Synopses & Reviews
This book considers the theory of partial differential equations as the language of continuous processes in mathematical physics. This is an interdisciplinary area in which the mathematical phenomena are reflections of their physical counterparts. The authors trace the development of these mathematical phenomena in different natural sciences, with examples drawn from continuum mechanics, electrodynamics, transport phenomena, thermodynamics, and chemical kinetics. At the same time, the authors trace the interrelation between the different types of problems--elliptic, parabolic, and hyperbolic--as the mathematical counterparts of stationary and evolutionary processes. This combination of mathematical comprehensiveness and natural scientific motivation represents a step forward in the presentation of the classical theory of PDEs, one that will be appreciated by students and researchers in applied mathematics and mathematical physics.
Review
"...A rigorous, systematic treatment of mathematics applied in classical physics." The American Mathematical Monthly
Review
"...a comprehensive account of the basic principles and applications of the classical theory of partial differential equations in mathematical physics...well-written for graduate students in physics, engineering, and applied mathematics sequences, and scientists and engineers whose projects require knowledge of equations of mathematical physics...extremely valuable appendixes review important mathematical concepts. Recommended." Choice
Review
"I enjoy paging through books like this one....[The authors] have succeeded in their goal of treating this field from an interdisciplinary, but unified standpoint. It is a nice book." SIAM Review
Synopsis
The book's combination of mathematical comprehensiveness and natural scientific motivation represents a step forward in the presentation of the classical theory of PDEs.
Synopsis
The unique feature of this book is that it considers the theory of partial differential equations in mathematical physics as the language of continuous processes, that is, as an interdisciplinary science that treats the hierarchy of mathematical phenomena as reflections of their physical counterparts. The bookâs combination of mathematical comprehensiveness and natural scientific motivation represents a step forward in the presentation of the classical theory of PDEs, one that will be appreciated by both students and researchers alike.
Synopsis
This text considers the theory of partial differential equations as the language of continuous processes in mathematical physics. Representing a step forward in the presentation of the classical theory of PDEs, it will be appreciated by mathematicians as well as physicists.
Table of Contents
Preface; 1. Introduction; 2. Typical equations of mathematical physics. Boundary conditions; 3. Cauchy problem for first-order partial differential equations; 4. Classification of second-order partial differential equations with linear principal part. Elements of the theory of characteristics; 5. Cauchy and mixed problems for the wave equation in R1. Method of travelling waves; 6. Cauchy and Goursat problems for a second-order linear hyperbolic equation with two independent variables. Riemann's method; 7. Cauchy problem for a 2-dimensional wave equation. The Volterra-D'Adhemar solution; 8. Cauchy problem for the wave equation in R3. Methods of averaging and descent. Huygens's principle; 9. Basic properties of harmonic functions; 10. Green's functions; 11. Sequences of harmonic functions. Perron's theorem. Schwarz alternating method; 12. Outer boundary-value problems. Elements of potential theory; 13. Cauchy problem for heat-conduction equation; 14. Maximum principle for parabolic equations; 15. Application of Green's formulas. Fundamental identity. Green's functions for Fourier equation; 16. Heat potentials; 17. Volterra integral equations and their application to solution of boundary-value problems in heat-conduction theory; 18. Sequences of parabolic functions; 19. Fourier method for bounded regions; 20. Integral transform method in unbounded regions; 21. Asymptotic expansions. Asymptotic solution of boundary-value problems; Appendix I. Elements of vector analysis; Appendix II. Elements of theory of Bessel functions; Appendix III. Fourier's method and Sturm-Liouville equations; Appendix IV. Fourier integral; Appendix V. Examples of solution of nontrivial engineering and physical problems; References; Index.