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Introduction to Partial Differential Equations with Applications by E. C. Zachmanoglou, Dale W. Thoe



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Introduction to Partial Differential Equations with Applications written by E. C. Zachmanoglou , Professor of Mathematics, Purdue University and Dale W. Thoe , Professor of Mathematics, Purdue University.
Throughout the book, the importance of the proper formulation of problems associated with partial differential equations is emphasized. Methods of solution of any particular problem for a given partial differential equation are discussed only after a large collection of elementary solutions of the equation has been constructed.
During the last five years, the book has been used in the form of lecture notes for a semester course at Purdue University. The students are advanced undergraduate or beginning graduate students in mathematics, engineering or one of the physical sciences. A course in Advanced Calculus or a strong course in Calculus with extensive treatment of functions of several variables, and a very elementary introduction to Ordinary Differential Equations constitute adequate preparation for the understanding of the book. In any case, the basic results of advanced calculus are recalled whenever needed.
The book begins with a short review of calculus and ordinary differential equations. A new elementary treatment of first order quasi-linear partial differential equations is then presented. The geometrical background necessary for the study of these equations is carefully developed. Several applications are discussed such as applications to problems in gas dynamics (the development of shocks), traffic flow, telephone networks, and biology (birth and death processes and control of disease).
The method of probability generating functions in the study of stochastic processes is discussed and illustrated by many examples. In recent books the topic of first order equations is either omitted or treated inadequately. In older books the treatment of this topic is probably inaccessible to most students.

Introduction to Partial Differential Equations with Applications written by E. C. Zachmanoglou, Dale W. Thoe cover the following topics.

  • I. SOMECONCEPTS FROM CALCULUS AND ORDINARY DIFFERENTIAL EQUATIONS
    1. Sets and functions
    2. Surfaces and their normals. The implicit function theorem
    3. Curves and their tangents
    4. The initial value problem for ordinary differential equations and systems

  • II. INTEGRAL CURVES AND SURFACES OF VECTOR FIELDS
    1. Integral curves of vector fields
    2. Methods of solution of dx/P = dy/Q = dz/R
    3. The general solution of + + = 0
    4. Construction of an integral surface of a vector field containing a given curve
    5. Applications to plasma physics and to solenoidal vector fields

  • III. THEORY AND APPLICATIONS OF QUASI-LINEAR AND LINEAR EQUATIONS OF FIRST ORDER
    1. First order partial differential equations
    2. The general integral of + = R
    3. The initial value problem for quasi-linear first order equations. Existence and uniqueness of solution
    4. The initial value problem for quasi-linear first order equations. Nonexistence and nonuniqueness of solutions
    5. The initial value problem for conservation laws. The development of shocks
    6. Applications to problems in traffic flow and gas dynamics
    7. The method of probability generating functions. Applications to a trunking problem in a telephone network and to the control of a tropical disease

  • IV. SERIES SOLUTIONS. THE CAUCHY-KO VALE VSKY THEOREM
    1. Taylor series. Analytic functions
    2. The Cauchy-Kovalevsky theorem

  • V. LINEAR PARTIAL DIFFEREN TIAL EQUATIONS. CHARACTERISTICS, CLASSIFICATION AND CANONICAL FORMS
    1. Linear partial differential operators and their characteristic curves and surfaces
    2. Methods for finding characteristic curves and surfaces. Examples
    3. The importance of characteristics. A very simple example.
    4. The initial value problem for linear first order equations in two independent variables
    5. The general Cauchy problem. The Cauchy-Kovalevsky theorem and Holmgren
    s uniqueness theorem
    6. Canonical form of first order equations
    7. Classification and canonical forms of second order equations in two independent variables
    8. Second order equations in two or more independent variables
    9. The principle of superposition

  • VI. EQUATIONS OF MATHEMATICAL PHYSICS
    1. The divergence theorem and the Green's identities
    2. The equation of heat conduction
    3. Laplace's equation
    4. The wave equation
    5. Well-posed problems

  • VII.LAPLACE'S EQUATION
    1. Harmonic functions
    2. Some elementary harmonic functions. The method of separation of variables
    3. Changes of variables yielding new harmonic functions. Inversion with respect to circles and spheres
    4. Boundary value problems associated with Laplace
    s equation
    5. A representation theorem. The mean value property and the maximum principle for harmonic functions
    6. The well-posedness of the Dirichlet problem
    7. Solution of the Dirichlet problem for the unit disc. Fourier series and Poisson
    s integral
    8. Introduction to Fourier series
    9. Solution of the Dirichlet problem using Green
    s functions.
    10. The Green
    s function and the solution to the Dirichlet problem for a ball in R3
    11. Further properties of harmonic functions
    12. The Dirichlet problem in unbounded domains
    13. Determination of the Green
    s function by the method of electrostatic images
    14. Analytic functions of a complex variable and Laplace
    s equation in two dimensions
    15. The method of finite differences
    16. The Neumann problem

  • VIII. THE WAVE EQUATION
    1. Some solutions of the wave equation. Plane and spherical waves
    2. The initial value problem
    3. The domain of dependence inequality. The energy method.
    4. Uniqueness in the initial value problem. Domain of dependence and range of influence. Conservation of energy
    5. Solution of the initial value problem. Kirchhoff
    s formula. The method of descent
    6. Discussion of the solution of the initial value problem. Huygens
    principle. Diffusion of waves
    7. Wave propagation in regions with boundaries. Uniqueness of solution of the initial-boundary value problem. Reflection of waves
    8. The vibrating string
    9. Vibrations of a rectangular membrane
    10. Vibrations in finite regions. The general method of separation of variables and eigenfunction expansions. Vibrations of a circular membrane

  • IX. THE HEAT EQUATION
    1. Heat conduction in a finite rod. The maximum-minimum principle and its consequences
    2. Solution of the initial-boundary value problem for the onedimensional heat equation
    3. The initial value problem for the one-dimensional heat equation
    4. Heat conduction in more than one space dimension
    5. An application to transistor theory

  • X. SYSTEMS OF FIRST ORDER LINEAR AND QUASI-LINEAR EQUATIONS
    1. Examples of systems. Matrix notation
    2. Linear hyperbolic systems. Reduction to canonical form
    3. The method of characteristics for linear hyperbolic systems. Application to electrical transmission lines
    4. Quasi-linear hyperbolic systems
    5. One-dimensional isentropic flow of an inviscid gas. Simple waves

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