Lectures on Partial Differential Equations

By I. G. Petrovsky

The field of partial differential equations is an extremely important component of modern mathematics. It has great intrinsic beauty and virtually unlimited applications. This book, written for graduate-level students, grew out of a series of lectures the late Professor Petrovsky gave at Moscow State University. The first chapter uses physical problems to introduce the subjects and explains its division into hyperbolic, elliptic, and parabolic partial differential equations. Each of these three classes of equations is dealt with in one of the remaining three chapters of the book in a manner that is at once rigorous, transparent, and highly readable.
Petrovsky was a leading figure in Russian mathematics responsible for many advances in the field of partial differential equations. In these masterly lectures, his commentary and discussion of various aspects of the problems under consideration will prove valuable in deepening students’ understanding and appreciation of these problems.

• Language: English
• Publisher: Dover Publications
• Release date: Dec 13, 2012
• ISBN: 9780486155081

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Library of Congress Cataloging-in-Publication Data

, I. G. (Ivan Georgievich)

s chastnymi proizvodnymi. English]

Lectures on partial differential equations / I.G. Petrovsky ; translated by A. Shenitzer.

p. cm.

s chastnymi proizvodnymi.

Previously published: New York: Interscience Publishers, Inc., 1954.

9780486155081

1. Differential equations, Partial. I. Shenitzer, Abe. II. Title.

QA377.P433 1991

515’.353—dc20

91-39231

CIP

Manufactured in the United States by Courier Corporation

66902503

www.doverpublications.com

FOREWORD

Professor Petrovsky is an outstanding and leading representative of the great mathematical tradition and of the most impressive active mathematical life of Russia. Great progress is due to him in the field of partial differential equations. It will, therefore, be highly welcome to English speaking students that Petrovsky’s masterly lectures on this important subject are now being made accessible through the present translation from the Russian original.

R. COURANT

TRANSLATOR’S NOTE

The work of an eminent mathematician, Petrovsky’s Partial Differential Equations, has many qualities which should make it popular with graduate students desiring to get acquainted with partial differential equations, a field of mathematics of great intrinsic beauty and of vast importance in applications.

The presentation of the material is rigorous and at the same time transparent and readable. Petrovsky recognizes the value of heuristic and physical considerations but at the same time maintains high mathematical standards. His comments on and discussion of various aspects of the problems under consideration cannot but deepen the students’ understanding and appreciation of these problems.

A. SHENITZER

PREFACE

The present book grew out of a series of lectures which I presented a number of times to students of mathematics at the department of mathematics and mechanics of the Moscow State University. These lectures were to some extent supplemented before being printed.

In §§ 24 and 35, integral equations are used in presenting the material covered in §§ 23 and 31 through 33. Consequently, the material exclusive of §§ 24 and 35 (as well as 34) is self-contained.

In the preparation of this book I was given a great deal of assistance. K. S. Kusmin prepared notes of my lectures. Z. Ya. Shapiro was particularly helpful in that she edited the manuscript and wrote §§ 22 through 25 and portions of other sections. Without her help the publication of this book would have been considerably delayed. A. D. Myshkis and M. I. Vishik read the manuscript and made a number of very useful remarks. A. D. Myshkis also wrote §§ 34, 35, and part of § 4. B. M. Levitan wrote para. 3 of § 26. I wish to express my deep gratitude to all of them.

I. PETROVSKY

9 April 1950

Table of Contents

DOVER BOOKS ON MATHEMATICS

Title Page

Copyright Page

FOREWORD

TRANSLATOR’S NOTE

PREFACE

CHAPTER I - INTRODUCTION. CLASSIFICATION OF EQUATIONS

CHAPTER II - HYPERBOLIC EQUATIONS

CHAPTER III - ELLIPTIC EQUATIONS

CHAPTER IV - PARABOLIC EQUATIONS

CHAPTER I

INTRODUCTION. CLASSIFICATION OF EQUATIONS

§ 1. Definitions. Examples

1. An equation containing partial derivatives of unknown functions u1, u2, ..., uN is said to be of the nth order if it contains at least one partial derivative of the nth order and no partial derivatives of order higher than n. By the order of a system of equations containing partial derivatives we mean the order of the highest order equation of the system.

A partial differential equation is called linear if it is linear in the unknown functions and in their derivatives; it is called quasi-linear if it is linear in the highest order derivatives of the unknown functions. Thus, for instance, the equation

is quasi-linear of the second order with respect to the unknown function u. The equation

is linear of the second order with respect to u. The equation

is neither linear nor quasi-linear with respect to u.

By a solution of an equation containing partial derivatives we mean a system of functions which, when put in the equation in place of the unknown functions, turns the equation into an identity in the independent variables. A solution of a system of equations is defined in an analogous manner.

We shall be primarily interested in linear equations of the second order in one unknown function. The following are examples of equations of this type:

(1)

(2)

(3)

Many problems in physics reduce themselves to partial differential equations, in particular, to the partial differential equations listed above.

2. Example 1. The heat equation.. This equation describes the flow of heat. Its derivation is based on the following physical law (Newton’s law):

Let the temperature of a body at a point (x1, x2, x3) in the body at the time t be given by a continuous and differentiable function u(t, x1, x2, x3). Then the formula

gives the amount of heat flowing through a small area ΔS in the body during the time Δt to within terms of order higher than the order of the product ΔS Δt. Here ∂u/∂n denotes the normal derivative in the direction of the flow of heat evaluated at an arbitrary point (x1, x2, x3) of ΔS. The positive coefficient k(x1, x2, x3) is called the coefficient of inner thermal conductivity at the point (x1, x2, x3).

If the element of area under consideration forms part of the boundary of the body, then the same law in somewhat different form holds. Again, let u(t, x1, x2, x3) denote the temperature of the body G at the point (x1, x2, x3) and let u1(t, x1, x2, x3) denote the temperature at an arbitrary point (x1, x2, x3) outside the body. Then, the amount of heat entering the body across the element of area ΔS in the time interval Δt is given, to within higher order terms, by the expression

ΔQ ≈ k1(x1, x2, x3) [u1(t, x1, x2, x3) − u(t, x1, x2, x3)] ΔS Δt.

Here (x1, x2, x3) denotes an arbitrary point of the element of area, and the values of the functions u1 and u at that point are limiting values obtained by approaching the point from the outside and from the inside of the body, respectively. The coefficient k1(x1, x2, x3) is referred to as the coefficient of outer thermal conductivity relative to the given medium.

We shall assume that the body is isotropic with respect to heat conduction, i.e. k(x1, x2, x3) is independent of the orientation of the element of area. We shall also assume that the function k(x1, x2, x3) has continuous partial derivatives with respect to x1, x2 and x3.

For the purpose of our derivation we imagine in the interior of the considered region G a cube D defined by the inequalities

xi ≤ ξi ≤ xi + Δxi.

Here (x1, x2, x3) denotes a point in the interior of G, (ξ1, ξ2, ξ3) denotes a point of the cube D, and Δx1 = Δx2 = Δx3 are small increments of the coordinates. We assume that the function u(t, x1, x2, x3) is twice continuously differentiable. We now compute the increase in the amount of heat inside our cube over the time interval Δt. By Newton’s law, the amount of heat flowing into the cube across the face lying in the plane ξ1 = x1 is given, to within terms of higher order, by

Similarly, for the parallel face ξ1 = x1 + Δx1 we have

and, according to the mean-value theorem,

Carrying out analogous computations for the remaining faces of the cube, we find that the total increase in the amount of heat in the cube D in time Δt is given by

(1, 1)

On the other hand, this amount of heat can be estimated by considering the increase in temperature in the cube in time Δt:

(2, 1)

Here ρ(x1, x2, x3) denotes the density of the body at the point (x1, x2, x3) and c(x1, x2, x3) its specific heat at that point. ¹ Dividing (1, 1) and (2,1) by ΔtΔx1Δx2Δx3, equating the results of division, and passing to the limit by letting Δt → 0, Δxi → 0, we get an exact equality at the point (x1, x2, x3) at time t:

(3,1)

This equation is called the heat equation for a non-homogeneous isotropic body. If the body is homogeneous, then

k(x1, x2, x3) = const., c(x1, x2, x3) = const., ρ(x1,x2,x3) = const.,

and equation (3,1 ) becomes

(4, 1)

Replacing (k/cp) t with t and denoting t by t, we obtain (4, 1) in the form

(5, 1)

Equations (3, 1) and (5, 1) have many solutions. In order to single out one of the many solutions of these equations, it is necessary to prescribe supplementary conditions which are the counterpart of initial conditions as prescribed in ordinary differential equations. Such supplementary conditions are most often so-called boundary conditions, i.e. conditions prescribed on the boundary of the region G of the space (x1, x2, x3) in which we are to find a solution of an equation with partial derivatives, and initial conditions which refer to some definite moment in time. It is clear from the physical point of view that the knowledge of the temperature of the body at some moment in time and of the distribution of heat at the boundary of the body should completely determine the temperature of the body at any later moment in time. Likewise, it is clear that the distribution of heat can be prescribed in a variety of ways. If the region G coincides with all of space, then a bounded solution of the heat equation for t > t0 is uniquely determined by prescribing initial conditions only, namely, the value of the function u(t, x1, x2, x3) at t = t0. In case of a bounded domain G one could, for example, prescribe the temperature at each point of the body at t = t0 and, in addition, the temperature at the boundary points of the body for all t > t0. It turns out that these conditions suffice to determine a unique, bounded solution for t > t0 and (x1, x2, x3) ∈ G.

Instead of prescribing u(t, x1, x2, x3) at the boundary of G for t > to, we could prescribe the values of the outer normal derivative ∂u/∂n of the solution function u. This condition also determines a unique solution of the heat equation. We arrive at this formulation of the problem if we study the temperature in the interior of G under the assumption that we know at all times the amount of heat which flows in time Δt from outside the body across each small surface element ΔS of the boundary of the body. This quantity must be equal to the amount of heat transmitted from the surface of the body to its interior. If this were not so, heat would keep on accumulating at the boundary of G and, since the mass of the boundary of the body is equal to zero, the temperature at the boundary would tend towards infinity. According to Newton’s law, the amount of heat mentioned is equal to

k(∂u/∂n)ΔS Δt.

Here k > 0 denotes the coefficient of thermal conductivity at the boundary point under consideration. Knowing the law of heat output at each boundary point of the region G, we could find the value of ∂u/∂n. In particular, if the body does not give up heat at the boundary, ∂u/∂n = 0 at the boundary.

Finally, it is possible to prescribe as boundary conditions for t ≥ t0 the values

k(∂u/∂n) + k1u.

Here k1 is the coefficient of outer thermal conductivity and k the coefficient of inner thermal conductivity. The values of these coefficients are assumed to be known. We arrive at a mathematical problem of this kind if we study the temperature in the interior of the body G under the assumption that we know the temperature u1 of the body surrounding G. Then, according to Newton’s law, we shall find that:

(1) The amount of heat flowing from the surrounding medium across a surface element ΔS in the time interval (t, t + Δt) equals, apart from terms of higher order,

k1(u1 − u) ΔS Δt (k1 > 0).

(2) The amount of heat transmitted during that time from the surface element ΔS to the interior of the body equals, to within terms of higher order,

k(∂u/∂n) ΔS Δt (k > 0),

so that, in view of the equality of the two amounts,

k1u + k(∂u/∂n) = k1 u1.

In particular, if u1 ≡ 0, this condition becomes

k(∂u/∂n) + k1u = 0.

Let us now assume that the temperature at each interior point (x1, x2, x3) of the body does not vary with increasing time. Then ∂u/∂t = 0 and equations (3,1) and (5,1) go over respectively into

(6, 1)

To determine u(x1, x2, x3) it is not necessary, in this case, to prescribe initial conditions. It suffices to prescribe boundary conditions which we shall regard as being independent of time. The physical picture in this case is as follows. If the boundary conditions do not change in time, then, regardless of the prescribed initial conditions, the temperature u(t, x1 x2, x3) at each point (x1, x2, x3) of the body goes to a definite limiting value u(x1, x2, x3). The limit function u(x1, x2, x3) satisfies the steady-state equations (6, 1) as well as the boundary conditions assumed to be independent of time.

The problem of determining a function satisfying either one of the equations (6, 1) as well as prescribed boundary conditions is called the Dirichlet problem or the first boundary-value problem.

The problem of heat conduction in 3-space has its counterparts in 2-space and in 1-space. Thus, consider a rod of which we assume that its temperature is the same for all points of a given cross-section and that no heat exchange takes place between the surrounding medium and the rod across the lateral surface of the rod. Then the temperature u of the rod will be a function of the time t and of one space coordinate x only. With a proper choice of units, the equation satisfied by the function u(t, x) will, in this case, be of the form

(7,1)

If the temperature u(t, x1, x2, x3) of a three-dimensional body were to depend on only one space coordinate, x1 = x, say, then u would also satisfy equation (7, 1). This would take place if the temperature of the body were the same for all points of the plane x = const., for an arbitrary value of that constant. Likewise, the study of conduction of heat in a thin plate leads to the equation

(8, 1)

3. Example 2. Equation of equilibrium and vibration of a membrane. We shall consider a membrane which when at rest is in the (x1, x2) plane. By a membrane we mean a stretched film which resists stretching but does not resist bending, i.e. changes of form which do not alter the area of an arbitrary piece of the membrane. The work of the outside force which produces the change in form is proportional to the change itself. The positive coefficient of pro-portionalty T does not depend on the form or position of the membrane element. We call T the tension of the membrane.

The following considerations clarify the term tension.

If we cut the membrane along an arbitrary straight line segment l and consider the part of the membrane on one side of that segment, then, as will be shown, the action on it of the other part of the membrane is the same as the action of a force Tl applied at l and perpendicular to it.

We first show that the force Tl, acting on l is perpendicular to l. To this end we consider a small piece of the membrane in the form of a plane parallelogram (cf. fig. 1). We denote one side of this parallelogram by l. Keeping the side of the parallelogram opposite to l fixed, we deform the membrane so that the side l remains on one line. At the same time the remaining two sides of the parallelogram remain parallel to each other and its area will stay fixed. We now replace the action of the membrane on our parallelogram by forces applied at its sides. Since the change in area of the parallelogram is zero, the work of the sum of the forces is also zero. Since the side AD (opposite to l) is fixed, the work of the force applied at AD is zero. Again, since our parallelogram is small, the forces applied at the parallel sides AD and BC are almost equal and almost oppositely directed so that the work of these forces is zero to within terms of order higher than that of the work of the force Tl. Hence, the work of the force applied at the side l is also equal to zero. Consequently, the force Tl must be perpendicular to l.

Figure 1

We now show that Tl = Tl. For this purpose we consider a piece of the membrane in the form of a rectangle. Let l be one side of this rectangle. If we now deform the membrane so that the side l is moved a distance λ parallel to itself and the position lines of the remaining sides stay fixed, then the area of our rectangle is increased by λl. It follows from the definition of tension that the work associated with this deformation of our rectangle is λlT. If, on the other hand, we again replace the action of the membrane on our rectangle by forces applied at its sides and denote by Tl, the force applied at the side l, we find that the work considered is equal to Tlλ. Equating the two expressions for the work done in deforming our rectangle, we get

Tl=Tl.

It is, therefore, in order to call T the tension of the membrane.

Now let the membrane, when at rest, coincide with a plane region G in the (x1, x2) plane. Let L denote the boundary of G.

We assume that each element Δx1 Δx2 (see fig. 2) of the membrane is acted on by a force given, to within infinitesimals of higher order, by ƒ(x1, x2) Δx1 Δx2 and having the direction of the normal to the plane (x1, x2). The membrane will bend under the action of this force and will assume the form of some surface

u = u(x1, x2),

say. The u-axis is assumed to be perpendicular to the plane (x1, x2).

We shall derive the equation satisfied by the function u(x1, x2) under the following restrictions. First, when in equilibrium, the surface of the membrane is bent little, i.e. it approximates a portion of the plane. This means that the derivatives ∂u/∂x1 and ∂u/∂x2 are small and that we may disregard powers of these derivatives higher than the first. Secondly, under the action of the force ƒ(x1, x2), the points of the membrane move only in directions normal to the plane (x1 x2) so that their coordinates (x1, x2) do not vary with the motion.

The derivation of the equation is based on one of the fundamental principles of mechanics, the principle of virtual work. This principle asserts that if a system is in a state of equilibrium, then in any small displacement of the system compatible with the constraints imposed on the system² the total work done by all the forces acting on the system is zero.

We first compute the work done by the forces acting on the element Δx1 Δx2 (fig. 2) when this element is displaced from its initial position in the (x1, x2) plane to a position described by the equation u=u(x1, x2). The forces acting on this element are as follows: the outwardly directed (with respect to our element) force of tension and the force ƒΔx1 Δx2. The work of the force of tension is proportional to the increase in area accompanying the displacement of the membrane. The area of the surface element ΔS whose projection is the square Δx1 Δx2 is approximately equal to

In the state of equilibrium this element coincides with the square Δx1 Δx2. Consequently, the increase in area is equal to

It follows that the work of the (outwardly directed) forces of tension is given by

Figure 2

and disregard higher order terms in