diffusion%20equations
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1: 30.2 Differential Equations
§30.2 Differential Equations
►§30.2(i) Spheroidal Differential Equation
… ► … ►The Liouville normal form of equation (30.2.1) is … ►§30.2(iii) Special Cases
…2: 15.10 Hypergeometric Differential Equation
§15.10 Hypergeometric Differential Equation
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15.10.1
►This is the hypergeometric differential equation.
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►The connection formulas for the principal branches of Kummer’s solutions are:
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3: 31.2 Differential Equations
§31.2 Differential Equations
►§31.2(i) Heun’s Equation
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31.2.1
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§31.2(v) Heun’s Equation Automorphisms
… ►Composite Transformations
…4: 29.2 Differential Equations
§29.2 Differential Equations
►§29.2(i) Lamé’s Equation
… ►§29.2(ii) Other Forms
… ►Equation (29.2.10) is a special case of Heun’s equation (31.2.1).5: 32.2 Differential Equations
§32.2 Differential Equations
►§32.2(i) Introduction
►The six Painlevé equations – are as follows: … ►§32.2(ii) Renormalizations
… ► …6: 28.2 Definitions and Basic Properties
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§28.2(i) Mathieu’s Equation
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28.2.1
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►This is the characteristic equation of Mathieu’s equation (28.2.1).
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§28.2(iv) Floquet Solutions
… ► …7: 28.20 Definitions and Basic Properties
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§28.20(i) Modified Mathieu’s Equation
►When is replaced by , (28.2.1) becomes the modified Mathieu’s equation: ►
28.20.1
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28.20.2
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►Then from §2.7(ii) it is seen that equation (28.20.2) has independent and unique solutions that are asymptotic to as in the respective sectors , being an arbitrary small positive constant.
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8: 20.13 Physical Applications
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►with .
►For , with real, (20.13.1) takes the form of a real-time
diffusion equation
…with diffusion constant .
…Theta-function solutions to the heat diffusion equation with simple boundary conditions are discussed in Lawden (1989, pp. 1–3), and with more general boundary conditions in Körner (1989, pp. 274–281).
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►This allows analytic time propagation of quantum wave-packets in a box, or on a ring, as closed-form solutions of the time-dependent Schrödinger equation.
9: 10.73 Physical Applications
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►See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25).
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►For applications of the Rayleigh function (§10.21(xiii)) to problems of heat conduction and diffusion in liquids see Kapitsa (1951a).
10: 28.33 Physical Applications
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►The wave equation
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►The equation of motion is given by
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►In particular, the equation is stable for all sufficiently large values of .
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Initial-value problems, in which only one equation (28.2.1) or (28.20.1) is involved. See §28.33(iii).
McLachlan (1947, Chapters XVI–XIX) for applications of the wave equation to vibrational systems, electrical and thermal diffusion, electromagnetic wave guides, elliptical cylinders in viscous fluids, and diffraction of sound and electromagnetic waves.