A PDE of first or second order for a function u of two variables x, y, for instance, is an expression of the form

(1)

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.

PDE Overview TU Bergakademie Freiberg 157

A function u, which is defined on a domain Ω

⊆R²

, 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

R²

is 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

Rⁿ

is

Xn

j=1

a

j

(x)u

xj

+ a

0

(x)u = f (x)

or

Xⁿ

j,k=1

a

j,k

(x)u

xj,xk

+

Xn

j=1

a

j

(x)u

xj

+ a

0

(x)u = f (x), (1) where a

j

, a

j,k

and f a given functions defined in

x∈Rⁿ

.

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.

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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

1,1

a

2,2−

a

²_1,2

> 0, (1) hyperbolic

⇔

a

1,1

a

2,2−

a

²_1,2

< 0, (1) parabolic

⇒

a

1,1

a

2,2−

a

²_1,2

= 0,

A comparision with the characterization of cone sections, i. e. second order algebraic equations of type

a

1,1

x

²

+ 2a

1,2

xy + a

2,2

y

²

+ a

1

x + a

2

y + b = 0 explains the terminology.

Characterize the PDEs ∆u(x) = f, u

tt

(t, x) = c

²

u

xx

(t, x) and u

t

(t, x) = κu

xx

(t, x) (c, κ > 0).

4.2 PDE in Mathematical Physics The elliptic Poisson

¹

equation

∆u(x) = f(x) (x

∈

Ω

⊂Rⁿ

)

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

²

equation.

The parabolic heat or diffusion equation

u

t

(t,

x) =

κ∆

x

u(t,

x)

(t > 0,

x∈

Ω

⊂Rⁿ

)

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)

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The hyperbolic wave equation

u

tt

(t,

x) =

c

²

∆

x

u(t,

x) (t >

0,

x∈

Ω

⊂Rⁿ

) 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

:

R³⊃

Ω

→R

div

u

=

∇ ·u

:= (u

1

)

x1

+ (u

2

)

x2

+ (u

3

)

x3

is 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

]

^T

is 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

⁴

equation:

vt

+ (v

· ∇

)v =

−¹_ρ∇

p + ν∆v +

f,

∇ ·v

= 0.

From fluid mechanics, we come to know the Korteweg

⁵

–de Vries

⁶

equation,

u

t

+ κu

x

+

^3κ_4η

u

xx

+

^κη₆²

u

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

⁷

equation

u

t

+ uu

x

= µu

xx

is 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

⁸

equations

∇ ·E

= 0,

∇ ·H

= 0,

Ht

+ c(∇ ×

E) =0, Et−

c(∇ ×

H) =0

with electric field

E

=

E(x, t), magnetic fieldH

=

H(x, t)

and speed of light c.

7J. M. Burgers (1895–1981)

8James Clerk Maxwell (1831–1879)

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A linear-elastic body is described by Lamé’s

⁹

equation:

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

∆

²

u = 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

¹¹

boundary condition):

u(x) = g(x) for

x∈

∂Ω.

We prescribe the heat flux on the boundary (Neumann

¹²

boundary condition):

∂

n

u(x) = h(x) for

x∈

∂Ω,

where ∂

n

u denotes the derivative of u in outer normal direction.

(∂

n

u(x)

≡

0 means perfect insulation of the boundaries).

11Peter Gustav Lejeune Dirichlet (1805–1859)

12Carl Gottfried Neumann (1832–1925)

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A Robin

¹³

boundary condition is a linear combination of Dirichlet and Neumann boundary conditions:

∂

n

u(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

xx

on the square Ω :=

{

(x, t)

∈R²

:

−

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.

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Example 2.

The solution of Laplace’s equation u

xx

+ u

yy

= 0 on Ω :=

R×

[0,

∞

)

⊂R²

with boundary conditions

u(x, 0) = 0 and u

y

(x, 0) = 1 n sin(nx) is given by

¹⁴

u

⁽ⁿ⁾

(x, y) = 1

n

²

sin(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) := (e^t−e⁻^t)/2

Goals of Chapter 4

You should be familiar with the basic terminology (PDE, solution, boundary and initial values).

You should be able to recognize linear PDEs and to classificate linear second order problems.

You should know some of the most important PDEs occuring in mathematical physics.

You should know what is meant by a well-posed problem in the sense of Hadamard.

You should know usual types of boundary and initial conditions and have a rough idea, which boundary/initial conditions lead to well-posed problems (for Poisson, heat and wave equation).