Liouvillean solutions
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1: 31.8 Solutions via Quadratures
§31.8 Solutions via Quadratures
… ►the Hermite–Darboux method (see Whittaker and Watson (1927, pp. 570–572)) can be applied to construct solutions of (31.2.1) expressed in quadratures, as follows. … ►are two independent solutions of (31.2.1). … ►For more details see Smirnov (2002). ►The solutions in this section are finite-term Liouvillean solutions which can be constructed via Kovacic’s algorithm; see §31.14(ii).2: 31.14 General Fuchsian Equation
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31.14.3
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31.14.4
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►An algorithm given in Kovacic (1986) determines if a given (not necessarily Fuchsian) second-order homogeneous linear differential equation with rational coefficients has solutions expressible in finite terms (Liouvillean solutions).
The algorithm returns a list of solutions if they exist.
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3: Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)
Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)
… ►A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3). Such a solution is given in terms of a Riemann theta function with two phases. …The agreement of these solutions with two-dimensional surface water waves in shallow water was considered in Hammack et al. (1989, 1995).4: 31.13 Asymptotic Approximations
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►For asymptotic approximations of the solutions of Heun’s equation (31.2.1) when two singularities are close together, see Lay and Slavyanov (1999).
►For asymptotic approximations of the solutions of confluent forms of Heun’s equation in the neighborhood of irregular singularities, see Komarov et al. (1976), Ronveaux (1995, Parts B,C,D,E), Bogush and Otchik (1997), Slavyanov and Veshev (1997), and Lay et al. (1998).
5: 31.18 Methods of Computation
§31.18 Methods of Computation
►Independent solutions of (31.2.1) can be computed in the neighborhoods of singularities from their Fuchs–Frobenius expansions (§31.3), and elsewhere by numerical integration of (31.2.1). Subsequently, the coefficients in the necessary connection formulas can be calculated numerically by matching the values of solutions and their derivatives at suitably chosen values of ; see Laĭ (1994) and Lay et al. (1998). Care needs to be taken to choose integration paths in such a way that the wanted solution is growing in magnitude along the path at least as rapidly as all other solutions (§3.7(ii)). …6: 16.25 Methods of Computation
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►There is, however, an added feature in the numerical solution of differential equations and difference equations (recurrence relations).
This occurs when the wanted solution is intermediate in asymptotic growth compared with other solutions.
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7: 29.17 Other Solutions
§29.17 Other Solutions
►§29.17(i) Second Solution
►If (29.2.1) admits a Lamé polynomial solution , then a second linearly independent solution is given by …For properties of these solutions see Arscott (1964b, §9.7), Erdélyi et al. (1955, §15.5.1), Shail (1980), and Sleeman (1966b). … ►Algebraic Lamé functions are solutions of (29.2.1) when is half an odd integer. …8: 32.13 Reductions of Partial Differential Equations
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►The Korteweg–de Vries (KdV) equation
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►The sine-Gordon equation
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►The Boussinesq equation
…has the traveling wave solution
…where is an arbitrary constant and satisfies
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9: 28.29 Definitions and Basic Properties
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►Then (28.29.1) has a nontrivial solution
with the pseudoperiodic property
…A solution satisfying (28.29.7) is called a Floquet solution with respect to
(or Floquet solution).
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