asymptotic solutions of difference equations
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11—20 of 36 matching pages
11: Bibliography W
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Uniform asymptotic expansion of via a difference equation.
Numer. Math. 91 (1), pp. 147–193.
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Asymptotic expansions for second-order linear difference equations with a turning point.
Numer. Math. 94 (1), pp. 147–194.
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Asymptotic expansions for second-order linear difference equations. II.
Stud. Appl. Math. 87 (4), pp. 289–324.
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Asymptotic expansions for second-order linear difference equations.
J. Comput. Appl. Math. 41 (1-2), pp. 65–94.
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Asymptotic solutions of a fourth order differential equation.
Stud. Appl. Math. 118 (2), pp. 133–152.
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12: Bibliography V
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A Mathieu equation for ships rolling among waves. I, II.
Norske Vid. Selsk. Forh., Trondheim 22 (25–26), pp. 113–123.
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Symbolic evaluation of coefficients in Airy-type asymptotic expansions.
J. Math. Anal. Appl. 269 (1), pp. 317–331.
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A note on the asymptotic expansion of generalized hypergeometric functions.
Anal. Appl. (Singap.) 12 (1), pp. 107–115.
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Integral representations for products of Lamé functions by use of fundamental solutions.
SIAM J. Math. Anal. 15 (3), pp. 559–569.
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On the rational solutions of the second Painlevé equation.
Differ. Uravn. 1 (1), pp. 79–81 (Russian).
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13: Bibliography J
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Calculus of Finite Differences.
Hungarian Agent Eggenberger Book-Shop, Budapest.
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Calculus of Finite Differences.
3rd edition, AMS Chelsea, Providence, RI.
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Derivation of Green-type, transitional and uniform asymptotic expansions from differential equations. V. Angular oblate spheroidal wavefunctions and for large
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Proc. Roy. Soc. London Ser. A 321, pp. 545–555.
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On Boutroux’s tritronquée solutions of the first Painlevé equation.
Stud. Appl. Math. 107 (3), pp. 253–291.
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The Dirichlet boundary value problem for real solutions of the first Painlevé equation on segments in non-positive semi-axis.
J. Reine Angew. Math. 583, pp. 29–86.
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14: 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(iii) Solutions
… ►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. …15: Bibliography S
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Asymptotic solutions of nonlinear evolution equations and a Painlevé transcendent.
Phys. D 3 (1-2), pp. 165–184.
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Asymptotic Solutions of the One-dimensional Schrödinger Equation.
American Mathematical Society, Providence, RI.
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Error bounds for asymptotic solutions of differential equations. I. The distinct eigenvalue case.
J. Res. Nat. Bur. Standards Sect. B 70B, pp. 167–186.
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Error bounds for asymptotic solutions of differential equations. II. The general case.
J. Res. Nat. Bur. Standards Sect. B 70B, pp. 187–210.
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The relation between asymptotic properties of the second Painlevé equation in different directions towards infinity.
Differ. Uravn. 23 (5), pp. 834–842 (Russian).
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16: Bibliography B
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Integral equations and exact solutions for the fourth Painlevé equation.
Proc. Roy. Soc. London Ser. A 437, pp. 1–24.
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Bäcklund transformations and solution hierarchies for the fourth Painlevé equation.
Stud. Appl. Math. 95 (1), pp. 1–71.
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An Introduction to Linear Difference Equations.
Dover Publications Inc., New York.
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Problem of two Coulomb centres at large intercentre separation: Asymptotic expansions from analytical solutions of the Heun equation.
J. Phys. A 30 (2), pp. 559–571.
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Uniform asymptotic solutions of a class of second-order linear differential equations having a turning point and a regular singularity, with an application to Legendre functions.
SIAM J. Math. Anal. 17 (2), pp. 422–450.
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17: 33.23 Methods of Computation
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§33.23(ii) Series Solutions
… ►§33.23(iii) Integration of Defining Differential Equations
… ►On the other hand, the irregular solutions of §§33.2(iii) and 33.14(iii) need to be integrated in the direction of decreasing radii beginning, for example, with values obtained from asymptotic expansions (§§33.11 and 33.21). … ►This implies decreasing for the regular solutions and increasing for the irregular solutions of §§33.2(iii) and 33.14(iii). … ►Bardin et al. (1972) describes ten different methods for the calculation of and , valid in different regions of the ()-plane. …18: 28.8 Asymptotic Expansions for Large
§28.8 Asymptotic Expansions for Large
… ►Barrett (1981) supplies asymptotic approximations for numerically satisfactory pairs of solutions of both Mathieu’s equation (28.2.1) and the modified Mathieu equation (28.20.1). …It is stated that corresponding uniform approximations can be obtained for other solutions, including the eigensolutions, of the differential equations by application of the results, but these approximations are not included. … ►Dunster (1994a) supplies uniform asymptotic approximations for numerically satisfactory pairs of solutions of Mathieu’s equation (28.2.1). … ►With additional restrictions on , uniform asymptotic approximations for solutions of (28.2.1) and (28.20.1) are also obtained in terms of elementary functions by re-expansions of the Whittaker functions; compare §2.8(ii). …19: 11.13 Methods of Computation
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