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    Applied Partial Differential Equations
    Applied Partial Differential Equations
    Applied Partial Differential Equations
    Ebook1,049 pages12 hours

    Applied Partial Differential Equations

    By Paul DuChateau and David Zachmann

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    Superb introduction devotes almost half its pages to numerical methods for solving partial differential equations, while the heart of the book focuses on boundary-value and initial-boundary-value problems on spatially bounded and on unbounded domains; integral transforms; uniqueness and continuous dependence on data, first-order equations, and more. Numerous exercises included, with solutions for many at end of book. For students with little background in linear algebra, a useful appendix covers that subject briefly.
    LanguageEnglish
    PublisherDover Publications
    Release dateOct 30, 2012
    ISBN9780486141879
    Applied Partial Differential Equations

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      Applied Partial Differential Equations - Paul DuChateau

      Index

      Preface

      The importance of partial differential equations as a component of applied mathematics has long been recognized, and the increasing complexity of today’s technology requires engineers and applied scientists to understand this subject on a level previously attained only by specialists. In addition the accelerating development of computers, both supercomputers and powerful personal computers, has had a significant impact on the way in which we approach problems in partial differential equations. In spite of all this, the teaching of the elementary courses in this subject has changed very little in three or four decades.

      In writing this book we have tried to reflect some of the modern attitudes toward the subject of partial differential equations without losing sight of the fact that this is intended to be an introductory text. The only prerequisites we assume are a good foundation in calculus and an introductory course in ordinary differential equations. Some background in linear algebra would be helpful, but we recognize that it is likely that many of the students in a course on partial differential equations have little knowledge of linear algebra. For that reason we have included a linear algebra appendix and integrated additional linear algebra facts into the text when they are needed.

      EMPHASIS

      Throughout we have tried to emphasize the development of only a few mathematical tools, but we concentrate on learning to use them very well. In this way, with only a modest mathematical background we are able to consider a number of interesting physical applications including flow in a porous medium, dispersive and nondispersive wave propagation, advection with a random coefficient, traffic flow, and waves in a ripple tank. We know from experience that not only can students understand these examples, they appreciate them more than the traditional but bland examples like the vibrating string.

      This book differs from other elementary books on partial differential equations in a number of ways. Most obvious is the fact that approximately one half of the book is devoted to numerical methods for solving partial differential equations. In part this is in recognition of the need to train applied mathematicians in computational mathematics [see A National Computing Initiative, the agenda for leadership, SIAM workshop, Feb. 1987]. In addition it is a reflection of the belief that, at an elementary level, it no longer makes sense to study the classical aspects of partial differential equations disjoint from the numerical side of the subject. We prefer to think of the two faces of a problem in partial differential equations as the continuous model and discrete model, respectively, for some underlying physical system. It is one of the principal themes of this book that intelligent use of numerical solution to the discrete version of a partial differential equation is not possible without a good understanding of the behavior of the solution of the associated continuous problem. Conversely it is often the case that insight into the workings of the continuous problem can be obtained by studying the associated discrete model. This is particularly evident in Chapter 1, in which we formulate the discrete model for some example physical systems. In the discrete setting it is quite clear what sort of auxiliary data in the form of initial and boundary conditions must be imposed in addition to the basic equation in order to obtain a well-posed problem. This motivates the selection of auxiliary conditions in formulating the associated continuous problem.

      CONTENT

      We begin in Chapter 1 and continue throughout the book to promote the point of view that in the modeling of physical systems there are three entities to be considered and that these should be accorded equivalent status. The three entities are the physical system, the continuous model, and the discrete model. In order to illustrate the relations that exist between the continuous and discrete model for a system, we very often will solve a continuous problem in the first half of the book by means of an eigenfunction expansion or integral transform and then solve the associated discrete problem in the second part of the book by an entirely parallel method. Each model provides information of some sort about the physical system and each model requires a specific set of tools to uncover that information. The tools of linear algebra are used to study the discrete model while the continuous model is analyzed by methods that have their roots in calculus. We hope to create in the students the impression that linear algebra and analysis (calculus) each has its place in the toolbox of the applied mathematician.

      Other differences between this text and most other elementary books on partial differential equations are more subtle. For example, we do not think that the student should leave this course with the feeling that once he or she has constructed a solution to a continuous problem in the form of an eigenfunction expansion or an integral representation the job is finished; in truth it is only the routine part of the problem that is finished. At that point begins the task of extracting information about the behavior of the system being modeled. Examples of the types of information available and how that information may be obtained include the following:

      a boundary-value problem for the Laplace equation that shows how the shape of the region can affect the smoothness of the solution

      initial-boundary-value problems for the heat equation that illustrate the influence of the lower-order terms in the equation (i.e., conduction with convection and conduction with dissipation)

      initial-boundary-value problems for the wave equation that illustrate the influence of lower-order terms (i.e., dispersive and dissipative wave propagation)

      initial-boundary-value problems for the heat equation that illustrate both time-dependent and time-independent steady-state solutions

      initial-boundary-value problems for the heat equation and wave equation that illustrate the difference between diffusionlike and wavelike evolution

      an advection equation with a random coefficient that induces diffusive behavior in the solution

      a traffic-flow model to illustrate the solution of first-order conservation laws

      a model of waves in a ripple tank to illustrate systems of first-order hyperbolic equations

      The treatment of first-order partial differential equations is found in Chapter 7, following the material on second-order problems. We have found that this material can be more easily grasped after an introduction to second-order partial differential equations. Chapter 7 is more than just a cursory introduction to first-order equations and includes discussions of shocks, fans, and generalized solutions. In addition to a thorough treatment of first-order equations and conservation laws, this chapter contains many more examples of the applications of first-order equations than is usual in a text at this level.

      ARRANGEMENT OF MATERIAL

      There are other ways in which the arrangement of material in this book differs from most other texts. In Chapter 1 on modeling of physical systems, the discrete model is derived before the associated continuous model. The reasons for this have already been mentioned. Chapter 2 develops the classical theory of Fourier series followed immediately by a discussion of generalized Fourier series and Sturm–Liouville problems. Here we generalize the usual pointwise notion of a function slightly and introduce the class of square integrable functions. The chapter discusses in detail the convergence of Fourier series but does not include a proof of the Fourier convergence theorem. We feel that this proof belongs more appropriately in a course on real analysis. We also omit any mention of special functions from this text. Although special functions have a place in solving partial differential equations, to include anything more than a superficial treatment would force us to exclude more relevant material.

      We begin solving partial differential equations in Chapter 3. This chapter focuses on problems on bounded regions so that the apropriate solution method is then eigenfunction expansion. We begin with static problems described by Laplace’s equation and proceed to time-dependent problems governed by the heat and wave equations. Chapter 4 presents the essentials of the Fourier and Laplace transforms, and in Chapter 5 these are applied to the solving of partial differential equations on unbounded sets. The problems in Chapter 5 are covered in the same order as in Chapter 3, and whenever possible we draw attention to similarities and differences between problems on bounded regions and their analogues on regions that are not bounded.

      Chapter 6 is devoted to some of the qualitative aspects of the problems we have studied in Chapters 1 through 5. In particular energy integral methods and max-min principles are introduced and used to expose various facts about the solutions to some of the examples considered in previous chapters. Chapter 7 presents a thorough treatment of first-order equations including a discussion of weak solutions, fans, and shocks.

      Chapters 8, 9, and 10 cover the solution of discrete models of parabolic, hyperbolic, and elliptic types, respectively. Parabolic problems are the focus of Chapter 8, in which a number of the standard topics associated with finite difference methods are introduced. Some of the less usual topics covered in this chapter include discrete Fourier methods and discrete conservation laws. Discrete material balance laws are the basis for developing conservative difference methods, a notion of considerable importance in many applications.

      Hyperbolic equations are treated in Chapter 9, which contains much material that is not to be found in any other text at this level. Examples include conservation law equations, numerical dispersion and dissipation, and the numerical method of characteristics.

      Chapter 10 concludes the book with a presentation of numerical methods for elliptic problems. As in the previous two chapters, the methods are illustrated with physical applications and an effort is made to emphasize the parallels between the discrete problem being discussed and the analogous continuous problem. Algorithms in the form of pseudocode are included in each of the last three chapters, and we strongly urge users of this book to engage in extensive computer experimentation. We have done our best to provide exercises to leave the student with the impression that the subject of numerical methods for solving partial differential equations is something far more subtle than just crunching numbers.

      There is more material in this book than can be covered in a one-semester or even full-year course in partial differential equations. However, by selecting certain portions of the book there are various courses and sequences of courses that can be offered using this book as a text. Some of these are indicated in flowcharts following the preface. Note that there is sufficient material in each chapter that, by selecting or omitting examples, an instructor can adapt the level of the course to suit the class.

      ACKNOWLEDGMENTS

      We should like to express our appreciation to Professor C. W. Groetsch of the University of Cincinnati, Professor Stephen Krantz of Washington University, Professor John Palmer of the University of Arizona, Professor Ed Landesman of the University of California, Santa Cruz, Professor Gilbert N. Lewis of Michigan Technological University, Professor Gary Walls of the University of Southern Mississippi, and Professor Bernard Marshall of McGill University. Each of them read various versions of the manuscript and made suggestions that led to improvements. We should also like to thank Professor John Hunter and Professor Jim Thomas of Colorado State University for many helpful conversations and Professor Ralph Niemann also of Colorado State for testing a preliminary version of this text in his junior/senior-level partial differential equations class. A number of rough spots were made smooth as a result of his input. Of course we are also grateful to the many students who diligently found and reported errors typographical and otherwise. Finally, we should like to express our appreciation to Peter Coveney and the staff at Harper & Row for their cooperation and support during this project.

      Paul DuChateau

      David Zachmann

      The above applied PDE course and the following course in finite differences can be taught as a sequence or independently.

      PART ONE

      EXACT METHODS

      Chapter 1

      Mathematical Modeling and

      Partial Differential Equations

      This is a book about solving partial differential equations. More specifically, it is a book about solving physically motivated problems in partial differential equations. Before we can proceed to solve such problems, we should first know something about how the problems arise, how they are properly formulated, and the notation and terminology used to discuss them. This is the subject matter of Chapter 1.

      Mathematical modeling at an elementary level is illustrated for three different physical systems. Examples of properly posed problems for the discrete model are given for each system, and these problems are used to motivate the formulation of a corresponding well-posed problem for the continuous model. In Chapter 1 we are only trying to demonstrate by example the meaning of well-posedness for problems in partial differential equations, and none of the examples in this chapter are proved to be well posed; this is done in Chapter 6 for some of the examples.

      The chapter concludes with a brief discussion of the notions of classification of equations.

      1.1  MATHEMATICAL MODELING OF PHYSICAL SYSTEMS

      The term mathematical model refers to a mathematical problem whose solution allows us to describe or predict the behavior of an associated physical system as it responds to a given set of inputs. The physical system is governed by a well-defined set of physical principles that are then translated into corresponding mathematical statements. These statements often take the form of equations in which the state of the physical system plays the role of the unknown in the problem. The mathematical model is considered to be well formulated if the output or response of the physical system is uniquely determined by the input for the problem.

      We intend to demonstrate the process of developing a mathematical model for a physical system by means of some examples. In each example we describe the physical system and the state variables that characterize its behavior. In addition, we explain how the governing physical principles may be expressed as mathematical equations. Finally, we try to illustrate in these examples the role that simplifying assumptions play in the development of a mathematical model.

      Continuous and Discrete Models of Physical System

      In each of the examples we present, we develop two distinct versions of a mathematical model for the physical system. One of the versions, the continuous model, treats the physical system as a continuous medium. A consequence of this point of view is that the governing physical principles translate into differential equations in the mathematical setting; if the number of independent variables is more than one, then these differential equations are partial differential equations.

      In addition to the continuous model, we discuss a discrete version of the mathematical model for each of the physical systems we consider. This model arises from viewing the physical system as composed of discrete entities each of finite rather than infinitesimal size. In this case, the mathematical statement of the governing physical principles takes the form of a system of algebraic equations.

      Relationship of Models to Each Other

      The continuous and discrete models for a given physical system are related to each other in the following way. The continuous model may be obtained from the discrete model by allowing the size of the discrete entities comprising the system to shrink to zero. Properly applied limiting procedures then cause the algebraic expressions to become differential expressions. Conversely, the discrete model may be obtained from the continuous model by approximating the derivatives in the continuous model by suitable difference quotients. If this is done correctly, the differential equations in which the unknowns are functions are replaced by algebraic equations in which the unknowns are the values of these same functions evaluated at discrete points in the domain of interest.

      The continuous and the discrete models each contribute to the understanding of a physical system and each can contribute to the understanding of the other. Currently it is more usual to rely on the continuous model for qualitative information about the physical system and to resort to the discrete model for quantitative (numerical) results. The motivation for this point of view is practical. The large-scale algebraic problems arising in connection with the discrete model are well suited to treatment by computer where numerical results are readily generated. On the other hand, even when no solution can be explicitly constructed, the continuous-model problem will often yield information of a qualitative nature using methods that have their roots in the calculus.

      Practical though it might be, this point of view propagates the following false impressions about the discrete model:

      (a)  The discrete model is just an approximation of the continuous model (this reduces the discrete model to the status of an approximation to an approximation of the physical system).

      (b)  The discrete model is obtained from the continuous model by simply replacing derivatives in the continuous-model equations by finite-difference expressions based on Taylor series expansions of the unknown functions.

      The impression created is that the continuous model stands between the physical system and the discrete model. We aim to illustrate in the pages to follow that the discrete and continuous models are equally valid alternative descriptions for a physical system and that neither should be viewed as any more approximate than the other. Moreover, while the continuous model can always be obtained from the discrete model by a passing to the limit, it often happens that there is no approximation of derivatives by Taylor series expansions that will lead from the continuous model to the discrete. More precisely, we should say that replacing derivatives in the continuous-model equation by finite-difference expressions based on Taylor series expansions does not always lead to correct discrete models. In order to ensure that the discrete model is a correct one, the derivation should be based on discrete versions of the physical principles used to develop the continuous model. A detailed discussion of the process of obtaining discrete models from continuous ones is found in Section 8.1.

      Qualitative versus Quantitative Information

      Another of the goals of the presentation in this text will be to alter the perception that the discrete model must serve a purely quantitative role in the study of a physical system while the continuous model is to be used only for qualitative purposes. In the chapters to come we provide examples of how the continuous-model solution can be used for quantitative purposes. These examples show that there are certain limitations in the responses (outputs) that can be modeled discretely. For example, in any discrete model, all responses of sufficiently high frequency are modeled in an ambiguous way. This is the phenomenon of aliasing mentioned in Chapter 2 in connection with discrete Fourier series. Deficiencies of this sort, resulting from the very discreteness of the discrete model are, of course, not present in the continuous model. This is one example then of a situation where the continuous model is to be preferred as a source, of quantitative information about the physical system.

      On the other hand, particularly in this first chapter, we use the discrete model as a source of qualitative information about the continuous model. Here, and again in the later chapters where the methods for constructing the solutions to the discrete model problem are developed, we make extensive use of a few basic principles from linear algebra. For those whose background in this area is lacking, we have collected most of the results we need in an appendix at the end of this text. Throughout the development we strive to emphasize the parallel between the treatment of the continuous problem by methods having their roots in the calculus (analysis) and the treatment of the discrete problem by the techniques of linear algebra. As a by-product of this approach, we hope to create the impression that for an applied mathematician, strong foundations in both analysis and linear algebra are essential.

      Summarizing what we have said so far, we have described three distinct entities: the physical system, the discrete model, and the continuous model (see Figure 1.1.1). Each of these entities provides information about the other two. In this text we are primarily concerned with extracting information about the physical system from the two mathematical models. Chapters 2–7 are devoted to applying the tools of analysis to solving the partial differential equations of the continuous model and to using those solutions to provide information about the physical system. Chapters 8–10 describe how the techniques of linear algebra can draw out from the discrete model information about the behavior of a physical system. A secondary theme throughout this text is the exchange of information between the continuous and discrete models.

      Figure 1.1.1

      The Role of Simplifying Assumptions

      We must also recognize that whether we are pursuing the continuous or the discrete approach, the final form of the mathematical model we develop depends on the exact nature of the simplifying assumptions we make. For example, although the state of a physical system under consideration may depend on a very large number of factors, we generally decide to take into account only those factors we consider to be of primary importance. Other simplifying assumptions may take the form of omitting certain terms from the equations that represent the mathematical expression of the system’s governing principles. These terms can be omitted for reasons of expediency (this is often done when the equation is more easily solvable when they are not present) or some terms in the equation may actually be negligible with respect to the other terms in the equation.

      In any case, since there may be several ways in which a given model may be simplified, it can happen that a single physical system is described by more than a single mathematical model. This does not necessarily represent an ambiguous situation. It may be that different models can be associated with different levels of refinement, and the solutions that result from solving these models can be thought of as analogous to viewing the physical system under differing degrees of magnification. On the other hand, a single physical system may be accurately represented by one model under one set of conditions while another set of conditions requires that a different model be used.

      We now illustrate the meaning of these remarks with some examples. Each of the examples leads to a partial differential equation that is of second order. Since so many physical problems do lead to equations of second order, and since the treatment of second-order problems is somewhat more systematic than it is for first-order equations, we discuss the solution of equations of order two before discussing equations of order one. Chapters 3 and 5 are primarily devoted to second-order problems while the first-order problems are considered in Chapter 7.

      1.2    EQUATION OF HEAT CONDUCTION

      We begin by considering a three-dimensional region Ω that we imagine to be filled with a heat-conducting material. Let u = u(x, y, z, t) denote the temperature at the position (x, y, z) in Ω at time t, and our aim is to be able to compute the unknown function u(x, y, z, t). The conduction of heat is governed by certain physical principles that we translate into mathematical equations. Since these equations are statements relating to the rate of heat flow, it will be convenient to define a quantity Φ that we call the heat flux. The heat flux Φ(x, y, z, t) is a vector quantity whose magnitude equals the rate of heat flow at the point (x, y, z) at the time t and whose direction indicates the direction of heat flow. We have then

      where i, j, k denote unit vectors in the x, y, z coordinate directions. Imagine now a small cubical cell in Ω situated so that each of its faces is perpendicular to one of the coordinate axes and its center is at the generic point (x, y, zand number the faces 1–6 such that if Ni denotes the unit outward normal to face number i, then

      Figure 1.2.1 shows Ω viewed along the z axis. Now if we consider a generic interval of time (t0, t1), the amount of heat that flows out of face number i during this period of time can be expressed as

      Here ΔQ is expressed in units of heat since Φ is expressed in units of heat per unit time per unit area. The total amount of heat flowing out of the cubical cell during this time interval is then the sum of these six terms:

      Figure 1.2.1

      Here we are making use of the fact that

      Now we bring to bear the first of the principles governing the physical system. This principle states that in the absence of internal heat sources or heat sinks, when an amount of heat is added to or taken from a heat-conducting body, a temperature change occurs. More precisely, the relationship between the quantity of heat, ΔQ, and the corresponding temperature change, Δu, is given by

      That is, ΔQ is proportional to Δu with the proportionality factor equal to the product (C Δm). Here C denotes a material-dependent parameter called the heat capacity and Δm denotes the mass of the conductor. Then Δm ³ for the cubical cell). We make the simplifying assumptions here that both C and σ are constants.

      We can express (1.2.1) and (1.2.2) more compactly if we introduce the notation

      for the differences in the flux components appearing in (1.2.1). Then equating the two expressions for ΔQ and using the notation (1.2.3), we obtain the equation

      ² from the equation, this reduces to the discrete heat balance equation

      The value of the time argument t appearing on the right side of (1.2.4) is indeterminate. It follows from the mean-value theorem that equality in (1.2.4) occurs for some t value between t0 and t1. For simplicity, we assume this value to be t0.

      Discrete Heat Equation

      Equation (1.2.4) is a statement (in a discrete setting) of the principle that the physical process of heat conduction proceeds so as to conserve thermal energy. This principle has been expressed in terms of the temperature and the heat flux (the state variables). This is one equation for the four unknown functions u, Φ1, Φ2, Φ3. What we are seeking, then, is an equation involving only the temperature function, and in order to achieve this, we need an additional equation relating heat flux to temperature. This equation is based on an empirical physical principle known as Fourier’s law of heat conduction. Fourier’s law is named for Joseph Fourier, a French mathematician of the Napoleanic era who carried out some of the early work on the mathematical modeling of heat conduction. The law states that heat flows from hot regions in a conductor to cool ones and that the rate of heat flow is proportional to the temperature difference between the hot and cold regions. Expressed in mathematical terms, this becomes

      where K denotes another material-dependent parameter called thermal conductivity. We assume that, as for density and heat capacity, K is a constant. Using (1.2.5) in (1.2.4) leads to

      We refer to (1.2.6) as the discrete version of the heat conduction equation. The temperature function u(x, y, z, t) must satisfy this equation at each point (x, y, z) in the conductor and for all times t0 and t1 during the process of heat conduction.

      Equation (1.2.6) is not, by itself, enough to allow us to compute the time evolution of the temperature in the conductor. Additional information is required, and the discrete version of the heat conduction equation is particularly well suited to illustrating by example what this information should be and how it is used.

      A Special Case: Discrete Model in One Dimension     For ease of illustration we suppose that we are dealing with a one-dimensional conductor, a thin rod whose lateral surface has been insulated against the flow of heat (Figure 1.2.2). Insulating the lateral surface of the rod ensures that the heat flows axially along the rod, and if the rod is sufficiently thin, it is an acceptable approximation of reality to suppose that the temperature does not vary across the thickness of the rod.

      If the rod lies with its center line along the x axis, the temperature in the rod depends only on x and t and (1.2.6) reduces to

      This can be rewritten in the more explicit form

      where β = K Δt/(C²).

      Now in order to be able to relate Equation . If we introduce the notation

      Figure 1.2.2

      then each of the cells can be thought of as being centered at one of the points xn and (1.2.8) can be written as

      This equation is satisfied in each cell (i.e., at each of the discrete points xn) in the one-dimensional conductor for every one of the discrete times tj. Equation (1.2.8*) is an example of the type of equation treated in more detail in Chapter 8. We now consider two example problems for the rod in order to see what sort of additional information will be required in order to be able to compute the temperatures in the rod from Equation (1.2.8*). In the first example we consider a rod of infinite length. Of course, practically speaking, there is no such thing, but if the rod is long, there will be a portion of the rod near the middle where for a finite amount of time the influence of the ends of the rod may be neglected. Then for this finite amount of time, the middle portion of the rod may be thought of as a rod of infinite length. This artifice provides a way of treating the problem of finding the temperature in the rod when we do not wish to deal with the effects of the ends of the rod.

      EXAMPLE 1.2.1

      An Infinitely Long Rod

      Suppose the temperature in the rod is everywhere known at tis known for every nfor every n from in the same way, and the procedure may (theoretically) be continued for as long as we like (see Problem 1).

      versus n for successive values of k versus n is often referred to as a "temperature profile" at time tj. A detailed description of how this procedure may be carried out is the subject of Sections 8.2 and 8.6.   

      EXAMPLE 1.2.2

      A Rod of Finite Length

      Suppose the rod occupies the interval (0, L) on the x axis and let x0 = 0 and xN = N = Lis known for n = 0, 1, . . . , N , for j may be calculated for n = 1, 2, . . . , N from may be computed in the same way (for n = 1, . . . , N), and the process may (again theoretically) be continued for as long as we like in order to predict the evolution of the temperature in the rod at the discrete points xnversus n for successive values of j in order to obtain a time lapse picture of the evolution of the temperature in the rod with time.

      Figure 1.2.3

      In particular, for a rod of finite length L = 0 for all j= 1 for n from (1.2.8*) and plot it against n for several values of j in order to obtain Figure 1.2.3.

      Specifying the temperature everywhere in the rod at t = 0 is referred to as imposing an initial condition on the temperature. Evidently, Equation (1.2.8*) together with an initial condition are sufficient to determine subsequent states of the temperature in an infinitely long rod. In a rod of finite length we must impose boundary conditions as well as an initial condition in order to be able to use (1.2.8*) to calculate subsequent tem perature states. Conditions on u(x, t) imposed at the ends (i.e., the boundaries) of the rod are referred to as boundary conditions.

      Semidiscrete Heat Equation

      The mean-value theorem for derivatives can be applied to Equation (1.2.7) in order to rewrite it in the form

      Then letting t1 approach t0,, we have the semidiscrete version of (1.2.8),

      where un(t) = u(xn, t) for xn = n (n an integer) and t > 0.

      We may apply (1.2.8’) in the case of a rod of finite length L with N = L. Suppose

      for known functions p(t), q(t) and suppose

      for specified constants Cn. Then we have a system of linear ordinary differential equations for the N − 1 unknown functions u1(t), . . . , uN − 1(t),

      In matrix notation this is written as

      where U(t) denotes the vector of unknown functions and F, C denote vectors containing the data of the problem. In the simpler case where F = 0, we know from the theory of ordinary differential equations that this system has a general solution of the form

      where α1, . . . , αN − 1 denote the eigenvalues of the matrix [A] and W1, . . . , WN −1 denote the corresponding eigenvectors. The Γ’s denote arbitrary constants. See Theorem 8.4.1 for expressions for the eigenvalues αn and eigenvectors Wn.

      Our purpose in discussing the semidiscrete equation (1.2.9) here is to show that information about the qualitative character of the solution to the continuous problem can be deduced from analysis of the discrete or semidiscrete problems. Before we pursue this point further with this example, we introduce the continuous version of this problem.

      Partial Differential Equation of Heat Conduction

      The continuous model may be obtained from the discrete model, in general, by applying an appropriate limit procedure to the discrete equation as the size of the discrete elements in the model is allowed to decrease to zero. We have already allowed the time interval (t0, t1) to decrease to zero in going from decreases to zero in (1.2.8′) leads to the partial differential equation

      which is often referred to as the one-dimensional heat equation. Here we use the notation

      to indicate the partial derivaties in this equation.

      Equation (1.2.11) is the continuous version of the heat conduction equation in just one dimension. Applying similar limit procedures to (1.2.6) leads to

      This is the full three-dimensional continuous version of the heat conduction equation.

      Auxiliary Conditions for Continuous Problem     In Example 1.2.1 we saw that in the case of a discrete model of heat conduction in an infinitely long rod it was necessary to know the initial state of the temperature in the rod in order to compute all subsequent temperature states. In Example 1.2.2 we saw that when the rod is of finite length, in addition to the initial temperature, temperatures at the ends of the rod must be known in order to uniquely determine all subsequent temperature states in the rod. Based on this experience with the discrete problems, we guess that similar specifications of auxiliary information will be required in the continuous problems.

      An Infinite Rod.     If the initial temperatures u(x, 0) are known for all x, −∞ < x < ∞, then u(x, t) can be computed for all t > 0 from (1.2.11).

      A Rod of Finite Length. Suppose the rod occupies the interval (0, L) of the x axis. If the end temperatures u(0, t) and u(L, t) are known for all t > 0, in addition to knowing u(x, 0) for 0 < x < L, then u(x, t) can be computed for all t > 0 from (1.2.11).

      The first problem here is referred to as a pure initial-value problem while the second is called a mixed initial-boundary-value problem. Later in this chapter we state these problems with more precision, and in Chapters 3 and 5 we solve several examples of such problems. In Chapter 6 we show how to prove there can be no more than one solution to a problem for which the proper auxiliary conditions have been imposed.

      Qualitative Properties of Solutions     We shall see that the solutions to initial-value problems for the heat equation exhibit certain characteristic qualitative behavior, and it is interesting to note that we can anticipate some of this behavior by applying results from linear algebra to the simpler semidiscrete problem. These linear algebra results are collected in the appendix. Their application to this problem is described in detail in Section 8.4. Here we simply summarize the conclusions.

      The matrix [A] appearing in (1.2.9) is symmetric. This implies that the eigenvalues αn are all real and that the corresponding eigenvectors are an orthogonal basis for RN − 1. In addition, the symmetric matrix [A] can be shown to be negative definite, a property that implies that the eigenvalues αn are all negative. Then it follows from (1.2.10) that the solution to the semidiscrete heat equation decreases exponentially to zero with time (in the case that F = 0). We shall see later that the solution to the analogous problem for the continuous heat equation exhibits this same asymptotic behavior. In addition, results apply that are analogous to the negativity of eigenvalues and the existence of an orthogonal basis of eigenvectors.

      In this example we have chosen to derive the discrete model for the conduction process first and to then pass to the semidiscrete and finally the continuous model by applying a limiting procedure as the size of the elements in the discrete model decreases to zero. We could have as well derived the continuous model first, using methods similar to those leading to the discrete model. This is in fact what is usually done in a text of this sort. The discrete model is then obtained by replacing the derivatives in the continuous model by finite-difference expressions resulting from a Taylor series development for the unknown function. As we have remarked previously, deriving the discrete model from the continuous model in this way can lead to discrete models that are inaccurate or even incorrect unless considerable care is taken. It is safer to derive the discrete model directly from discrete versions of the applicable conservation principles.

      Exercises: Heat Conduction Equation

      1.  Consider an infinitely long rod for which the parameters K, σ, C are such that β = 0.1. Then Equation (1.2.8*) becomes

      Suppose

      Then use for n = −5, … , 15 for j = 1, … , 5. For each value of j, for how many n different from zero?

      2.  Repeat Exercise 1 for the situation in which the rod is of finite length L = L. Suppose

      and

      Then use for n = 1, … , 9 and j = 1, … , 5.

      3.  Repeat Exercise 2 in the case that

      4.  Let the rod be as in Exercise 2 but suppose that at the ends of the rod we have the conditions

      Show that these conditions are consistent with the condition of no heat flow near the ends of the rod; the heat flux is zero at x and x = L for n = 0, 1, … , 10 for j = 1, 2, … , 5 if

      5.  Repeat Exercise 4 for a rod of length L = π. Use the values u(x, 0) = sin x evaluated at the nodes xn as the initial values,

      Then use for n = 1, … , 9.

      6.  Let k denote a positive integer and let

      Then verify that uk(x, t) satisfies

      7.  Let uk(x, t) be as in Exercise 6. Then what equation, what initial condition, and what boundary conditions are satisfied by the function v(x, t) = u1(x, t) + 2u2(x, t)?

      8.  Use the values of the function u(x, 0) = sin(10x) evaluated at the nodes xn as the initial values for the discrete model for the problem in = sin(10xn) = sin(nπ), n for n = 1, … , 9. Compare these results with the exact solution

      9.  For all positive integers n, show that the functions

      all satisfy

      In addition, each satisfies at least one of the following boundary conditions:

      (a)  u(0, t) = 0       (b)  u(1, t) = 0

      (c)  ux(0, t) = 0      (d)  ux(1, t) = 0

      Match each function with all of the conditions it satisfies.

      10.  Consider the semidiscrete heat equation (1.2.9) in the case of a rod of finite length. Let the parameters K, Cbe such that μ = 1. Suppose N = 2 and that

      Then write and solve the initial-value problem (1.2.9) for u1(t).

      1.3  STEADY-STATE CONDUCTION OF HEAT

      Equations (1.2.6) and (1.2.11) are, respectively, the discrete and continuous models for an evolution process and as such are called evolution equations. This name derives from the fact that these equations describe a physical process that is evolving with time. Physical processes that are stationary with respect to time lead to somewhat different problems in partial differential equations. For example, suppose the temperature u(x, y, z, t) in (1.2.6) is independent of time. This would be the case if the temperature were observed after all transient behavior had died out and the system had reached a steady state, that is, a state of thermal equilibrium. Then (1.2.6) reduces to

      where we continue to assume that K is a constant. Equation (1.3.1) is known as the discrete Laplace equation for the unknown function u = u(x, y, z). Problems of this type are taken up again in Chapter 10 where they are considered in more detail.

      EXAMPLE 1.3.1

      Two-dimensional Conduction Problem

      Here we only want to illustrate the type of auxiliary data needed in connection with Equation (1.3.1), and for that purpose we consider, for simplicity, an example in which the temperature does not depend on z. Then in determining u = u(x, y), it will be sufficient to consider not all of the region Ω but rather just a planar cross section taken such that the plane is normal to the z axis. Let the plane cross section be denoted by D. Since u is a function only of x and y, Equation (1.3.1) reduces to

      where

      and δyδyu(x, y) is similarly defined. Then Equation (1.3.2) may be rearranged to read

      For the purpose of relating Equation (1.3.2*) to the region D, we must imagine that D , and that the cells are oriented so that their sides lie parallel to the coordinate axes. Each cell has four neighboring cells with which it shares a side, and Equation (1.3.2) is just the statement that the net flow of heat between any cell and its four neighbors is equal to zero. The array of cells is called a grid and the center points of the cells in the grid are called nodes. Suppose the nodes in the array covering D are arranged and numbered as in Figure 1.3.1.

      Let us assume that the function u = u(x, y) is known at all points on the boundary of D, which means then that u is known at the nodes that lie on (or approximately on) the boundary (i.e., nodes 1–10). Then we can seek to determine u at each of the nodes interior to D (i.e., nodes 11–16). It is evident from the figure that placement of the grid on D is somewhat arbitrary and that the nodes considered to be boundary nodes may not actually lie precisely on the boundary. A node is considered to be a boundary node if it lies outside of D but is the center of a cell that shares a side with a cell whose center is an interior node of D.

      Figure 1.3.1

      We apply Equation (1.3.2*) with (x, y) located at each of the nodes 11-16; that is, (1.3.2*) must be satisfied for each of the six cells centered at an interior node of D. For example, at node 11 the equation states

      Similarly

      We can rearrange these equations so that all the unknown quantities (u values corresponding to interior nodes) are on the left side of the equation and all known quantities (u values corresponding to boundary nodes) are on the right. In matrix notation this becomes

      This is a system of linear algebraic equations of the form

      where N denotes the number of interior nodes (in this example N = 6), uN denotes the vector of unknowns (containing the unknown values of u at the interior nodes), and fN denotes the data vector (containing the known values of u at the boundary nodes). Finally, [LN] denotes the N × N coefficient matrix. This matrix can be seen to have some special structure that is the result of the form of the equation (1.3.2). For example, the matrix entries are symmetric about the diagonal and the largest entry in each row is the number 4 that appears on the diagonal. In Chapter 10 we show the properties of [LN] are sufficient to imply that the system (1.3.2’) has a unique solution uN for each data vector fN. We conclude that if u is known at each of the boundary nodes of the grid covering D, then the equation (1.3.2) uniquely determines u at the interior nodes of the grid. This is a discrete version of a well-posed boundary value problem. Chapter 10 contains a more complete discussion of discrete boundary value problems. In that chapter we not only learn how to solve these problems, we will begin to see how the physical properties of the modeled system are reflected in the algebraic properties of the matrix [LN].

      We conclude that if u is known at each of the boundary nodes of the grid covering D, Equation (1.3.2) uniquely determines u at the interior nodes of the grid. Then this is a discrete version of a well-posed boundary value problem. Section 10.3 contains ad ditional information on discrete boundary-value problems.

      Qualitative Properties of Solution

      An interesting property of the solution to the discrete Laplace equation becomes evident when we write (1.3.2*) in the form

      This says that at each node in D, u(x, y) equals the arithmetic average of the values of u at the four nodes adjacent to the node (x, y). Then u at (x, y) cannot exceed the largest of these four values nor can it be smaller than the smallest of the four values. Since this equation applies at each of the nodes in the grid that covers D, it follows that at each interior node the value of u is not greater than the maximum u value attained at a boundary node nor is it smaller than the minimum u value attained at a boundary node. We state this result formally as

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