delta wing equation
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1: 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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►This equation has regular singularities at , with corresponding exponents , , , , respectively (§2.7(i)).
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►Next, satisfies (31.2.1) if is a solution of (31.2.1) with transformed parameters ; , , .
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2: 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
…3: 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).4: 15.10 Hypergeometric Differential Equation
§15.10 Hypergeometric Differential Equation
►§15.10(i) Fundamental Solutions
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15.10.1
►This is the hypergeometric differential equation.
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5: 32.2 Differential Equations
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►The six Painlevé equations
– are as follows:
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►with , , , and arbitrary constants.
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►If in , then set and , without loss of generality, by rescaling and if necessary.
If and in , then set and , without loss of generality.
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►If in , then set , without loss of generality.
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6: 28.20 Definitions and Basic Properties
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§28.20(i) Modified Mathieu’s Equation
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28.20.1
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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.
…as with , and
…as with .
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7: 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
… ► …8: 29.19 Physical Applications
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§29.19(ii) Lamé Polynomials
►Ward (1987) computes finite-gap potentials associated with the periodic Korteweg–de Vries equation. …Hargrave (1978) studies high frequency solutions of the delta wing equation. …Roper (1951) solves the linearized supersonic flow equations. Clarkson (1991) solves nonlinear evolution equations. …9: 1.17 Integral and Series Representations of the Dirac Delta
§1.17 Integral and Series Representations of the Dirac Delta
►§1.17(i) Delta Sequences
… ►Sine and Cosine Functions
… ►Coulomb Functions (§33.14(iv))
… ►Equations (1.17.12_1) through (1.17.16) may re-interpreted as spectral representations of completeness relations, expressed in terms of Dirac delta distributions, as discussed in §1.18(v), and §1.18(vi) Further mathematical underpinnings are referenced in §1.17(iv). …10: Bibliography H
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Solving Ordinary Differential Equations. I. Nonstiff Problems.
2nd edition, Springer Series in Computational Mathematics, Vol. 8, Springer-Verlag, Berlin.
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High frequency solutions of the delta wing equations.
Proc. Roy. Soc. Edinburgh Sect. A 81 (3-4), pp. 299–316.
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Poncelet Polygons and the Painlevé Equations.
In Geometry and Analysis (Bombay, 1992), Ramanan (Ed.),
pp. 151–185.
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Estimates of the stability intervals for Hill’s equation.
Proc. Amer. Math. Soc. 14 (6), pp. 930–932.
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Differential Equations: A Modern Approach.
Holt, Rinehart and Winston, New York.
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