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 4p 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 x1, x2, x3, as fourth co-ordinate x4 (or x0) = 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 dt. Then is the discharge of heat energy from dt in unit time. There corresponds to it a decrease of heat content of dt 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 dt, 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 cv and cp). 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
Then (14) yields
This has the same form which is also assumed by the vibration equation (4) if we set their u=y?e(-iwt), 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
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
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, 2B, 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 jx, jy there applies Equation (10), whence the first term in (9) drops out. The factor of the second term reduces then, due to (11), to Ajx +Bjy 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 (Ajx+Bjy)?/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 P1, P2 and transferring the values of F1(x), F2(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 x0, y0 of it by power series in x - x0 and y - y0 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 F1, F2 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 (F1 or F2) 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, f2 = f*1, i.e., f2 must be the conjugate of f1,**,. 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 Du = 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 X0,Y0, 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 ds 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 Du=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 2pr0, 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 uQ = 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, 4p r? = surface of sphere)
In essence, this is the so-called Newtonian potential.
The denominator 4p 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 2p ?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 C1, C2 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 C1 - C2 = 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 KP and KQ. 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 KP and KQ, 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 PP1P2 of Fig. 12, formed by the two characteristic segments PP1, PP2 and the arc P1P2 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
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 dyds = kdtds extends over the three-dimensional mantle of this cylinder, which is directed on the surface elements ds 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[iw(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/2a
You know from the elementary calculus the Laplace integral
Hence the right hand side of (9), which we will denote by U, becomes
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
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 (4kt)-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=(4kt)-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 y0= kt, 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 -uQ. Thus, (17) yields, after subdividing the second integral into its parts for the bar ends x0, x1 and indicating this by ,
Here, V0 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 u satisfies at the bar's ends the condition G = 0 and thereby causes the term with in (18) to vanish. We will see in Chapter III how to construct G out of V in the case of a linear heat conductor by means of reflection. An application of (18) to laminar fluid friction is the task of Exercise 2.4.
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.