Book contents
- Frontmatter
- Contents
- Preface
- Chapter 1 Introduction
- Chapter 2 Typical equations of mathematical physics. Boundary conditions
- Chapter 3 Cauchy problem for first-order partial differential equations
- Chapter 4 Classification of second-order partial differential equations with linear principal part. Elements of the theory of characteristics
- Chapter 5 Cauchy and mixed problems for the wave equation in ℝ1. Method of traveling waves
- Chapter 6 Cauchy and Goursat problems for a second-order linear hyperbolic equation with two independent variables. Riemann's method
- Chapter 7 Cauchy problem for a 2-dimensional wave equation. The Volterra–D'Adhemar solution
- Chapter 8 Cauchy problem for the wave equation in ℝ3. Methods of averaging and descent. Huygens's principle
- Chapter 9 Basic properties of harmonic functions
- Chapter 10 Green's functions
- Chapter 11 Sequences of harmonic functions. Perron's theorem. Schwarz alternating method
- Chapter 12 Outer boundary-value problems. Elements of potential theory
- Chapter 13 Cauchy problem for heat-conduction equation
- Chapter 14 Maximum principle for parabolic equations
- Chapter 15 Application of Green's formulas. Fundamental identity. Green's functions for Fourier equation
- Chapter 16 Heat potentials
- Chapter 17 Volterra integral equations and their application to solution of boundary-value problems in heat-conduction theory
- Chapter 18 Sequences of parabolic functions
- Chapter 19 Fourier method for bounded regions
- Chapter 20 Integral transform method in unbounded regions
- Chapter 21 Asymptotic expansions. Asymptotic solution of boundary-value problems
- Appendix 1 Elements of vector analysis
- Appendix 2 Elements of theory of Bessel functions
- Appendix 3 Fourier's method and Sturm–Liouville equations
- Appendix 4 Fourier integral
- Appendix 5 Examples of solution of nontrivial engineering and physical problems
- References
- Index
Preface
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- Chapter 1 Introduction
- Chapter 2 Typical equations of mathematical physics. Boundary conditions
- Chapter 3 Cauchy problem for first-order partial differential equations
- Chapter 4 Classification of second-order partial differential equations with linear principal part. Elements of the theory of characteristics
- Chapter 5 Cauchy and mixed problems for the wave equation in ℝ1. Method of traveling waves
- Chapter 6 Cauchy and Goursat problems for a second-order linear hyperbolic equation with two independent variables. Riemann's method
- Chapter 7 Cauchy problem for a 2-dimensional wave equation. The Volterra–D'Adhemar solution
- Chapter 8 Cauchy problem for the wave equation in ℝ3. Methods of averaging and descent. Huygens's principle
- Chapter 9 Basic properties of harmonic functions
- Chapter 10 Green's functions
- Chapter 11 Sequences of harmonic functions. Perron's theorem. Schwarz alternating method
- Chapter 12 Outer boundary-value problems. Elements of potential theory
- Chapter 13 Cauchy problem for heat-conduction equation
- Chapter 14 Maximum principle for parabolic equations
- Chapter 15 Application of Green's formulas. Fundamental identity. Green's functions for Fourier equation
- Chapter 16 Heat potentials
- Chapter 17 Volterra integral equations and their application to solution of boundary-value problems in heat-conduction theory
- Chapter 18 Sequences of parabolic functions
- Chapter 19 Fourier method for bounded regions
- Chapter 20 Integral transform method in unbounded regions
- Chapter 21 Asymptotic expansions. Asymptotic solution of boundary-value problems
- Appendix 1 Elements of vector analysis
- Appendix 2 Elements of theory of Bessel functions
- Appendix 3 Fourier's method and Sturm–Liouville equations
- Appendix 4 Fourier integral
- Appendix 5 Examples of solution of nontrivial engineering and physical problems
- References
- Index
Summary
This book represents an attempt to implement a general approach that in essence views the theory of partial differential equations (PDEs) of mathematical physics as the language of continuous processes, that is, an interdisciplinary science that considers the hierarchy of mathematical phenomena as a reflection of their physical counterparts. A comprehensive, mathematically rigorous account of the classical theory of PDEs in mathematical physics is thus inseparably bound with the features of the corresponding natural continuum objects. We shall therefore endeavor to trace the simultaneous origins of some basic mathematical objects in different natural contexts (continuum mechanics, electrodynamics, transport phenomena, thermodynamics, and chemical kinetics). In parallel, we shall trace the interrelation between different types of problems (elliptic, parabolic, and hyperbolic) as mathematical counterparts of their natural prototypes: steady-state and evolutionary processes (dissipative and conservative). This will be done by an asymptotic analysis of the behavior of these processes in time and their dependence on the relevant governing parameters.
In view of the almost complete absence of a physics background in undergraduate and graduate curricula of mathematics and applied mathematics, it seems important, in a course of mathematical physics, to provide an introduction to the basic concepts of different natural sciences and their relation with PDEs in terms of certain typical boundary-value problems that recur in different scientific contexts. Chapters 1 and 2 are therefore addressed primarily to students of mathematics.
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- Information
- Partial Differential Equations in Classical Mathematical Physics , pp. xiii - xivPublisher: Cambridge University PressPrint publication year: 1994