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Partial differential equations separation of variables and other methods

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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
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