Partial Differential Equations

Contents Preface 1. First-order Partial Differential Equations 1.1 Linear PDEs 1.2 First-order PDEs and Characteristic Strips 1.2.1 Integral surfaces 1.2.2 Characteristic curves and characteristic strips 1.2.3 Solution of the first-order quasi-linear PDEs 1.3 Initial-value Problems of the First-order PDEs 1.4 Complete Integrals 1.4.1 Complete solutions, general solutions, and singular solutions 1.4.2 First-order PDEs with easily obtainable complete solutions 1.4.3 Integrable condition 1.4.4 Lagrange–Charpit method 1.5 Hamilton–Jacobi Theory 1.5.1 Extremum problems and the Euler–Lagrange equation 1.5.2 Lagrange's equation and canonical equation 1.5.3 Canonical transform 1.5.4 Hamilton–Jacobi equation 1.6 Integral of the Hamilton–Jacobi equation 2. Second-order Partial Differential Equations 2.1 Examples of Second-order PDEs 2.1.1 Wave equation 2.1.2 Diffusion equation 2.1.3 Laplace–Poisson equation 2.2 Classification of Second-order PDEs 2.2.1 Classification (hyperbolic, elliptic, parabolic) 2.2.2 Integral surfaces and initial-value problems 2.3 Separation of Variables and Eigenvalue Problems 2.3.1 Orthogonal curvilinear coordinates and separation of variables 2.3.2 Eigenvalues and eigenfunctions in the Sturm–Liouville equations 2.3.3 Eigenfunction expansions 2.3.4 Fourier–Bessel Expansions 2.4 Green's Function and the Boundary-value Problems 2.4.1 Delta function 2.4.2 Adjoint PDE and Green's formulae 2.4.3 Green’s function 2.4.4 Principal solution of the Laplace equation 2.4.5 Adjoint Green's function and reciprocity principle 2.4.6 Principal solution of the wave equation 2.4.7 Principal solution of the diffusion equation 2.5 Initial- and/or Boundary-value Problems in Rectangular Regions 2.5.1 Laplace equation 2.5.2 Wave equation 2.5.3 Diffusion equation 2.6 Initial- and/or Boundary-value Problems in Circular or Spherical Regions 2.6.1 Dirichlet's problems for circular or spherical regions 2.6.2 Vibration of a circular membrane 2.6.3 Diffusion in a circular region 2.6.4 Radiation and scattering of waves 3. Integral Transforms and their Applications 3.1 General Theory of Integral Transforms and their Applicability 3.2 Integral Transforms in a Finite Region 3.2.1 General theory 3.2.2 Further applications 3.3 Integral Transforms in an Infinite Region 3.3.1 General theory 3.3.2 Fourier transforms 3.3.3 Laplace transforms 3.3.4 Mellin transforms 3.3.5 Hankel transforms 3.3.6 Application of integral transforms to integral equations Appendix A A.1 Gamma Function and Beta Function A.1.1 Gamma function A.1.2 Beta function A.2 Bessel Functions A.2.1 Bessel functions and Neumann functions A.2.2 Hankel functions A.2.3 Spherical Bessel functions A.2.4 Modified Bessel functions A.3 Legendre Functions A.3.1 Legendre functions A.3.2 Associated Legendre functions A.3.3 Spherical surface harmonic functions Bibliography Index