Partial Differential Equations: An Introduction

By David Colton

Intended for a college senior or first-year graduate-level course in partial differential equations, this text offers students in mathematics, engineering, and the applied sciences a solid foundation for advanced studies in mathematics. Classical topics presented in a modern context include coverage of integral equations and basic scattering theory. This complete and accessible treatment includes a variety of examples of inverse problems arising from improperly posed applications. Exercises at the ends of chapters, many with answers, offer a clear progression in developing an understanding of this essential area of mathematics. 1988 edition.

• Language: English
• Publisher: Dover Publications
• Release date: Jun 14, 2012
• ISBN: 9780486138435

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Index

Chapter 1

Introduction

In this chapter we prepare the groundwork for our study of partial differential equations. We show how the wave equation, the heat equation, and Laplace’s equation arise in the modeling of certain basic physical phenomena and how these equations are in fact the simplest examples of the three main classes of second order partial differential equations. In addition, we introduce the elementary theory of first order linear partial differential equations as well as the tools beyond advanced calculus that are needed in our investigations—i.e., the elementary theory of Fourier series, the Fourier integral, and analytic functions. Section 1.5, Analytic Functions, is kept brief and is needed only for the starred sections of this volume. In particular, if you are willing to forgo Sections 3.5 and 3.6, and Chapter 6 on scattering theory, you need not read anything about analytic function theory at all. However, at the end of our presentation of analytic function theory, we do use the identity theorem to establish the existence of a linear partial differential equation that has no solution at all. We conclude by giving a brief history of the theory of partial differential equations, which will serve as a guide through the analysis contained in the following chapters. Since the area of partial differential equations encompasses almost all areas of analysis (and is intimately connected with the physical sciences), we can only touch upon the highlights of its history. For a more comprehensive history, we recommend the monumental and scholarly treatise by Kline, upon which we have based our own short presentation.

First, we need to introduce some notation and a few elementary definitions. We will denote n-dimensional Euclidean space by Rn. A point in Rn will be written as x = (x1,...,xn), where x1,x2,...,xn are the Cartesian coordinates of x. For n = 2 or n = 3, we may also use different letters for coordinates, e.g., (x,y) ∈ R², (x,y,z) ∈ R³, etc. The Euclidean distance between two points x,y E Rn is denoted by |x − y|. The boundary of a set S will be denoted by ∂S . The complement of a set S in Rn is written as Rn\S. An open interval in R¹ will be written as (a,b) and its closure as [a,b]. The set of functions defined on a set S and having all its partial derivatives of order less than or equal to k continuous in S will be denoted by Ck(S). If the function f(x) is continuous in S, we write f ∈ C(S), whereas if f(x) has continuous derivatives of all orders in S, we write f ∈ C∞(S). A surface S in Rn is said to be in class Ck if for any point x0 ∈ S there exists an n-dimensional neighborhood N(x0) of x0 and a function f ∈ Ck(N(x0)) such that ∇ f(x0) ≠ 0 and the set S ∩ N(x0) is described by the equation f(x) = 0. The partial derivatives of a function f(x) with respect to xi will be denoted by ∂f/∂xi . If f = f(x) is a function of a single variable x, we will denote the derivative of f(x) by df/dx, f’, or f. Finally, if f(x) is a complex valued function, its complex conjugate will be written as f(x).

A partial differential equation is an equation relating an unknown function u(x1,...,xn), the independent variables x1,...,xn, and a finite number of the partial derivative of u, i.e.,

The order of a partial differential equation is that of the derivative of the highest order. A partial differential equation is called quasilinear if it is linear in all the highest order derivatives of the unknown function, e.g.,

is a quasilinear second order equation. A partial differential equation is called linear if it is linear in the unknown function and its partial derivatives, e.g.,

is a second order linear equation. We shall be mainly concerned with second order linear equations, especially the wave equation

Laplace’s equation

and the heat equation

Except for Sections 1.5.4, 4.8, and 6.3-6.6, we shall always assume that solutions of initial or boundary value problems for partial differential equations are real valued.

1.1 PHYSICAL EXAMPLES

We shall now briefly examine how the wave equation, Laplace’s equation, and the heat equation apply to certain physical situations. This discussion is merely illustrative rather than representative of the variety of applications of these equations in the physical and biological sciences.

We first consider the theory of heat conduction. Let D be a body in R³ with boundary ∂D. Let T(x,t), x ∈ D, denote the temperature in the body at the point x and at time t. A difference in temperature in D creates a heat flow given by

(1.1)

where κ is the coefficient of heat conductivity in D, which is assumed to be constant, and the gradient ∇ is taken with respect to the space variables. Then through a surface dσ with unit normal ν the amount of heat

(1.2)

flows in time ∆t in the direction ν. Let c denote the specific heat of D, where c is assumed to be constant. If we consider a portion D1 of D which is bounded by ∂D1, then the heat content of D1 is given by

(1.3)

and from (1.2) we have

(1.4)

if we assume that the heat content of D1 can be changed only by a flow of heat through ∂D1. From (1.3) and (1.4) we have

(1.5)

where ν is the unit outward normal to ∂D1. Using the divergence theorem

we can rewrite (1.5) as

(1.6)

where we have used the identity

Since D1 is arbitrary, we have

(1.7)

throughout D, so T satisfies the heat equation. We note that the change of variables t’ = (κ/c) t reduces (1.7) to

which is the form of (1.7) we shall later investigate. On ∂D we can either prescribe the temperature T or the heat flux −κ (∂T/∂ν), and in addition we must prescribe the initial temperature at time t = 0.

If T is independent of time, then T satisfies Laplace’s equation

(1.8)

For example, this will be true when the temperature of the body has reached a steady state. In such situations, it is no longer necessary to prescribe the initial temperature. Laplace’s equation also appears in problems in fluid dynamics and electrostatics (cf. Bergman and Schiffer).

We now look at the propagation of sound waves of small amplitude viewed as a problem in fluid dynamics. Let v(x,t) be the velocity vector of a fluid particle in an inviscid fluid and let p(x,t) and ρ(x,t) denote the pressure and density, respectively, of the fluid. If no external forces are acting on the fluid, Euler’s equations of motion are

(equation of continuity)

(equation of conservation of momentum)

(equation of state),

where f(ρ) is a function depending on the fluid. Assuming v(x,t), p(x,t), and ρ(x,t) are small, we perturb these quantities around an equilibrium state v = 0, p = p0, and p = ρ0 (with ρ0 = f(ρ0)) and write

(1.9)

where 0 < ∈ << 1 and the dots refer to higher order terms in ∈. Inserting (1.9) into Euler’s equations and retaining only terms of order ∈ gives (assuming that ρ0 is a constant) the linearized Euler equations

(1.10a)

(1.10b)

(1.10c)

Assume that at time t = 0 the fluid flow is irrotational, i.e., curl v1(x,0) = 0. Then there exists a potential ψ(x) such that v1(x;0) = — Δψ(x) and hence from (1.10b) we have

(1.11)

where φ(x,t) is the term in brackets in (1.11). Inserting (1.11) into (1.10b) gives

and hence we can choose

(1.12)

Equation (1.10c) now implies that

and from (1.10a) we have

Setting c² = f‘(ρ0) and assuming that f’(ρ0) > 0 gives the wave equation

(1.13)

Note that from (1.12) the pressure p1(x,t) also satisfies the wave equation (1.13). The change of variables t’ = ct reduces (1.13) to

(1.14)

which is the form of (1.13) we shall consider in the next chapter.

In order to determine φ(x,t) from (1.13) or (1.14), we must prescribe initial and boundary data. In particular, at time t = 0 we must be given the initial velocity and acceleration, i.e. φ(x,0) and ∂φ(x,0)/∂t. On the boundary of the region containing (or contained in) the fluid there are various possibilities. For example, if the boundary is rigid and impenetrable, then v1 · ν = 0 on the boundary where ν is the normal vector, i.e., ∂φ/∂ν = 0. At the other extreme, if the boundary is a fixed surface which is a site of pressure release so that the pressure vanishes there, then it suffices to set φ = 0 on the boundary. Note that if φ(x,t) is time harmonic, i.e.,

then φ(x) satisfies the Helmholtz equation

where k = ω/c. In this case it is no longer necessary to prescribe initial data.

The wave equation also appears in the theory of vibrations and electromagnetic wave motion (cf. Baldock and Bridgeman).

1.2 FIRST ORDER LINEAR EQUATIONS

The simplest type of partial differential equation in two independent variables is the first order linear equation. Our reason for studying such equations right at the beginning is for the pedagogical value in introducing characteristic curves and their importance in solving initial value problems, as well as for the fact that our results on first order equations will be needed shortly when we want to reduce second order equations in two independent variables to canonical form. Our aim is to show how the first order linear partial differential equation

(1.15)

can be solved by reducing it to an ordinary differential equation. We make the assumption that the coefficients a(x,y), b(x,y), c(x,y), and f(x,y) are continuously differentiable functions of x and y in some domain D, and that a(x,y) and b(x,y) never both vanish at the same point. We further assume that the coefficients a(x,y), b(x,y), c(x,y), and f(x,y) have real values. We shall show that by a change of variables we can reduce (1.15) to an equation of the form

(1.16)

which is an ordinary differential equation in ξ depending on the parameter η, where ξ and η are new independent variables. If a(x,y) or b(x,y) is identically zero, then (1.15) is already in the form (1.16), and hence we suppose that neither a(x,y) nor b(x,y) is identically zero.

Let φ E C¹(D), ψ E C¹(D) be such that the Jacobian

is not identically zero, and define

Then if u(x,y) = w(ξ,η) we have

Then (1.15) becomes

(1.17)

Hence if we choose ψ(x,y) such that

(1.18)

then we arrive at an equation of the form (1.16). The problem now is to construct a solution ψ(x,y) of (1.18) and a function φ(x,y) such that the Jacobian J(φ,ψ) is not identically zero.

Suppose for the moment that a solution ψ(x,y) of (1.18) exists such that ∂ψ/∂y is not identically zero. Define the curve y = y(x) implicitly by ψ(x,y) = γ where γ is a constant. Then

which implies that

Thus ψ(x,y) = γ implicitly defines a solution of the ordinary differential equation

(1.19)

Conversely, if ψ(x,y) = γ implicitly defines a solution of (1.19) such that ∂ψ/∂y is not identically zero, then ψ satisfies (1.18). Equation (1.19) is called the characteristic equation of the partial differential equation (1.15) and defines a one-parameter family of curves called the characteristic curves of the partial differential equation (1.15).

Having found ψ(x,y) as the solution of (1.19), we can choose the function φ(x,y) arbitrarily such that the Jacobian doesn’t vanish. For example, choose φ(x,y) = x. The ordinary differential equation (1.17) for w is then

and from the elementary theory of ordinary differential equations the solution of this equation is

where d(η) is an arbitrary function of η and a(ξ,η) = a(ξ,y(ξ,η)), c(ξ,η) = c(ξ,y(ξ,η)), etc., where the function y(ξ,η) is determined by solving for y in the equations ξ = x, η = ψ(x,y). Hence, the general solution of (1.15) is

(1.20)

where

□ EXAMPLE 1

Find the general solution of xux − yuy + u = x.

■ Solution. The characteristic equation is

This is a separable ordinary differential equation, i.e.,

Integrating both sides gives

where c is an arbitrary constant, or

where γ is (another) arbitrary constant. Hence, setting

in our first order partial differential equation yields

whose solution is

where d(η) is an arbitrary function of η. The solution of xux − yuy + u = x is now given by

In order to guarantee that u is continuously differentiable we require that the arbitrary function d be continuously differentiable. ■

We now turn to the problem of solving initial value problems for first order linear partial differential equations. The Cauchy problem for the partial differential equation (1.15) is to find a solution of (1.15) taking on prescribed values φ(x) on a specified curve C: y = y(x) in the xy plane. The curve C must not be a characteristic curve of (1.15) (or tangent to such a curve). On a characteristic curve we have ψ(x,y) = γ where γ is a constant and hence on this curve (1.20) becomes

i.e., u(x,y) does not equal φ(x) on C: y = y(x) unless φ(x) is of the special form

where k is a constant. On the other hand, if φ(x) is of this form, there exist infinitely many solutions of (1.15) given by (1.20) where d(η) is any differentiable function such that d(γ) = k.

□ EXAMPLE 2

Find the solution of xux − yuy + u = x such that u(x,y) = x on the curve y = x².

■ Solution. As we have shown in the previous example, the general solution of xux − yuy + u = x is given by

where d is an arbitrary differentiable function. Hence, on y = x² we want

i.e.,

or

Therefore

Note that we must not allow (x,y) to be on the lines x = 0 or y = 0 because there u(x,y) is not continuously differentiable. In particular, no solution to our Cauchy problem exists in a neighborhood of the origin. This is a consequence of the fact that at the origin the curve y = x² is tangent to the characteristic curve y = 0.

For a more complete exposition of first order partial differential equations, including quasilinear equations, the reader is referred to Courant and Hilbert 1961, Garabedian, and John.

1.3 CLASSIFICATION OF SECOND ORDER EQUATIONS AND CANONICAL FORMS

1.3.1 Types of Second Order Equations

We now consider second order quasilinear partial differential equations in Euclidean n space Rn and show how at a given point in Rn they can be classified into four distinct types. As we shall see later, the type of a partial differential equation dictates to a large extent the behavior of solutions to these equations. In this sense our classification procedure can be compared in spirit to that of a zoologist who classifies animals into mammals, birds, fishes, and reptiles and then proceeds to study representative examples from each of these classes.

Consider the second order partial differential equation

(1.21)

where aij = aij(x1,...,xn) are given real valued functions defined in a domain D C Rn and without loss of generality assume that aij = aji. be a fixed point of D and consider the quadratic form

(1.22)

Then

Equation (1.21) is of elliptic if at this point the quadratic form (1.22) is non-singular and definite, i.e., it can be reduced by a real linear transformation to a sum of n squares all of the same sign.

Equation (1.21) is of hyperbolic if at this point (1.22) is non-singular, indefinite, and can be reduced by a real linear transformation to the sum of n squares, n − 1 of which are the same sign.

Equation (1.21) is of ultra-hyperbolic if at this point (1.22) is non-singular, indefinite, and can be reduced by a real linear transformation to the sum of n (n ≥ 4) squares with more than one coefficient of either sign.

Equation (1.21) is of parabolic if at this point (1.22) is singular, i.e., it can be reduced by a real linear transformation to the sum of fewer than n squares, not necessarily all of the same sign.

Equation (1.21) is of one of these types in D if it is of that type at each point in D. There is no similar classification scheme for partial differential equations of order greater than two and in the remainder of this book we shall consider only second order equations. (This does not imply that higher order equations are not of interest from both a physical and mathematical viewpoint!)

□ EXAMPLES 3-6

The wave equation

is hyperbolic in any domain, the Laplace equation

is of elliptic type in any domain, and the heat equation

is of parabolic type in any domain. The equation

is of ultra-hyperbolic type in any domain. □

1.3.2 Reduction of Second Order Equations with Constant Coefficients to Canonical Form

This section shows how a partial differential equation with constant coefficients can be reduced to a particularly simple form by a linear change of variables. In particular, consider the partial differential equation with real valued constant coefficients

(1.23)

Define the linear change of variables

where the real constants cki are to be chosen later such that det |cki| does not vanish.

Then (1.23) becomes

(1.24)

where

can be expressed in terms of the bi and f(x1,...,xn), respectively. From linear algebra it is known that we can always choose the cij such that

where λk = 0, 1, or −1 and (1.24) becomes

(1.25)

The signs and/or vanishing of the λk clearly determine the type of equation (1.25) and hence of equation (1.23), since from the definition of type a real linear change of variables cannot change the type of a partial differential equation. Equation (1.25) is called the canonical form of D. An exception is the case n = 2, if we allow our transformation to be nonlinear. We shall now show how this can be done.

1.3.3 Reduction of Second Order Equations in Two Independent Variables to Canonical Form

Consider the partial differential equation

(1.26)

where A = A(x,y), B = B(x,y), and C = C(x,y) are real-valued twice continuously differentiable functions of x and y in some domain D C R² and A, B, and C do not all vanish at the same point. The factor of two appearing in the coefficient of ∂²u/∂x∂y is purely for notational convenience to avoid having a factor of four appearing repeatedly in our subsequent formulas. (The reader is warned to remember this factor when doing the exercises at the end of the chapter.) The quadratic form corresponding to (1.26) is

(1.27)

(1.26) is of hyperbolic type if B² − AC > 0, of parabolic type if B² − AC = 0, and of elliptic type if B² − AC < 0.

Now make the change of variables

(1.28)

such that ξ(x,y), η(x,y) E C¹(D) and the Jacobian

does not vanish for (x,y) ∈ D. Then (1.26) becomes

(1.29)

where