Freiling G., Yurko V. Lectures on Differential Equations of Mathematical Physics: A First Course

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2 G.

Freiling and V. Yurko

1. Equation

of a vibrating string.

Consider a stretched string which lies along the x -axis, and suppose for simplicity that

the oscillations take place in a plane, and that all points of the string move perpendicularly

to the x -axis. Then the oscillation process can be described by a function u(x,t), which

characterizes the vertical displacement of the string at the moment t. Suppose that the string

is homogeneous, inextensible, its thickness is constant, and it does not resist bending. We

consider only ”small” oscillations with (u

)

<< 1. Then the function u(x,t) satisﬁes the

following equation (see [1, p.23], [2, p.56]):

= a

+ f (x,t), (1.1.4)

where a > 0 is the wave velocity, and f (x,t) represents a known external force. More

general equations for one-dimensional oscillations have the form

ρ(x)u

= (k(x)u

)

−q(x)u + f (x,t), (1.1.5)

where ρ(x),k(x) and q(x) are speciﬁed by properties of the medium. For example, the

equation of longitudinal oscillations of a rod has the form (1.1.5), where u(x,t) is the

displacement of the point x from the equilibrium at the moment t , ρ(x) is the density of

the rod, k(x) is the elasticity coefﬁcient, q(x) = 0 and f (x,t) is the density of external

forces per unit length [1, p.27].

2. Equation of transverse oscillations of a membrane.

Consider a thin homogeneous inextensible membrane. Suppose that the membrane does not

resist bending, and that all its points move perpendicularly to the (x, y) -plane. Let u(x, y,t)

be the vertical displacement of the membrane at the moment t. We consider ”small” trans-

verse oscillations with (u

)

+(u

)

<< 1. Then the function u(x,y,t) satisﬁes the follow-

ing equation (see [1, p.31]):

= a

+ u

) + f (x,y,t),

where a > 0 , and f (x,y,t) represents a known external force.

3. Wave equation.

Multidimensional oscillation processes in the space of n spatial variables

x = (x

,. .. ,x

) are described by the so-called wave equation

= a

∆u + f (x,t), (1.1.6)

where

∆u :=

∑

k=1

∂

∂x

Introduction 3

is the

Laplace operator (or Laplacian), a is the speed of wave propagation, and f (x,t)

represents external forces. For n = 1 equation (1.1.6) coincides with the equation of a vi-

brating string (1.1.4). Therefore, equation (1.1.4) is also called the one-dimensional wave

equation. For n = 2 equation (1.1.6) describes, for example, two-dimensional oscillations

of a membrane [1, p.31]. Propagation of acoustic waves in the space of three spatial vari-

ables is described by the wave equation for n = 3. More general wave equations have the

form

ρ(x)u

= div(k(x)gradu) −q(x)u + f (x,t), (1.1.7)

where

div(k(x)gradu) :=

∑

k=1

∂

∂x

k(x)

∂u

∂x

where

the

functions ρ(x),k(x) and q(x) describe the properties of the medium.

4. Telegrapher’s equation.

An important particular case of the one-dimensional wave equation is the so-called telegra-

pher’s equation. Consider the system of differential equations

+Cv

+ Gv = 0,

+ Li

+ Ri = 0,

)

(1.1.8)

which governs the transmission of signals in telegraph lines. Here v denotes the voltage

and i denotes the current; L,C, R and G represent inductance, capacity, resistance and

conductivity, respectively (per unit length) [2, p.88]. After elimination of i (or v ) from

(1.1.8) we arrive at the equation

= a

+ 2b

+ c

w, (1.1.9)

where w = i or v (the same equation is obtained for the voltage and the current), and

= LC, 2b

= RC + GL, c

= GR.

Equation (1.1.9) is called the telegrapher’s equation. The replacement

w = u exp

−

reduces

(1.1.9)

to the simpler form

= a

+ Bu, (1.1.10)

where

a =

√

, B =

−a

note that B = 0 if and only if RC = GL (Heaviside’s condition) which corresponds to

lines without distortion.

4 G.

Freiling and V. Yurko

5. Equation

of oscillations of a rod.

In a standard course devoted to partial differential equations of mathematical physics we

usually deal with second order equations. However, there are a number of problems of

mathematical physics which lead to equations of higher order. As an example we consider

the problem of free oscillations of a thin rectangular rod when one of its ends is ﬁxed. In

this case the oscillation process is described by the following fourth order equation (see [1,

p.143]):

∂

∂t

+ a

∂

∂x

= 0.

This

equation

is also used for studying the stability of rotating shafts, vibrations of ships

and for other problems of mathematical physics (see [1, Ch.2]).

Many other equations of science and engineering are of order higher than two. In plane

elasticity one meets the fourth order biharmonic equation ∇

u = 0 , in shell analysis eight

order equations and in dynamics of three-dimensional travelling strings sixth order equa-

tions - below we shall not study higher order equations in detail.

6. Heat equation.

Consider a long slender homogeneous rod of uniform cross sections which lies along the

x - axis. The lateral surface of the bar is insulated. The propagation of heat along the bar is

described by the equation [1, p.180]

= a

+ f (x,t), (1.1.11)

which is called the heat conduction equation or, shorter, heat equation. Here the variable t

has the signiﬁcance of time, x is the spatial variable, u(x,t) is the temperature, a > 0 is a

constant which depends on physical properties of the rod, and f (x,t) represents the density

of heat sources. Equation (1.1.11) describes also various diffusion processes, therefore it is

also called the diffusion equation. A more general one-dimensional heat equation has the

form

ρ(x)u

= (k(x)u

)

−q(x)u + f (x,t).

Propagation of heat in the space of n spatial variables x = (x

,. .. ,x

) is described by the

multidimensional heat conduction equation

= a

∆u + f (x,t), (1.1.12)

or, in the more general case,

ρ(x)u

= div(k(x)gradu) −q(x)u + f (x,t). (1.1.13)

Introduction 5

7. Diffusion

equation.

Consider a medium which is ﬁlled non-uniformly by a gas. This leads to a diffusion of the

gas from domains with higher concentrations to domains with less ones. Similar processes

take place in non-homogeneous chemical solutions.

Consider an one-dimensional diffusion process for a gas in a thin tube which lies along

the x -axis, and suppose that the diffusion coefﬁcient is a constant. Denote by u(x,t) the

concentration of the gas at the moment t. Then the function u(x,t) satisﬁes the equation

(see [1, p.184]):

= a

, a > 0.

8. Laplace equation.

Let x = (x

,. .. ,x

) be the spatial variables, and let u(x) be an unknown function. The

equation

∆u = 0 (1.1.14)

is called the Laplace equation. Many important physical and mathematical problems are

reduced to the solution of the Laplace equation. Usually the Laplace equation describes

stationary ﬁelds. For example, suppose we study a temperature distribution in some do-

main of an n -dimensional space of variables x = (x

,. .. ,x

). In the simplest case the heat

propagation is described by the equation u

= ∆u. A stationary (steady-state) temperature

distribution is a temperature ﬁeld which is independent of time. Such a state may be real-

ized if a sufﬁciently long time has elapsed from the start of heat ﬂow. Then u

= 0, and the

heat equation u

= ∆u reduces to the Laplace equation ∆u = 0.

The non-homogeneous Laplace equation

∆u = f (x) (1.1.15)

is called the Poisson equation. For example, the gravitational (electro-statical) potential

u(x), generated by a body with the mass (charge) density ρ(x), satisﬁes equation (1.1.15)

for f (x) = −4πρ(x). More general partial differential equations, which describe stationary

processes (ﬁelds), have the form

div(k(x)gradu) −q(x)u = f (x).

Other examples of problems in physics and mechanics, which lead to partial differential

equations, and the derivation of the main equations of mathematical physics one can ﬁnd,

for example, in [1]-[3].

6 G.

Freiling and V. Yurko

1.2. Classiﬁcation

of Second-Order Partial Differential Equa-

tions

1. Classiﬁcation of second-order equations with two variables

We consider quasi-linear second-order partial differential equations with two independent

variables x,y of the form:

+ 2a

+ a

+ F(x,y,u,u

) = 0. (1.2.1)

Here u(x,y) is an unknown function and a

= a

(x,y), 1 ≤ j, k ≤ 2, are given con-

tinuous functions which are not zero simultaneously. The properties of the solutions of

equation (1.2.1) depend mainly on the terms containing second derivatives, i.e. on the co-

efﬁcients a

(x,y). Let us give a classiﬁcation of equations of the form (1.2.1) with respect

to a

(x,y).

Deﬁnition 1.2.1.

1) Equation (1.2.1) is called hyperbolic if a

−a

> 0 .

2) Equation (1.2.1) is called parabolic if a

−a

= 0 .

3) Equation (1.2.1) is called elliptic if a

−a

< 0 .

The type of equation (1.2.1) is deﬁned point-wise. If equation (1.2.1) has the same type

in all points of a certain domain, then by a suitable change of variables (i.e. by changing the

coordinate system) one can achieve that the equation will be in the ”simplest” (canonical)

form. Let us make the following change of variables:

ξ = ϕ(x, y), η = ψ(x,y), (1.2.2)

where ϕ and ψ are twice continuously differentiable functions, and

6= 0. (1.2.3)

We note that (1.2.3) is the condition for the local invertibility of the transformation (1.2.2).

Then

= u

+ u

, u

= u

+ u

= u

ξξ

+ 2u

ξη

+ u

ηη

+ u

= u

ξξ

+ u

ξη

(ϕ

+ ϕ

) + u

ηη

+ u

= u

ξξ

+ 2u

ξη

+ u

ηη

+ u

Substituting this into (1.2.1) we obtain

˜a

ξξ

+ 2 ˜a

ξη

+ ˜a

ηη

F(ξ, η,u, u

) = 0, (1.2.4)

where

˜a

= a

+ 2a

+ a

˜a

= a

+ a

(ϕ

+ ψ

) + a

˜a

= a

+ 2a

+ a











(1.2.5)

Introduction 7

Remark 1.2.1. One

can easily check that

˜a

− ˜a

˜a

= (a

−a

)Ω

where Ω := ϕ

−ψ

6= 0. This means the invariance of the type with respect to the

replacement (1.2.2).

We choose the replacement (1.2.2) such that equation (1.2.4) takes the ”simplest”

(canonical) form. In (1.2.2) we have two arbitrary functions ϕ and ψ. Let us show that

they can be chosen such that one of the following conditions is fulﬁlled:

1) ˜a

= ˜a

= 0;

2) ˜a

= ˜a

= 0 (or symmetrically ˜a

= ˜a

= 0 );

3) ˜a

= ˜a

, ˜a

= 0.

Let for deﬁniteness a

6= 0. Otherwise, either a

6= 0 (and then interchanging places of

x and y, we get an equation in which a

6= 0 ), or a

= a

= 0 , a

6= 0 (and then

equation (1.2.1) already has the desired form). So, let a

6= 0. We consider the following

ordinary differential equation

−2

+ a

= 0

, (1.2.6)

which is called the characteristic equation for (1.2.1). The characteristic equation can be

also written in the symmetrical form:

(dy)

−2a

dydx + a

(dx)

= 0. (1.2.6

)

Solutions of the characteristic equation are called the characteristics (or characteristic

curves) of (1.2.1). Equation (1.2.6) is equivalent to the following two equations

−a

. (1.2.6

)

According

Deﬁnition 1.2.1, the sign of the expression a

−a

deﬁnes the type of

equation (1.2.1).

Lemma 1.2.1. If ϕ(x,y) = C is a general integral ( or solution surface ) of equation

(1.2.6) , then

+ 2a

+ a

= 0. (1.2.7)

Proof. Take a point (x

) from our domain. For deﬁniteness let ϕ

) 6= 0,

and let ϕ(x

) = C

. Consider the integral curve y = y(x,C

) of equation (1.2.6) passing

through the point (x

). Clearly, y(x

) = y

. Since ϕ(x,y) = C is a general integral,

we have on the integral curve:

+ ϕ

= 0;

hence,

= −

8 G.

Freiling and V. Yurko

Then

0 = a

−2

+ a

+ 2a

+ a

y=y(x,C

)

particular

, taking x = x

, we get (1.2.7) at the point (x

). By virtue of the arbitrari-

ness of (x

), the lemma is proved. 2

Case 1: a

−a

> 0 - hyperbolic type.

In this case there are two different characteristics of equation (1.2.6) passing through each

point of the domain. Let ϕ(x,y) = C and ψ(x,y) = C be the general integrals of equations

( 1.2.6

), respectively. These integrals give us two families of characteristics of equation

(1.2.6). We note that (1.2.3) holds, since

= −

−a

= −

−

−a

and

consequently

(1.2.1)

we make the substitution (1.2.2), where ϕ and ψ are taken from the general

integrals of equations ( 1.2.6

), respectively. It follows from (1.2.5) and Lemma 1.2.1 that

˜a

= ˜a

= 0. Therefore, equation (1.2.4) is reduced to the form

ξη

= Φ(ξ, η,u, u

), (1.2.8)

where Φ = −

˜a

This is the canonical form of equation (1.2.1) for the hyperbolic case.

There also exists another canonical form of hyperbolic equations. Put

ξ = α + β, η = α −β,

i.e.

α =

ξ + η

, β =

ξ −η

Then

+ u

, u

−u

, u

ξη

αα

−u

ββ

and

consequently

, equation (1.2.8) takes the form

αα

−u

ββ

= Φ

(α,β,u,u

). (1.2.9)

Introduction 9

This is

the second canonical form of hyperbolic equations. The simplest examples of

an hyperbolic equation are the equation of a vibrating string u

= u

and the more general

equation (1.1.4).

Case 2: a

−a

= 0 - parabolic type.

In this case equations ( 6

) and ( 6

−

) coincide, and we have only one general integral of

equation (1.2.6): ϕ(x,y) = C. We make the substitution (1.2.2), where ϕ(x,y) is a general

integral of equation (1.2.6), and ψ(x,y) is an arbitrary smooth function satisfying (1.2.3).

By Lemma 1.2.1, ˜a

= 0. Furthermore, since a

√

, we

ve according to

(1.2.5),

˜a

= (

√

)

and

consequently

√

= 0.

Then

˜a

(

√

)

(

√

)

= 0

and equation (1.2.1) is reduced to the canonical form

ηη

= Φ(ξ, η,u, u

), (1.2.10)

where Φ = −

˜a

The simplest example of a parabolic type equation is the heat equation

= u

Case 3: a

−a

< 0 - elliptic type.

In this case the right-hand sides in ( 1.2.6

) are complex. Let ϕ(x,y) = C be the complex

general integral of equation ( 1.2.6

). Then

ϕ(x,y)

= C is

the general integral of equation

( 1.2.6

−

). We take the complex variables ξ = ϕ(x, y) , η =

ϕ(x,y), i.e.

mak

e the substitu-

tion (1.2.2), where ψ =

ϕ. Repeating the arguments from Case 1, we deduce that equation

(1.2.1) is reduced to the form u

ξη

= Φ(ξ,η,u, u

). But this equation is not canonical for

elliptic equations since ξ, η are complex variables. We pass on to the real variables (α,β)

by the replacement ξ = α + iβ , η = α −iβ. Then α = (ξ + η)/2 , β = (ξ −η)/(2i). We

calculate u

= (u

−iu

)/2 , u

= (u

+iu

)/2 , u

ξη

= (u

αα

ββ

)/4. Therefore, equation

(1.2.1) is reduced to the canonical form

αα

+ u

ββ

= Φ

(α,β,u,u

). (1.2.11)

The simplest example of an elliptic equation is the two-dimensional Laplace equation

∂

∂x

∂

∂y

= 0.

Remark

1.2.2. Since

we used complex variables, our arguments for Case 3 are valid

only if a

i j

(x,y) are analytic functions. In the general case, the reduction to the canonical

form for elliptic equations is a more complicated problem. For simplicity, we conﬁned

ourselves here to the case of analytic coefﬁcients.

10 G.

Freiling and V. Yurko

2. Classiﬁcation

of second-order equations with an arbitrary

number of variables.

We consider the equation

∑

i, j=1

i j

,. .. ,x

)

∂

∂x

+ F

.. ,x

,u,

∂u

∂x

.. ,

∂u

∂x

= 0. (1.2.12)

ithout

loss of generality, let a

i j

= a

. We make the substitution

= ξ

,. .. ,x

), k =

1,n. (1.2.13)

Denote α

∂ξ

∂x

. Then

∂u

∂x

∑

k=1

∂u

∂ξ

∂

∂x

∑

k,l=1

∂

∂ξ

∑

k=1

∂

∂ξ

∂

∂x











(1.2.14)

Substituting

(1.2.14)

into (1.2.12) we get

∑

k,l=1

˜a

(ξ

,. .. ,ξ

)

∂

∂ξ

.. ,ξ

,u,

∂u

∂ξ

.. ,

∂u

∂ξ

= 0,

where

˜a

∑

i, j=1

Consider the quadratic form

∑

i, j=1

i j

, (1.2.15)

where a

i j

= a

i j

,. .. ,x

) at a ﬁxed point (x

,. .. ,x

). The replacement

∑

k=1

reduces (1.2.15) to the form

∑

k,l=1

˜a

, where ˜a

∑

i, j=1

i j

Thus, the coefﬁcients of the main part of the differential equation are changed similarly to

the coefﬁcients of the quadratic form. It is known from linear algebra (see [10]) that the

quadratic form (1.2.15) can be reduced to the diagonal form:

∑

i, j=1

i j

∑

l=1

(±q

), m ≤ n,

Introduction 11

and the

numbers of positive, negative and zero coefﬁcients are invariant with respect to

linear transformations.

Deﬁnition 1.2.2. 1) If m = n and all coefﬁcients for q

have the same sign, then

equation (1.2.12) is called elliptic at the point x

2) If m = n and all signs except one are the same, then equation (1.2.12) is called hyperbolic

at the point x

3) If m = n and there are more than one of each sign ” + ” and ” −”, then equation (1.2.12)

is called ultrahyperbolic at the point x

4) If m < n, the equation is called parabolic at the point x

For example, the Laplace equation ∆u = 0 is elliptic, the wave equation u

= ∆u is

hyperbolic, and the heat conduction equation u

= ∆u is parabolic in the whole space.

1.3. Formulation of Problems of Mathematical Physics

From the theory of ordinary differential equations it is known that solutions of an ordinary

differential equation are determined non-uniquely. For example, the general solution of an

n -th order equation

(n)

= F(x,y,y

,. .. ,y

(n−1)

)

depends on n arbitrary constants:

y = ϕ(x,C

,. .. ,C

In order to determine the unique solution of the equation we need additional conditions, for

example, the initial conditions

|x=x

= α

,. .. ,y

(n−1)

|x=x

= α

n−1

For partial differential equations solutions are also determined non-uniquely, and the

general solution involves arbitrary functions. Consider, for example, the case of two inde-

pendent variables (x,y) :

1) the equation

∂u

∂y

= 0

has

the

general solution

u = f (x),

where f (x) is an arbitrary function of x ;

2) the equation

∂

∂x∂y

= 0

has

the

general solution

u = f (x) + g(y),

where f (x) and g(y) are arbitrary smooth functions of x and y, respectively.