associated coe cien t matrix should be singular,
dx
(III.1.15)det (a b) = 0:
dt
This characteristic equation should be seen as a polynomial equation of degreen fordx=dt. The
classi cation of rst order partial di eren tial equations is based on the above spectral analysis.
Classi cation of rst order linear PDEs
- if the nb of real eigenvalues is 0, the system is said elliptic;
- if the eigenvalues are real and distinct, or
if the eigenvalues are real and the system is not defective, the system is said hyperbolic;
- if the eigenvalues are real, but the system is defective, the system is said to be parabolic.
nLet us recall that a system of size n is said non defective if its eigenvectors generate ,
that is, the algebraic and geometric multiplicities of each eigenvector are identical.
Characteristic curves and Riemann invariants
Each eigenvalue dx=dt de nes a curve in the plane (x;t) called characteristic. To each
characteristic is associated a curvilinear abscissa , de ned by its di eren tial,
d dt @ dx @
= +
d d @t d @x
(III.1.16) dt @ dx @
= + :
d @t dt @xBenjamin LORET 61
Inserting (III.1.12) into (III.1.13) yields,
du dt
(III.1.17) a + c = 0:
d d
Quantities that are constant along a characteristic are called Riemann invariants.
A simple, but subtle and tricky issue
1. Please remind that the left and right eigenvalues of an arbitrary square matrix are identical,
but the left and right eigenvectors do not, if the matrix is not symmetric. The left eigenvectors
Tof a matrix a are the right eigenvectors of its transpose a .
2. The generalized (left) eigenvalue problem (adx=dt b) = 0 becomes a standard (left)
eigenvalue problem when b = I, i.e. (adx=dt I) = 0. The left eigenvectors of the pencil
T T(a;b) are also the right eigenvectors of the pencil (a ;b ).
3. Note the subtle interplay between the sets of matrices (a;b), and the variables (t;x). The
above writing has made use of the ratio dx=dt, and not of dt=dx: we have broken symmetry
without care. That temerity might not be without consequence. Indeed, an immediate question
comes to mind: are the eigenvalue problems (adx=dt b) = 0 and (a bdt=dx) = 0
equivalent? The answer is not so straightforward, as will be illustrated in Exercise III.2.
Some further terminology
If the system of PDEs,
@u @u
(III.1.18)a + b + c = 0;
@t @x
can be cast in the format,
@F(u) @G(u)
(III.1.19)+ = 0;
@t @x
it is said to be of divergence type. In the special case where the system can be cast in the
format,
@u @G(u)
(III.1.20)+ = 0;
@t @x
it is termed a conservation law.
III.2 Second order partial di eren tial equations
The analysis addresses a single equation, delineating the case of constant coe cien ts from that
of variable coe cien ts.
III.2.1 A single equation with constant coe cien ts
Let us start with an example. For the homogeneous wave equation,
2 2@ u 1 @ u
(III.2.1)Lu = = 0;
2 2 2@x c @t
the change of coordinates,
=x ct; =x +ct; (III.2.2)
transforms the canonical form (III.2.1) into another canonical form,
2@ uLu = = 0: (III.2.3)
@@62 Classi cation of PDEs
Therefore, the solution expresses in terms of two arbitrary functions,
u(;) =f() +g(); (III.2.4)
which should be prescribed along a non characteristic curve.
But where are the characteristics here? Well, simply, they are the lines constant and
constant.
Let us try to generalize this result to a second order partial di eren tial equation for the
unknownu(x;y),
u =u(x;y) unknown;
(III.2.5)
(x;y) variables;
with constant coe cien ts,
2 2 2@ u @ u @ u @u @u
(III.2.6)Lu =A + 2B +C +D +E +Fu +G = 0:
2 2@x @x@y @y @x @y
The question is the following: can we nd characteristic curves, so as to cast this PDE into an
ODE? The answer was positive for the wave equation. What do we get in this more general
case?
Well, we are on a moving ground here. To be safe, we should keep some degrees of freedom.
So we bet on a change of coordinates,
= x +y; = x +y; (III.2.7)1 2
where the coe cien ts and are left free, that is, they are to be discovered.1 1
Now come some tedious algebras,
@u @u @ @u @ @u @u
= + = 1 2@x @ @x @ @x @ @
(III.2.8)
@u @u @ @u @ @u @u
= + = + ;
@y @ @y @ @y @ @
and
2 2 2 2@ u @ u @ u @ u2 2= + 2 + 1 21 22 2 2@x @ @@ @
2 2 2 2@ u @ u @ u @ u
(III.2.9)= + 2 +
2 2 2@y @ @@ @
2 2 2 2@ u @ u @ u @ u
= ( + ) :1 1 2 22 2@x@y @ @@ @
Inserting these relations into (III.2.6) yields the PDE in terms of the new coordinates,
2 2@ u @ u2 2Lu = (A 2B +C) + (A 2B +C)1 21 22 2@ @
2@ u
(III.2.10)+2( A ( + )B +C)1 2 1 2
@@
@u @u
+( D +E) + ( D +E) +Fu +G = 0:1 2
@ @6
6
Benjamin LORET 63
Let us choose the coe cien ts to be the roots of
2A 2B +C = 0; (III.2.11)
namely, pB 1
2 = B AC: (III.2.12)1;2
A A
Therefore we are led to distinguish three cases, depending on the nature of these roots. But
before we enter this classi cation, we can make a very important observation:
the nature of the equation depends only on the coe cien ts of the second order
terms. First order terms and zero order terms do not play a role here.
2III.2.1.1 Hyperbolic equation B AC > 0, e.g. the wave equation
If the discriminant of the quadratic equation (III.2.11) is strictly positive, the two roots are real
distinct, and the equation is said hyperbolic. The coe cien t of the mixed second derivative of
the equation does not vanish, p4
2 (III.2.13)2( A ( + )B +C) = B AC = 0:1 2 1 2
A
The equation can then be cast in the canonical form,
2@ u @u @u0 0 0 0 (III.2.14)(H) +D +E +F u +G = 0;
@@ @ @
where the superscript ’ indicates that the original coe cien ts have been divided by the non
zero term (III.2.13).
Another equivalent canonical form,
2 2@ u @ u @u @u00 00 00 00 (III.2.15)(H) +D +E +F u +G = 0;
2 2@ @ @ @
is obtained by the new set of coordinates,
1 1
(III.2.16) = ( +); = ( ):
2 2
00The superscript in (III.2.15) indicates another modi cation of the original coe cien ts.
2III.2.1.2 Parabolic equation B AC = 0, e.g. heat di usion
A single family of characteristics exists, de ned by
B
(III.2.17) = = :1 2
A
A second arbitrary coordinate is introduced,
= x +y; = x +y; =; (III.2.18)
which allows to cast the equation in the canonical form,
2@ u @u @u0 0 0 0(P) +D +E +F u +G = 0; (III.2.19)
2@ @ @
where the superscript ’ indicates a modi cation of the original coe cien ts.64 Classi cation of PDEs
2III.2.1.3 Elliptic equation B AC < 0, e.g. the laplacian
There are no real characteristics. Still, one may introduce the real coordinates,
1 1
(III.2.20) = ( +) = ax +y; = ( ) = bx;
2 2i
with real coe cien ts a andb,
pB B
2 (III.2.21) =aib; a = ; b = AC B ;1;2
A A
so as to cast the equation in the canonical form,
2 2@ u @ u @u @u00 00 00 00 (III.2.22)(E) + +D +E +F u +G = 0;
2 2@ @ @ @
where the superscript " indicates yet another modi cation of the original coe cien ts.
III.2.2 A single equation with variable coe cien ts
When the coe cien ts of the second order equation are variable, the analysis becomes more
complex, but, fortunately, the main features of the constant case remain. Moreover, the anal-
2ysis below shows that this nature relies entirely on the sign of B AC, like in the
constant coe cien t equation.
The characteristics are sought in the more general format,
=(x;y); =(x;y); (III.2.23)
whence
@u @u @ @u @
= +
@x @ @x @ @x
(III.2.24)
@u @u @ @u @
= + ;
@y @ @y @ @y
and
2 2 2 2 2 2 2 2@ u @ @ u @ @ @ u @ @ u @ @u @ @u
= + 2 + + +
2 2 2 2 2@x @x @ @x @x @@ @x @ @x @ @x @
2 2 2 2 2 22 2@ u @ @ u @ @ @ u @ @ u @ @u @ @u
= + 2 + + +
2 2 2 2 2@y @y @ @y @y @@ @y @ @y @ @y @
2 2 2 2 2 2@ u @ @ @ u @ @ @ @ @ u @ @ @ u @ @u @ @u
= + + + + + ;
2 2@x@y @x @y @ @y @x @x @y @@ @x @y @ @x@y@ @x@y@
(III.2.25)
yielding nally ,
2 2 2@ u @ u @ u @u @u0 0 0 0 0 0 0Lu =A + 2B +C +D +E +F u +G = 0: (III.2.26)
2 2@ @@ @ @ @
The coe cien ts of higher order,
0 0 0A =Q(;); B =Q(;); C =Q(;); (III.2.27)