Partial differential equations separation of variables and other methods
by taratuta
Comments
Transcript
Partial differential equations separation of variables and other methods
CONTENTS 18.6 18.7 18.8 18.9 18.10 18.11 18.12 18.13 18.14 Spherical Bessel functions Laguerre functions Associated Laguerre functions Hermite functions Hypergeometric functions Confluent hypergeometric functions The gamma function and related functions Exercises Hints and answers 614 616 621 624 628 633 635 640 646 19 19.1 Quantum operators Operator formalism 648 648 Commutators 19.2 Physical examples of operators 656 Uncertainty principle; angular momentum; creation and annihilation operators 19.3 19.4 Exercises Hints and answers 671 674 20 20.1 Partial differential equations: general and particular solutions Important partial differential equations 675 676 The wave equation; the diffusion equation; Laplace’s equation; Poisson’s equation; Schrödinger’s equation 20.2 20.3 General form of solution General and particular solutions 680 681 First-order equations; inhomogeneous equations and problems; second-order equations 20.4 20.5 20.6 The wave equation The diffusion equation Characteristics and the existence of solutions 693 695 699 First-order equations; second-order equations 20.7 20.8 20.9 Uniqueness of solutions Exercises Hints and answers 705 707 711 21 Partial differential equations: separation of variables and other methods Separation of variables: the general method Superposition of separated solutions Separation of variables in polar coordinates 713 713 717 725 21.1 21.2 21.3 Laplace’s equation in polar coordinates; spherical harmonics; other equations in polar coordinates; solution by expansion; separation of variables for inhomogeneous equations 21.4 Integral transform methods 747 xii