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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.
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.
Excerpted from APPLIED PARTIAL DIFFERENTIAL EQUATIONS by Paul DuChateau, David Zachmann. Copyright © 1989 Harper & Row, Publishers, Inc.. Excerpted by permission of Dover Publications, Inc..
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