Chapter 1 Introduction 1.1 Equations of nmthematieal physics 1.2 Basic concept and definition 1.3 Linear operator Exercises Chapter 2 Mathematical models and problems for defining solutions 2.1 Typical equations 2.2 String oscillation 2.3 Membrane oscillation 2.4 Heat conduction in solid 2.5 Gravitation potential 2.6 The conditions and problems for defining solutions 2.7 Principle of superposition Chapter 3 Classification and simplification for linear partial differential equations of second order 3.1 Linear second order partial differential equations with two variables 3.2 Simplification and standard forms 3.3 Examples Exercises Chapter 4 Integral method on characteristics 4.1 D'Alembert formula of Cauchy problem for string oscillation' 4.2 Small oscillations of semi-infinite and finite strings with rigidly fixed or free ends, method of prolongation 4.3 Three-dimensional wave equation 4.4 The method for descending dimension 4.5 Cauchy problem for non-homogeneous wave equation 4.6 Integral method on characteristics for second order hyperbolic equations with two variables Exercises Chapter 5 The method of separating variables on finite region 5.1 Separation of variables 5.2 The process by separation of variables for solving mixed problel on string oscillation 5.3 The application of the method on separating variables 5.4 Non-homogeneous problems 5.5 Uniqueness of the solutions for two mixed problems Exercises Chapter 6 Eigenvalue problems and special functions 6.1 Sturm-Liouville problem 6.2 Eigenfunctions 6.3 The boundary value problem of ordinary differential equation and Green function 6.4 The construction of Green function 6.5 Eigenvalue problem and Green function 6.6 Bessel function 6.7 Singular Sturm-Liouville problem 6.8 Legendre function Exercises Chapter 7 Multidimensional boundary value problems 7,1 Dirichlet problem in cube 7.2 Dirichlet problem in cylindrical body 7.3 Boundary value problems in a sphere 7.4 Membrane oscillation on rectangular region 7.5 Heat conduction on rectangular plate 7.6 Wave in three-dimensional cube 7.7 Heat conduction in cube 7.8 The problem on hydrogen atom 7.9 Forced vibration on membrane Exercises Chapter 8 Integral transformations 8.1 Fourier integral transformation 8.2 The properties of Fourier transformation 8.3 Application of Fourier integral transformation 8.4 Laplace integral transformation 8.5 Application of Laplace integral transformation Exercises Chapter 9 Basic properties of harmonic functions 9.1 Convex, linear, and concave functions in R1 9.2 Superhamonic, harmonic, and subharmonic functions in multidimen- sional regions 9.3 Hopf lemma and strong maximum principle 9.4 Green formulas, uniqueness theorems 9.5 Integral identity, mean value theorem, inverse mean value theorem Chapter 10 Green function and their application to PDEs 10.1 Definition and main properties concerning Laplace operator 10.2 The method of superposition of sources and sinks 10.3 Poisson integral Supplement Exercises Selected answers for exercises Appendix A Appendix B Appendix C
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