Partial Differential Equations in Classical Mathematical Physics

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.