**About partial
differential equations**

**2.7 Occurrence of the simplest
partial differential equations****:**** **We
know the potential equation

as an expression of a field action view in the **gravitation theory**
in contrast to Newton's point of view of **distance action**.
The meaning of the **Laplace operator****
**is known to us:

The same equations (1) and (2) also occur in the theory of **static
and magnetic fields**, (1) in **empty space**,
(1a) in the **presence of sources of density**; we have
placed the factor 4*p* in (1a) in brackets, because it can be
removed by a rational choice of units.

(1) also occurs in **hydrodynamics of incompressible and
irrotational flow**; then *u *is the **velocity
potential**. We also note the **two-dimensional
potential equation**

as the foundation of **Riemann's
function theory**, which we can briefly characterize as the **field
theory of the analytic functions** *f*(*x + iy*).

On the other hand, we know the **wave
equation**

It rules in **acoustics** (*c*
is the velocity of sound), **electrodynamics** of varying
fields (*c* is the velocity of light) and especially in **optics**.
Introducing into the **special theory of relativity****,**
in addition to the spatial co-ordinates *x*_{1}*, x*_{2}*, x*_{3}*, *as* *fourth
co-ordinate *x*_{4 }(or *x*_{0}) = *ict*, one can write (4) as **four-dimensional
potential equation**

In two dimensions, (4) appears with the **vibrating membrane**,
in one dimension with the **vibrating string**. In
the last case, we write

setting temporarily *y = ct *(not *y* = *ict*!). Membranes as well as strings do not have
elasticity of their own; their constant *c* is computed from the externally imposed tension and their density per area or
length unit.

As a special case of the **general ****theory
of elasticity**** **, there arises the
differential equation for the **transverse ****vibrations
of a thin plate**:

naturally, for dimensional reasons, *c *is here not, as in acoustics, the **velocity
of sound** of the elastic material, but is computed in terms of the elasticity, density and thickness
of the plate. Similarly, one has for the differential equations of a **vibrating
elastic bar**

It will be derived in Exercise 2.1,
where we compare the corresponding **eigen-oscillations**
with the acoustical vibrations of open and closed pipes.

As a third type, we place beside the equilibrium states [(1) - (3)] and the vibration states [(4) - (8)] the
equations of **compensation processes**.
We consider here in somewhat greater detail as its most prominent representative **heat
conduction**** **(compensation of energy differences), but note immediately that also **diffusion****
**(compensation of differences in material density), **friction in fluids**
(compensation of impulse differences) as well as **conduction of electricity****
**(compensation of voltage differences) follow the same scheme.

Let denote a vector of the magnitude and
direction of a heat flow. We surround the point *P *under consideration with an infinitesimal space element *d**t**.
*Then is the discharge of heat energy from *d**t*
in unit time. There corresponds to it a decrease of heat content of *d**t*
which we denote, also computed per time unit, by . Then

Hence, we view our heat conductor as a rigid body, so that we can abstract it from expansion and work
performance; heat content is then its energy content. Now, every supply of heat *dQ *causes a temperature increase of *d**t*,
every reduction in heat -*dQ *a reduction in temperature. Denoting the temperature by *u*, we have

Here *c* is the **specific heat **(in
the case of a rigid solid, we need not make a difference between *c*_{v} and *c*_{p}).
The factor *dm* with *c *is due to the fact that *c* is related to the mass unit. From (9) and (10) follows

At this stage. there enters **Fourier's contribution**
which fixes the link between and *u*. It states for an **isotropic
medium**

heat flux occurs in the direction of the temperature gradient and is proportional to the strength of this gradient.

With (12), there arises from (11) the differential equation for **heat
conduction**

The proportionality factor *k* is called the** ****heat
conductivity****.**

The German physiologist August Fik (1833 - 1916) transferred Fourier's idea to **diffusion**.
The variable *u* is then the concentration of the dissolved substance in the solvent,
the material flux of the dissolved substance, *k *the **diffusivity**.
In the case of **internal friction** of an incompressible
fluid, *k* is the **kinematical viscosity**, (13)
the **Navier-Stokes equation****
of a laminar fluid**, i.e., of a fluid flowing in a fixed direction; due to the tensor character of
this process, in general, (12) does not apply here. The **electric**
case corresponds to Fourier's formulation of **Ohm's law**,
when *u *is the **voltage**,
the **specific electric current** (= current per unit
area of the conductor), *k *the **specific resistance of the conductor**.
(13) reflects the type of **Maxwell's equation **in the
case of pure Ohm conduction.

Formally speaking, this scheme also includes **Schr?inger's
equation**** of wave mechanics**, especially
in the **force-free case**,
on which we will concentrate here for the sake of brevity:

Due to the imaginary character of the constant
instead of the real *k* in (13), Equation (14) does not describe a compensation, but a **vibration
process**. For example, you see this during the transition to the case of periodicity in time, when
you set

This has the same form which is also assumed by the vibration equation (4) if we set their *u=**y*?*e*^{(-i}^{w}^{t)},
when we would have *c = **w*?/*c*?.

We will deal in Chapter III with
the so-called **linear heat conduction**
when the thermal state only depends on a single variable *x*. In order to simplify the comparison with Equations (3) and
(6a), we rewrite the differential equation in the form

Surveying this brief overview, we note the **family similarity **of
the differential equations of physics. It is obviously due to the required **invariance
with respect to rotation and displacement **of the co-ordinate system; this invariance brings with it
as second order differential operator Laplace's *D*; in the case
of the space invariance of the relativity theory, its place is taken by the corresponding four-dimensional operator of
(15). In an **an-isotropy medium**, *D**
*must be replaced by a sum of all second order differential operators with factors, which are determined by the crystal
constants. In the case of a **non-homogeneous medium****,**
these factors become functions of the space co-ordinates. We will deal with such generalized differential expressions at the start
of the next section.

As has been indicated at the start of this section, our dealing with **partial
differential equations **is due to the fact that present day physics rests on a **field
action point of view**, according to which only neighbouring space elements can interact.

**2.8 Elliptic, hyperbolic, parabolic
types. Characteristics theory**: We will restrict ourselves
to two independent variables *x* and *y*. The most general form of a linear, second
order partial differential equation is then

*A, B, ? ? ? , F *are given, sufficiently often differentiable functions of *x* and *y*.
For the immediately following material, we can even consider the more general equation

where *F** *need not be linear in

We now examine the conditions for the solubility of the following problem which heads the mathematical theory of partial differential equations, although it is in physical applications behind those actual boundary value problems, to be treated below.

Let there be prescribed along a given curve *G**
*in the *xy*-plane the values of *u* and their derivative
along the normal to the curve; we ask: Is there a solution of (2) which satisfies these initial conditions?

We start with an observation: Simultaneously with *u*, we also know
along *G*; however, we can compute from the
derivatives , whence *u* as well as its first order derivatives are known
along the curve *G**.*

We now introduce the type of abbreviations, used in the plane theory:

In terms of *r*, *s*, *t*, (2) becomes

Moreover, in general, and*, *in particular, along *G,*

Since now *p* and *q *are known along the curve *G*,
(3) and (3a) are three linear equations for the determination of *r*, *s*, *t* along it. The determinant of
the system is

Only when
this determinant does not vanish, can *r, s, t *be computed from (3), (3a), (3b). However, in general, there are at each
point *x, y *two directions *dy*:*dx *in which this is not so, whence there exist also two (real or conjugate
imaginary) families of curves, along which *D** *= 0 and
which, following Gaspard Monge (1746 - 1818), are called **characteristics***
They are shown in broken lines in Fig. 9. Along each of these characteristics, in general, the second derivatives of *r, s, t *cannot
be computed through *u*, *p, q *or, when computable, are not definite. Hence, we shall impose a necessary condition
for the solubility of the present task, namely, that *G** *has
nowhere an arc element in common with a characteristic. The manner in which we must change the task, when *G**
*coincides with a characteristic, will be discussed in 2.91
when we will discuss the method of Jean Baptiste le Rond d'Alembert (1717 - 1783).

*A geometrically clear introduction to the theory of characteristics is given, for example, in B. Baule, Vol. VI (partial differential equations) of his "Mathematics for Naturalists and Engineers", Leipzig, 1944.

When , there must indeed be a solution of the
differential equation in the neighbourhood of *G*, because the
higher derivatives can be computed in the same manner as the second ones. Consider the third derivatives:

Differentiation of (3) and (3a,b) with respect to *x* yields

We denote here by ??? on the right hand side of the first of these equations all the terms which arise during
the differentiation, which do not involve third order derivatives, i.e., only contain known quantities. However, the determinant
of this system is again *D**.* The same statement applies
to the equations arising from differentiation with respect to *y*. However, our condition is also sufficient for the
computation of the third and all higher order derivatives, whence we can expand *u* at each point of* **G**
*into a Taylor series, the coefficients of which can be determined uniquely from the boundary conditions, prescribed on *G**.*

Next, consider the equation of the characteristics

where we restrict ourselves to the neighbourhood of an arbitrarily selected point of the *xy*-plane and
distinguish the cases:

elliptic
type: The
characteristics are conjugate complex |
||

hyperbolic
type: Characteristics form two real families |
||

parabolic
type: Single real family of characteristics |

Each of these three types can be given a special normal form by introduction of new co-ordinates into the equations of the characteristics: If these equations are

the transformation yields

the normal form for the **elliptic type**

by the transformation

the normal form for the **hyperbolic type**

and by

the normal form for the **parabolic type**:

Before turning to the proof, we compare (5a), (6a) and (7a) with (7.3),
(7.6a) and (7.16),
i.e., with the two-dimensional **potential equation**,
the **equation of the vibrating string** and the linear **heat
conduction equation**. We see that the left hand sides of (5a) and (7.3) agree with the exception of
the notation for the independent variables. Similarly, (7a) and (7.16) agree with each other. In the case of (6a), one need only
introduce the simple transformation

with the inverse

(8a)

A simple transformation yields

i.e., we encounter also here on the left hand side the same terms as in (6a) and (7.6a), whence we conclude: *The
two-dimensional potential equation, the equation of the vibrating string and the linear heat conduction equation are the simplest
examples of the elliptic, hyperbolic and parabolic types.*

Starting with the discussion of the **hyperbolic case**,
we will, first of all, show that (6a) results from (2) by the transformation (6). According to (6), changing the notation of the
derivatives to subscripts of the functions *j** *and *y*
,

whence follow the second derivatives

where ??? represent further terms involving first order derivatives. Multiplication of the last three equations
by *A*, 2*B*, *C *and their addition yield on the left hand side of (2)

However, here the coefficients of are zero,
because for the family of characteristics *j** *= const

whence, substituting *dx* : *dy *into the equation of the characteristics (4),

The derivatives of *y* also satisfy this
equation, whence indeed (9) assumes the hyperbolic normal form (6a) after one shifts the factor of
in (9) to the other side of the equation.

In the **parabolic case**, because *h*
= *x, *we must set in (9)

while for *j*_{x}*, **j*_{y}*
*there applies Equation (10), whence the first term in (9) drops out. The factor of the second term reduces then, due to (11),
to *A**j*_{x}* *+*B**j*_{y}*
*and is also zero, because, due to *AC-B*?=0, the left hand side of (10) becomes a complete square, whence we can
rewrite it as (*A**j*_{x}+*B**j*_{y})?/*A*
= 0. Finally, by (11), the third term in (9) becomes

and the parabolic form has been obtained.

We need not recalculate the **elliptic case**;
obviously, it can be reduced to the hyperbolic case by the transformation

analogous to (8a), whereby both sides of the equation become real.

**2.9 Differences between hyperbolic,
elliptic and parabolic equations. The analytic character of their solutions:**** **The
task of integration, shown in Fig. 9, is only applied to
hyperbolic equations of physics; in the case of elliptic equations, it is replaced by a quite different kind of problem
formulation: The **boundary value problem**.
For the moment, we will explain this fundamental difference in general terms and point to the more detailed explanation of the
following sections.

**2.9.1 Hyperbolic Differential
Equation:**** **As its simplest model, we will use the equation of the **vibrating
string**, written in its normal form

Here, the characteristics are the straight lines *x*
= const, *h** *= const, which in Fig.
10 intersect at 45? the *x,* *y *axes. The general solution of (1) involves a function of *x*
only and one of *h** *only:

In view of the significance of *x** *and*
**h*, this is **D'Alembert's
solution**. It would be simplest if*, *say, we were given *u* along the two
characteristic segments *AB *and *AD*, so that *u* would be determined throughout the rectangle *ABCD*.
We could then compute *u *at the point *P* by going parallel to the characteristics to *P*_{1}, *P*_{2}
and transferring the values of *F*_{1}(*x*), *F*_{2}(*h*),
prescribed there, to *P*. These values propagate along the characteristics. A possibly prescribed discontinuity also
propagates inwards from ABCD. Hence, the solution need not be, within its range of validity, an **analytic
function**** **of *x* and *y*.

A function of two real variables *x*, *y* is said to be **analytic
**in a given region, if it can be represented at each point *x*_{0}*, y*_{0}
of it by power series in *x - x*_{0}* *and *y - y*_{0}* *or by equivalent formulae.

In physics, we give instead the values of *u* and along
a segment *l* (*l *= length of string), in fact,

This segment takes the place of the curve *G**
*in Fig.9, where also the two values of were
to be prescribed, and satisfies that posed condition so that it is not touched by a characteristic.

In order to transfer the conclusions, just drawn from (2), to our present problem, we must compute the functions
*F*_{1}*, F*_{2} from our present *u*(*x*, 0), *v*(*x*,0). According to
(2), this is achieved by the readily understood equations

Hence, we can also now draw the conclusion: *The given initial values, together with their discontinuities,
propagate along the characteristics*. In general, the solution *u*(*x*, *y*) is not an analytic function
of *x* and *y*. It is only determined inside the characteristics rectangle, given by the length *l *of the
string of Fig. 10.

Physically
speaking, the solution must, of course, be determined for all subsequent times, i.e., for all *y*
> 0. This indicates that, apart from the initial values, we must still prescribe certain boundary values at the ends of the
string. These are the end conditions *u*=0 at *x*=0 and *x=l*. On the boundary of the thus formed strip, we
must, just as we had to prescribe along the *x*-axis at every point two values (),
prescribe on each of the borders, parallel to the *y*-axis, **two****
**values. This corresponds to the fact that our differential equation is of second order in both variables *x* and *y,
*and the only difference is that the two values relate along the *x*-axis to the same point *x*, 0, while for *y*
> 0 they are distributed to the two **different **points
0, *y* and *l*, *y*. Only the characteristics themselves are here an exception to the two boundary values to
be prescribed, in that on them, as we have seen, **one value (***F*_{1}**
or ***F*_{2}**) is sufficient**.

We will show in 2.11 that the results, which we have derived from our string model, can be transferred to the general hyperbolic type.

**2.9.2 Elliptic Equations:****
**Here, the characteristics are **imaginary**,
whence they have no role for the problems which we will handle. These do not relate to a segment of a curve *G*,
as in Fig. 9, but to a closed region *S* of the real *xy*-plane. At its boundary, not (or
a linear combination of these) are prescribed arbitrarily. Possible discontinuities of the boundary values do **not
propagate **inside *S*, but into the imaginary; moreover, the function *u* behaves
analytically everywhere inside *S*.

These theorems are known from the **function theory**
(**two-dimensional potential theory**),
The following statement presents their proof for arbitrary linear elliptic differential equations: In the potential theory, the
analogue to D'Alembert's solution (2) is

where, however, in order that *u *is real,* f*_{2 }= *f*^{*}_{1},
i.e., *f*_{2 }must be the conjugate of *f*_{1},**,.
whence we can write

where *f* is an arbitrary analytic function of the complex variable *z = x + iy. *However, this
general integral of the equation *D**u = *0 does not
assist us (in each case not directly) to solve generally our boundary value problem.

**We will use throughout the notation *f*^{*}
instead of , used in mathematics, because we want to reserve
this notation for purposes involving time. denote, as usual,
real and imaginary parts.

**2.9.3 Parabolic differential equation:****
**Here, the two families of characteristics become a single one. This takes place in the normal form, especially during
linear heat conduction parallel to the *x*-axis. As on the characteristics
of the hyperbolic differential equation, one must prescribe on it a single boundary condition. One can extract this also directly
from the form (7.16) of the heat conduction
equation: It determines directly when the dependence of *u*
on *x *is known. The same yields physical insight. The thermal behaviour of a bar of length *l* is for its entire
future determined, apart from suitable conditions at its ends (the surface of the bar, when heat is to flow only in the *x*-direction,
must be imagined to be adiabatically protected).

We will see in 2.12
that at an arbitrary, possibly even discontinuous initial temperature of the bar, its temperature distribution for *y* >
0 becomes an **analytic function**** **of *x*
and *y*. Hence, so far, the parabolic type joins the boundary value problem of the elliptic type. However, this task does
now not relate to a closed domain, but, as in the case of the hyperbolic type, to a strip, i.e., a region with an open end. Thus,
the parabolic type lies between the elliptic and hyperbolic type.

**2.10 Green's Theorem and Function for
Linear, especially Elliptic Differential Equations:**** **Equation
(2.8.1) shows the general form of a linear second
order differential equation; in order to join its three types, we avoid for the present the reduction to its normal form.

**2.10.1 Definition of the adjoint
diffential expression:**** **To start with, we introduce the apparently rather
formal concept of the differential form *M*(*v*), **adjoint**
to *L*(*u*). It is defined by the demand: The expression *vL*(*u*) - *uM*(*v*) should
be, in general, **integrable**,
or, as we could also say, a kind of **divergence*****.
In fact, we demand that

The objective is to determine the expressions of *M* and *X,Y *as* *functions of* v,*
of *u,* *v* , respectively.

*** Actually, the operation of divergence is only defined for a* ***vector**.
Since, as will follow from (5), *X* and *Y* are not vector components, we speak here of **a
kind of ****convergence**.

Obviously, *X* and *Y* are only defined apart from quantities *X*_{0},*Y*_{0},
the divergence of which vanishes, whence one can still change the expressions in (5) by
with *X *and with *Y*, where *F**
*can be* *an arbitrary function of *x*, *y *as well as of *u*, *v*.

The following identities serve this purpose:

Here, ??? indicates that there apply instead of (2) and (3) also corresponding equations with *y* instead
of *x *and* C*,* E *instead of *A*, *D*, and also that instead of the right hand side of (2a)
also the symmetrical expression, obtained by exchange of *x* and *y, *can be used. After simple
manipulations, we find /

It is seen that the relationship between *L *and *M* is **mutual**:
*L*(*v*) is also the **adjoint** of *M*(*u*).

For mathematical physics, special importance attaches to those differential expressions for which *L*(*u*)
= *M*(*u*). They are said to be **self-adjoint**.
A comparison of (4) and (6.1) yields readily the **condition
for self-adjointness**

**2.10.2 Normal Form of Green's
Theorem, especially for Elliptic Equations:**** **Consider now in the
*x*, *y *plane a region *S*, bounded by *C*, and integrate (1) over *S*. Denote the area
element of *S* by *d**s* and the line element of *S*
by *ds*; let the sense of direction of *C *be the same as from +*x *to +*y *(Fig. 11).

Gauss' theorem yields

We know that Gauss' Theorem applied to a

two-dimensional vector with components *X*, *Y *is

or in co-ordinates

We have transferred this in (7) as a formal rule to our pseudo-vector *X*, *Y*.

Equation (7) is the general formulation of **Green's theorem **which
is valid for the three types under consideration. If we set here *A*=*C*=1, it is specialized for the elliptic type
and is its normal form. Then

We recognize here the generalization of Green's theorem of the potential theory

which (7a) becomes for *D* = *E* = 0. (It is of no consequence here that in potential theory *F*
= 0.)

We will meet a second form of Green's theorem in Exercise 2.2.

In particular, if *u* and *v *satisfy inside *S*

when there vanish the left hand sides of (7), (7a) and therefore also the line integrals along *C* on the
right hand side; depending on whether one starts from (7) or (7a), one obtains

However, this is only true if *u* and *v *are continuous throughout *S *together with their
derivatives involved. In contrast, if *v* has at the point *Q* with co-ordinates *x* = *x*,
*y* = *h** *a discontinuity, we must, as always in
Green's theorem, exclude it from the domain of integration. Thus, we surround *Q* by a curve *K*, for which a circle
with arbitrarily small radius is best. Then, the integration in (7b,c) is extended to both boundaries:

where the direction in the integrals must be opposite and the direction of *n* is on both boundaries
outwards.

If we use, as is convenient, for *K *(7c) and for *C* (7b), we obtain

**2.10.3 Definition of Unit Source and
Principal Solution:**** **We will assume that the discontinuity of *v* at *Q
*is a **unit source**. This means that the strength *q
*of an arbitrary source *Q* is correctly defined by the outward gradient of its field *v*. Denoting the distance
from *Q* by *r**, *we write

where *K* is the same as before. By assuming that in the immediate neighbourhood of the source *v *depends
markedly only on *r**, *we also have

Thus, a unit source is given by

We write for this for arbitrary *r*:

in that we understand by *U *and *V *analytic
functions of *x*, *y *as well as of *x ,h*, where,
moreover, *U* is to take on the value *p*/2 during the
transition .

We call a function of this type a **principal solution**
of its differential equation *M*(*v*) = 0. In the same way, we will speak of the principal solution of the adjoint
equation *L*(*u*) = 0. Since it also should correspond to a unit source, it has the same form as (10) whenever the
meaning of *U *and *V* is changed. Here too, there is no objection to assuming that *U* and *V *are
analytic as long as the coefficients *D, E, F *of the differential equation are analytic. In the case of the potential
equation *D**u=*0, the thus defined principal solution
with the logarithmic potential is well known. If we omit here the unimportant additive constant, we have not only in the
neighbourhood of *Q*, but for every *r*

**2.10.4 The Analytic Character of the
Solution of an Elliptic Differential Equation:**** **After these preparations, we
return to Equation (8). Substituting (10) into (8), we see that only the term with

makes* *a finite contribution on integration over *K*, while all other terms on the left hand side
of (8) like *r* log* **r**
*or of higher order vanish. Since *u* is continuous at *Q* and the circumference of *K *equals *2**pr*_{0},
we obtain for the left hand side under consideration

whence , by (8),

In this formula, the dependence on *x* and* **h*
is of special interest;* *it is due to the factors and,
by (10), is given in analytic form. If *Q *lies **inside**,
i.e., not on the boundary of *S*, as we have assumed, log *r**
*is also a regular analytic function, since during the integration the point *P* = *x, y *is restricted to *C
*and does not coincide with *Q*. Hence, also *u*_{Q}* = u*(*x,
h*) is an analytic function of its argument inside *S*. The analytic character of the boundary
values is therefore not important, as altogether the dependence
of the integrand on *x *and *y *is removed by the integration with respect to *ds*. Moreover, possible
discontinuities of these boundary values are averaged. *Discontinuities whatsoever do not
propagate into the region inside S*. (Recall that the characteristics are imagined!) Thus, we have
proved the statements in 2.9.2.

In the case of a self-adjoint differential equation, one has *D* = *E =* 0 according to (6) which
is specialized into the normal form. Then, one obtains from (11), using the form (7c) of the line
integral,

Especially, in the case of the potential equation with (10a) for *v*, one has:

**2.10.5 Principal Solution in an
Arbitrary Number of Dimensions:**** **We insert here comments
regarding principal solutions in more (or less) than two dimensions, where, for the sake of brevity, we shall confine ourselves to
the potential equation.

In three dimensions, we have, analogous to (9b), (*r* = distance from the source *Q*, 4*p**
r*? = surface of sphere)

In essence, this is the so-called **Newtonian
potential**.

The denominator 4*p** *corresponds
exactly to the rational units of electro-dynamics. (Vol. III)

In four dimensions, we are concerned with the differential equation (7.5).
Here, with *R* the distance from *Q *and 2*p** *?*R*?
the surface of the hyper-sphere (cf. Volume III regarding the special theory of relativity)

The following table displays the continued weakening of the tendency to become infinite with decreasing
dimension number and leads in the case of one dimension to **continuity**
of *v* at the location of the unit source. Indeed, the potential equation in one dimension is *d*?*v*/*dx*?
= 0 and yields *dv*/*dx* = *C*, where the constant has different values *C*_{1}*, C*_{2}*
*to the right and to the left of the source. This follows from the condition of the unit source according to which one must
have *C*_{1 }*- C*_{2 }*= *1. The discontinuity has gone from *v* into the gradient of
*v*. (Exercise 2.3).

**2.10.6 Definition of Green's Function
for Self-adjoint Equations:**** **We will now discuss the
boundary value problem of 2.9.2. This is in
no way solved by our principal solution. Dealing with the simplest case of self-adjointness, we consider (11a).
In order to compute with *u the *assistance at the point *Q*, we would have to know the values of
along *C*, while we are given in the boundary value problem either .
We ask now: *How do we have to change the principal solution v so that the term with
(u, respectively) will vanish from *(11a)*.*
We shall call the altered function of the two pairs of variables *x*, *y *and *x,
h** *the* *Green function and denote it by *G*(*P,Q*). It
must meet the three conditions:

Conditions a) and c) are the same as for the earlier *v, *but b) has been added. If it is met, we then
have by substitution of this *G* for *v* into
(11a)

Thus, the boundary value problem has been solved in both cases (However,**
**the derivation of *G *becomes, due to the additional condition b), another boundary value problem, which is
simpler than the general boundary value problem for *u *and can in special cases, as we will see later on, be solved in an
elegant manner by the **reflection procedure**.
On the other hand, *G* is not, as we required, a function, regular inside *C*, but, as *v*, a function with a
prescribed unit source.

Equation (12) reduces the unsolved problem in (11a) to simple quadrature. The same role as here is taken by
Green's function in the general theory of integral equations, where it is called the **solving
kernel**.

We mention yet another interesting property of *G*, which follows from the conditions a), b). c), i.e.,
the **reciprocity relation**

It indicates that the source and object point, so to say, the **cause
**and **reaction****, **can
be interchanged.

In order to prove this statement, we set in (7a)

The point is called the **integration
point**. Since, for *I = P, *the function *u*, for *I = Q, *the function *v
*becomes infinite, these two points must be excluded from the integration by infinitesimal circles *K*_{P}*
*and *K*_{Q}. The region *S *between these circles and the boundary curve *C* yields 0,
according to the condition a) on the left hand side of 7a). Thus, there only remain the boundary integrals over the circles *K*_{P}*
*and *K*_{Q}, which together, by c), yield

Since also this contribution must vanish, the reciprocity theorem has been proved.

Equation (12) is the solution of the boundary value problem for the **homogeneous
**equation *L*(*u*)=0. We confront it with the solution of the boundary value problem
for the **non-homogeneous **differential equation

in which *r*(*x, y*) is a given,
arbitrary function which together with its first and second derivatives is continuous. If we start again from (7a) and set there *v*
= *G*(*P,Q*), the first term on the left hand side becomes

and is added to the contribution (12), which arises from the boundary *C*. Hence,
one has instead of (12)

and, if not *u* but is given on *C,*

These representations can be applied to every self-adjoint differential expression in the normal form *L*(*u*)
= *D*(*u*) + *Fu *and, especially also, to the
ordinary vibration equation (*F = k*? = const) as well as to the potential equation (*F* = 0).

In the case of a **non-self-adjoint****
**differential expression *L*(*u*), the representations (12) and (13a,b) are
maintained. However, Green's function *G* must then, as can be seen from (7a), satisfy in the variables *x*, *y *the
equation *M*(*G*) = 0, adjoined to *L*; moreover, as can be seen from the same equation, one must somewhat
modify the second condition b). In fact, we have instead of a) and b)

Condition c) remains the same, while the reciprocity theorem becomes

where *H* is the Green function for the adjoint equation of *M* = 0, which therefore satisfies in
the variables of *Q* the equation *L*(*H*) = 0.

**2.11 Riemann's Integration of the
Hyperbolic differential Equation:**** **(2.8.1)
yields the normal form of a linear, hyperbolic, second order differential equation, if one sets there *A* = *C = *0,
*B* = 1/2:

By (10.4), its adjoint equation is

At the same time, (10.5) yields

Substitute (1) - (3) into Green's theorem

In order to obtain a strict integration of the hydro-dynamic equations, Riemann selected as *S *for the
equivalent partial differential equation the triangle *PP*_{1}*P*_{2}* *of Fig. 12, formed by
the two characteristic segments *PP*_{1}, *PP*_{2} and the arc *P*_{1}*P*_{2}
of the boundary curve *G*. Let the values of be
prescribed on *G*;* *obviously they yield also the values
of . This curve *G**
*must satisfy the condition stated in 2.8
that it must not be touched by a characteristic. Now, Riemann determines the function *u *in (4) by the conditions

Note here that:

1. It would not be possible to pose instead of (5b) a discontinuity demand of the type (10.10),
because a hyperbolic differential equation does not have isolated singularities; these propagate along the characteristics. This
is the reason that we avoid here also the earlier terms **principal solution**
or** ****Green's function **for *v *and
rather call it the **characteristic function**.

2. Conditions (5c) prescribe for *v *at the two characteristics *x**=**x*
and *y*=*h** *only** **a** ****single****
**boundary condition, while we had to state for *u* along the curve *G***
****two**** **conditions. This
corresponds to the comment which was already made in 2.9
in the case of the vibrating string, namely that the characteristics with respect to the boundary value
problem to be stated occupy an exceptional position. It we call the integration task in the case of a characteristic two corner
shape a **boundary value problem****
of the second kind**, in contrast to the **boundary value
problem of the first kind**, which relates to a general curve *G**,
*we can say that Riemann's method involves a reduction of the boundary value problem of the first kind to a much simpler
boundary value problem of the second kind.

According to (4), we substitute in the condition (5a) *M*(*v*) = 0 and simultaneously set *L*(*u*)
= 0, whence

In the last integral, which, since cos (*n*,*y*) = 0, relates only to *X*, we change the
term with by integration by parts:

Combination of the result with the other terms yields

Correspondingly, one finds for the central integral in (6), which, due to cos (*n, x*) = 0, relates only
to *Y *and in which cos (*n, y*) = -1 (outward normal),

In (6a) and (6b), the two integrals on the right hand side vanish due to Conditions (5c), whence one obtains from (6), taking into consideration (5b),

The value of *u* at the arbitrarily selected point *P* is here represented by the
values of *u *entering into *X* and *Y *and their first derivatives on *G**
*(also belong to these values). Hence, we can state: Our
boundary value problem of the **first kind** for *u *is
reduced by (7) to a knowledge of *v, *i.e., to a boundary value problem for *v *of the **second
kind**, defined by (5a, b, c).

The determination of *v *encounters now no difficulties, especially in the case of the hydrodynamic
problem considered by Riemann. In this example,

Condition (5c) demands

Both of these demands and, at the same time, (5b) are met, if one sets

Finally, in order to meet also (5a), Riemann extends
(9) for *v* to

where

is the **hyper-geometric series**.
We will report in 4.24.4 on
the function theoretical properties of this series and prove in the Appendix
4.2 that our formulation (10) for *v *is indeed that demanded by (5a) and that it satisfies the
equation *M*(*v*) = 0, used above. We only need note here that for the characteristic *x* =* **x**
*or* y = **h**, *where *z *= 0 and
therefore *F* = 1, (10) becomes identical with (9). By the presentation (10) of *v* and the integral representation
(7) of *u*, our hyperbolic boundary value problem is completely solved in this example.

**2.12 Green's Theorem in Heat
Conduction. The Principal Solution of the Heat Conduction Equation****:****
**The differential equation of heat conduction (7.14)

**is not self-adjoint**. Its adjoint equation is

You see this from 2.8 by setting

at the same time, you obtain from (10.5)

By (1),
(2), (3), integration over the interior and its border of abounded region yields, as in the case of
elliptic and hyperbolic equations, from (10.2) **Green's theorem for heat conduction**.
Due to the incommensurable meaning of *x *and *y* (*x
= *space , *y *= time measurement), one need not consider here integration domains with a curved boundary, but merely
such as are shown in Fig. 13 with boundaries parallel to the *x*- and *y*-axes. In particular, one has along *AB
ds = dx*, *dn = - dy, *cos (*n,x*) = 0, cos (*n,y*) = -1, i.e.,

The same applies to the side *CD*, parallel to the *x*-axis, where the sign of *ds* as well
as that of cos (*n*, *y*) inverts. Hence Fig.13 yields for the two sides *BC *and *AD*, parallel to
the *y*-axis

With the values of *L. M, X, Y*, given by (1), (2), (3), one obtains Green's theorem in the form

where now the first integral on the right hand side is taken over the rectangle's sides parallel to the *x*-axis,
the second one over those parallel to the *y*-axis.

However, Equation (4) represents also Green's theorem of two- or three-dimensional heat conduction, if we replace on both sides of the equation

and, moreover, on the right hand side

and naturally on the left hand side

Hence, in the three-dimensional case, we must integrate on the left hand side over a four-dimensional cylinder,
the base of which is the three-dimensional heat conductor and the generators of which are parallel to the time axis; the
integration for the first term on the right hand side - indicated in (5b) by *dy* and now must be replaced by *dyd**s*
*= kdtd**s* extends over the three-dimensional mantle of
this cylinder, which is directed on the surface elements *d**s**
*of the heat conductor parallel to the time axis.

Before we start the evaluation of this general formula, we must make a decision regarding the selection of the
analogue to the **principal solution**. We will see that
the place of the **unit source **is taken by the **heat
pole ****of strength 1**.

To start with, we return to linear heat conduction and the differential equation *L *= 0; the transition
to the adjoint equation *M*** = **0 and to 2 and 3 dimensions will then become simpler.

Extend the linear heat conductor in both directions to infinity; let its temperature at *t* = 0 be given
as a function of *x*:

We now represent, according to (4.8),
the function *f*(*x*) by a Fourier integral

In order to obtain a solution of (1), we need only complement exp[*i**w*(*x
- **x*)] into the product

Substitution into (1) yields

Due to the obviously necessary condition *j*(0)
= 1, one has *C* = 1, i.e., we must replace in (6)

While apparently the Fourier integral thus becomes more complicated, it actually becomes much simpler. In fact,
the function *f*(*x*) in (6) had to vanish at infinity **sufficiently
strongly**** **for the integral with respect to *x*
to converge and this integral had to be evaluated ahead of the much simpler one with respect to *w*,
because otherwise the latter would yield a completely senseless and indefinite result. In contrast, the sequence of the
integrations is now unconditionally invertible and the function *f*(*x*) is far reachingly unconstrained with regard
to its behaviour at infinity. The factor , added in (7), acts
for all *y* > 0 as a convergence factor.****

Cf. the author's thesis K?igsberg, 1891: **The Arbitrary Functions in Mathematical Physics**,
where, instead of (6), there is considered, in general, the limit of a Fourier
integral with a convergence factor. The function *f*(*x*) may then have also **infinitely many maxima and
minima** or arbitrary discontinuities.

Hence, with *y = kt, *(6) and (7) yield

We write for the exponent in the second integral *-**aw*?
+ *bw** *and complement it into a complete square:

hen, since *a* = *kt*, *b*
=* i*(*x - **x*)*, *we obtain with the
substitution *p*=*w-b*/2*a*

You know from the elementary calculus the **Laplace
integral**

Hence the right hand side of (9), which we will denote by *U*, becomes

and (8)

This shows us: *The initial temperature at the point x = **x**
spreads, independently of the initial temperature at all other points, through space in agreement with the law, contained in U.*
(The reason for this are: The linearity of the differential equation and hence the super-position of its solutions.) For ,
where *u*(*x*, *t*) is to become *f*(*x*), (10a) says

Hence, we conclude that *U *has the **character of a ***d***
-function**, i.e., according to 1.6.,
*U *vanishes in the limit for all values of and,
however, becomes infinite in such a way that

(Naturally, these properties of *U *can also be discovered readily from the representation (10) of *U*.)
Thus, if for the moment we do not distinguish between heat energy and temperature, we can say: *U *describes the space-time
behaviour of a **unit-heat-source **or
of a **heat-pole of strength 1**.

Depending on how we generalize the position of the heat pole (initial state at *t* = *t*)
or specialize (heat pole at *x** *= 0), we obtain from
(10)

Ahead of a discussion of the deeper meaning of these formulae, we will extend them to 2 and 3 dimensions.

We have already pointed out in 1.4 the possibility of extending the Fourier integral into a fourfold or six fold integral. This is done step by step, writing at first (6) as

and then

the combination of (11) and (11a) yields

The same process, which led from (6) to (10a), yields in the present two-dimensional case from (11b)

with *U *as the product of two factors of the form (10):

In the same way, one obtains *U *in the three-dimensional case as the product of three factors of the
form (10):

The expressions (13) and (14) represent **unit
poles in the plane and space****. **Like (10), they display a link between **heat
conduction and probability**.

At first, we compare (10d) with **Gauss'
error law**

Here *dW* is the probability of an error occurring between *x *and *x *+ *dx *during
a measuring action, the accuracy of which is given by the **precision
factor*** **a*.
It is represented here by (4*kt*)^{-1}; **infinite precision **means
here *t* = 0, i.e., absolute concentration of heat at the point *x* = 0; **decreasing
precision**** **corresponds to increasing *t*. The known pattern of the bell
curves in Fig. 14 for decreasing *a** *represents at the
same time the behaviour of *U* during increasing *t*. Our *U* in (10d) is exactly equal to the **probability
density**** ***dW*/*dx.*

Correspondingly, you compare
(13) with measurements at a point in the *xy* - plane with exact location at *x, h*,
as well as (14) with the measurement of a space point with the exact location *x, h, z*.
In both cases, as before, the precision factor is *a**=*(4*kt*)^{-1}.
At this stage, the conclusion suggests itself that the physical reason of heat conduction does not have a **dynamic****,**but
a** ****statistical**** **character.
This becomes clear in the **kinetic gas theory**,
which would be better called **statistical gas theory****.
**This is also linked to the following fact which we will explain, for example, by the situation in space demonstrated by
Fig.. 14. For t = 0, the entire heat energy is located at the point *x,h,z*.
However, already after an arbitrarily short time, we find at a distant point *x*, *y*, *z *some temperature *U
*other than zero. Hence, heat propagates at an infinite velocity. Dynamically speaking, this is impossible, because here no
velocity may exceed the limit *c*.

We know from 2.7
that diffusion, electric conduction and viscosity like heat conduction obey the same differential equation. Also, in these
subjects, a statistical approach suggests itself. In fact, diffusion is based on the **Brownian
motion** of the individual molecules, dissolved in the solvent, the statistical origin of which
has been ascertained theoretically and experimentally. Moreover, the electron theory of metals demonstrates that during electric
conduction electrons interact with the atoms of the metal lattice, diffuse through it, etc.

For all these applications, we have now obtained with the function *U* in (10), (13), (14) the **principal
solution of their differential equation ***L*(*u*) = 0. Let us convert it now into the
principal solution *V* of the **adjoint equation****
***M*(*v*) = 0. As the comparison of (1) and (2) shows, this involves simply a change of sign of *y*=*kt*,
where we will then also invert the sign of *y*_{0}= *k**t*,
in order that the heat pole will again lie at *x *= *x*, *t
*= *t**. *This function*
V*,* *written* *following (10c), becomes

Hence, in essence, *V* becomes singular for *t *= *t**
*and can only be employed for the past of *t*, i.e. *t*<*t*,
in contrast to the principal solution *U*, which, according to (10), behaves regularly only for the future of *t*,
i.e., for *t >**t*.

Returning now to our Green's theorem (4), we will set there *v* = *V *and at the same time subject
*u *to the equation *L*(*u*) = 0 to obtain

Here, the two integrals must be extended over the pairs of sides of the rectangle in Fig. 13, the first over the two horizontal, the second over the two vertical sides.

Since *V *like *U *is a* **d**
***-function**, the first integral, extended over the side *t* = *t**
*, yields -*u*_{Q}. Thus, (17) yields, after subdividing the second integral into its parts for the
bar ends *x*_{0}*, x*_{1}* *and indicating
this by ,

Here, *V*_{0} is given by (16) for *t *= 0.

Since the point *Q* of the source can be placed at **any
point** *x = **x*, *t *=
*t*, this representation of *u* is always valid. However,
it does not yet solve the boundary value problem in 2.9.3,
since it assumes, apart from the initial values of *u*, also a knowledge of the boundary values of given
at the bar's ends, while we have seen above that only either
may be prescribed. In order to arrive at the solution of the boundary value problem, we must replace in (18) *V* by **Green's
function **** G**, which for given

Apparently, the preceding material is transferred smoothly from the one-dimensional case to two and three
dimensions. As above in connection with (5a,b), one needs only to extend the integration in the first integral of (18) over the
base, the second one over the mantle of the three- or four-dimensional space-time-cylinder. However, the construction of Green's
function *G* employing reflection only succeeds in exceptional cases in two- or three- dimensional problems (cf.(3.17)).

On the other hand, already Equation (18) (i.e., written with *V*, and not *G*) is sufficient for
controlling the analytic character of *u*, discussed at the end of 2.9.
In fact, the co-ordinates *x, t** *(*x,h,t**
*or *x,h,z,t,** *respectively, occur on the right hand
side only in the principal solution *V*, i.e., only analytically. Hence, the solutions of our parabolic equations, as in
the case of the elliptic equations, are **analytic functions **inside
the region of their independent variables. However, this domain is not, closed on all sides, as in the elliptic type, but is
upwards an open strip, as has been emphasized in 2.9. In this last respect, the parabolic boundary value problem resembles the
hyperbolic type.