4 Partial Differential Equations - Some background
4.1 Terminology and Classification
Partial differential equations (PDE) are the analogue of ODEs for functions of several variables.
A PDE of first or second order for a function u of two variables x, y, for instance, is an expression of the form
Φ(u
x, u
y, u, x, y) = 0
or Φ(u
xx, u
xy, u
yy, u
x, u
y, u, x, y) = 0
where Φ is a given (usually continuous) function of five (or eight) real
variables.
A function u, which is defined on a domain Ω ⊆ R
2, is called a solution of such a PDE (over Ω) when it is once (or twice) continuously
differentiable in every point of Ω.
Example
The general solution of the equation u
xy= 0 over R
2is u(x, y) = f (x) + g(y)
with arbitrary functions f, g ∈ C
1( R ).
For uniqueness (as in the case of ODE) we will have additional
conditions.
A PDE for a function u is linear, if it is linear in all occurring derivatives and in the function itself, too.
The linear equation of first (or second) order in R
nis
n
X
j=1
a
j(x)u
xj+ a
0(x)u = f (x)
or
n
X
j,k=1
a
j,k(x)u
xj,xk+
n
X
j=1
a
j(x)u
xj+ a
0(x)u = f (x), (1)
where a
j, a
j,kand f a given functions defined in x ∈ R
n. If f ≡ 0, the above equations are called homogeneous, otherwise inhomogeneous.
Homogeneous linear PDE fulfill (as their ODE analogues) the
superposition principle, i. e. each linear combination of solutions is again
a solution of the PDE.
W.l.o.g. consider the linear second-order PDE (1) with a symmetric coefficient matrix A(x) = [a
j,k(x)]
1≤j,k≤n.
Then this equation is said to be
elliptic at x, if all eigenvalues of A(x) are nonzero and have the same sign.
parabolic at x, if exactly one of the eigenvalues is zero, all the others are nonzero and have the same sign, and the matrix [A(x) [a
1(x) . . . a
n(x)]
T] has full rank n.
hyperbolic at x, if all eigenvalues are nonzero and one of them has the opposite sign of the (n − 1) others.
The PDE is said to be elliptic (parabolic, hyperbolic) in Ω if has this
property at any point in Ω.
In the case of two variables, therefore the following holds:
(1) elliptic ⇔ a
1,1a
2,2− a
21,2> 0, (1) hyperbolic ⇔ a
1,1a
2,2− a
21,2< 0, (1) parabolic ⇒ a
1,1a
2,2− a
21,2= 0,
A comparision with the characterization of cone sections, i. e. second order algebraic equations of type
a
1,1x
2+ 2a
1,2xy + a
2,2y
2+ a
1x + a
2y + b = 0
explains the terminology.
Characterize the PDEs ∆u(x) = f , u
tt(t, x) = c
2u
xx(t, x) and
u
t(t, x) = κu
xx(t, x) (c, κ > 0).
4.2 PDE in Mathematical Physics The elliptic Poisson
1equation
∆u(x) = f (x) (x ∈ Ω ⊂ R
n)
describes e. g. a time-invariant temperature distribution or a electrostatic potential. The function f characterizes heat sources or a charge density.
The Poisson equation with f ≡ 0 is called Laplace
2equation.
The parabolic heat or diffusion equation
u
t(t, x) = κ∆
xu(t, x) (t > 0, x ∈ Ω ⊂ R
n)
describes e. g. the variation in temperature in a given region over time.
The constant κ > 0 is the thermal conductivity.
1Siméon Denis Poisson (1781–1840)
2Pierre Simon Laplace (1749–1827)
The hyperbolic wave equation
u
tt(t, x) = c
2∆
xu(t, x) (t > 0, x ∈ Ω ⊂ R
n)
describes the propagation of waves in a medium with speed of light (wave) c.
Before we give some more examples, we repeat some terms from vector analysis. For a vector field u : R
3⊃ Ω → R
div u = ∇ · u := (u
1)
x1+ (u
2)
x2+ (u
3)
x3is the divergence of u and rot u = ∇ ×u := [(u
3)
x2−(u
2)
x3, (u
1)
x3−(u
3)
x1, (u
2)
x1−(u
1)
x2]
Tis the rotation of u.
Physically, the divergence can be interpreted as a source density and the rotation as a circulation density of the vector field u (think to
incompressible fluids).
The motion of incompressible fluid substances are described by the (nonlinear) Navier
3-Stokes
4equation:
v
t+ (v · ∇)v = −
1ρ∇p + ν∆v + f,
∇ · v = 0.
From fluid mechanics, we come to know the Korteweg
5–de Vries
6equation,
u
t+ κu
x+
3κ4ηu
xx+
κη62u
xxx= 0,
which represents a mathematical model of waves on the shallow water surface.
3Louis Marie Henri Navier (1785–1836)
4George Gabriel Stokes (1819–1903)
5Diederik Johannes Korteweg (1848–1941)
6Gustav de Vries (1866–1934)
The Burgers
7equation
u
t+ uu
x= µu
xxis a simple model for a one dimensional fluid dynamics and is also used in quantum mechanics or in traffic models.
The electromagnetic fields in vacuum are described by the Maxwell
8equations
∇ · E = 0, ∇ · H = 0, H
t+ c(∇ × E) = 0, E
t− c(∇ × H) = 0
with electric field E = E(x, t), magnetic field H = H(x, t) and speed of light c.
7J. M. Burgers (1895–1981)
8James Clerk Maxwell (1831–1879)
A linear-elastic body is described by Lamé’s
9equation:
ρu
tt= (λ + µ)∇∇ · u + µ∆u + ρf .
Here u = u(t, x) is the displacement of the point, which was originally at point x, at time t, ρ and f are mass and force densities, and ρ, µ are constants characterizing the elastic properties of material.
Small deformations of an originally flat, thin elastic plate can be described by the biharmonic equation
∆
2u = u
xxxx+ 2u
xxyy+ u
yyyy= 0.
9Gabriel Lamé (1795–1870)
4.3 Initial and Boundary Value Problems
Our goal is to formulate a well-posed problem in the sense of Hadamard
10, i. e. we would like to have
the existence of a solution, the uniqueness of this solution,
the continuous dependence of the solution on the given data.
(The latter point indicates that the solution should not have to change much if the data are slightly perturbated.)
To ensure these, additional conditions must be imposed to a PDE.
10Jacques Salomon Hadamard (1865–1963)
Examples
Consider Laplace’s equation ∆u = 0 for a stationary heat conduction problem. In a domain Ω it has infinitely many solutions. For uniqueness we need additional conditions, for example:
We fix the temperature on the boundary of the domain (Dirichlet
11boundary condition):
u(x) = g(x) for x ∈ ∂Ω.
We prescribe the heat flux on the boundary (Neumann
12boundary condition):
∂
nu(x) = h(x) for x ∈ ∂Ω,
where ∂
nu denotes the derivative of u in outer normal direction.
(∂
nu(x) ≡ 0 means perfect insulation of the boundaries).
11Peter Gustav Lejeune Dirichlet (1805–1859)
12Carl Gottfried Neumann (1832–1925)
A Robin
13boundary condition is a linear combination of Dirichlet and Neumann boundary conditions:
∂
nu(x) + α(u(x) − k(x)) = 0 for x ∈ ∂Ω.
It describes heat exchange with a heat transfer coefficient α and an ambient temperature k(x).
Of course there are many more possibilities to set well-posed problems.
For instance, on different parts of the domain boundaries, different boundary conditions can be applied.
13Gustave Robin (1855–1897)
For time-dependent problems, the boundary conditions can be time dependent. In general one sets additional initial conditions for u or/and its derivatives at time t
0(mostly t
0= 0).
For the diffusion equation u
t= κ∆u, one normally sets u(x, 0) = u
0(x), x ∈ Ω;
for the wave equation u
tt= c
2∆u one usually sets additionally u
t(x, 0) = u
1(x), x ∈ Ω
(note the different order of the time derivatives!).
The terms „homogeneous “ und „inhomogeneous “ are spread
analogously on boundary and initial conditions.
Warning: Not all additional requirements lead to well posed problems!
Example 1. The solution of wave equation u
tt= u
xxon the square Ω := {(x, t) ∈ R
2: −1 < x + t < 1, −1 < x − t < 1}
differs on opposite edges only by a constant (see later).
It is therefore impossible to impose arbitrary Dirichlet boundary
conditions.
Example 2. The solution of Laplace’s equation u
xx+ u
yy= 0 on Ω := R × [0, ∞) ⊂ R
2with boundary conditions
u(x, 0) = 0 and u
y(x, 0) = 1
n sin(nx) is given by
14u
(n)(x, y) = 1
n
2sin(nx) sinh(ny)
At y = 0 the boundary conditions for n → ∞ converge uniformly to zero. The „limit problem“ obviously has the solution u
(∞)≡ 0. Due to
|u
(∞)(π/2, y) − u
(n)(π/2, y)| = sinh(ny)/n
2≈ exp(ny)/n
2(n = 4k + 1 large), this solution does not continuously depend on the given data.
14sinh(t) := (et−e−t)/2