Summation of asymptotic expansions of multiple-valued functions using algebraic approximants: Application to anharmonic oscillators

Sergeev, A.V.; Goodson, David Z.

doi:10.1088/0305-4470/31/18/018

Cited by 73 publications

(57 citation statements)

References 49 publications

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“…The algebraic approximant enables us to obtain the solution branches while the dominant singularity or criticality in the problem is obtained easily using the differential approximant. For details on the above procedure, interested readers can see ([3], [4], [5]). Using the above procedure on the solution series in section (2), we obtained the results as shown in table (1) below: …”

Section: Bifurcation Studymentioning

confidence: 99%

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Exothermic explosions in a slab: A case study of series summation technique

Makinde¹

2004

International Communications in Heat and Mass Transfer

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This paper examine the steady-state solutions for the strongly exothermic decomposition of a combustible material uniformly distributed between symmetrically heated parallel plates under Bimolecular, Arrhenius and Sensitised reaction rates, neglecting the consumption of the material. Analytical solutions are constructed for the governing nonlinear boundary-value problem using perturbation technique together with a special type of Hermite-Padé approximants and important properties of the temperature field including bifurcations and thermal criticality are discussed.

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Section: Bifurcation Studymentioning

confidence: 99%

“…In the following sections, equation (3) is solved using both perturbation and multivariate series summation techniques ( [3], [4], [5]). …”

Section: Introductionmentioning

confidence: 99%

Exothermic explosions in a slab: A case study of series summation technique

Makinde¹

2004

International Communications in Heat and Mass Transfer

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“…(In the nineteenth century, Hermite and Padé defined ideas similar to Shafer's, but made no applications, and their generalization of ordinary Padé approximants were forgotten.) Shafer/Hermite-Padé approximants have been successfully used in oceanography [19] and quantum mechanics [21]. However, Hermite-Padé approximants have the disadvantage that they are highly nonuniform in the expansion variable because power series are highly nonuniform.…”

Section: Hermite-padé Methodsmentioning

confidence: 99%

Chebyshev expansion on intervals with branch points with application to the root of Kepler’s equation: A Chebyshev–Hermite–Padé method

Boyd

2009

Journal of Computational and Applied Mathematics

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When two or more branches of a function merge, the Chebyshev series of u(λ) will converge very poorly with coefficients a n of T n (λ) falling as O(1/n α ) for some small positive exponent α. However, as shown in [J.P. Boyd, Chebyshev polynomial expansions for simultaneous approximation of two branches of a function with application to the one-dimensional Bratu equation, Appl. Math. Comput. 143 (2002) 189-200], it is possible to obtain approximations that converge exponentially fast in n. If the roots that merge are denoted as u 1 (λ) and u 2 (λ), then both branches can be written without approximation as the roots of (u − u 1 (λ))(u − u 2 (λ)) = u 2 + β(λ)u + γ (λ). By expanding the nonsingular coefficients of the quadratic, β(λ) and γ (λ), as Chebyshev series and then applying the usual roots-of-a-quadratic formula, we can approximate both branches simultaneously with error that decreases proportional to exp(−σ N ) for some constant σ > 0 where N is the truncation of the Chebyshev series. This is dubbed the "Chebyshev-Shafer" or "Chebyshev-Hermite-Padé" method because it substitutes Chebyshev series for power series in the generalized Padé approximants known variously as "Shafer" or "Hermite-Padé" approximants. Here we extend these ideas. First, we explore square roots with branches that are both real-valued and complex-valued in the domain of interest, illustrated by meteorological baroclinic instability. Second, we illustrate triply branched functions via roots of the Kepler equation, f (u; λ, ) ≡ u − sin(u) − λ = 0. Only one of the merging roots is real-valued and the root depends on two parameters (λ, ) rather than one. Nonetheless, the Chebyshev-Hermite-Padé scheme is successful over the whole two-dimensional parameter plane. We also discuss how to cope with poles and logarithmic singularities that arise in our examples at the extremes of the expansion domain.

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“…(second) asymptotic approximation of the error integral for the superasymptotic approximation; 2. resurgence schemes or resummation of late terms [12,13,3]; 3. complex-plane matching of asymptotic expansions [29]; 4. isolation strategies, or rewriting the problem so the exponentially small thing is the only thing [6]; 5. special numerical algorithms, especially spectral methods [6]; 6. hybrid numerical/analytical perturbative schemes [6,18,16]; 7. sequence acceleration including Padé and Hermite-Padé approximants [1,34,31]. There is some overlap between these categories, but each is a whole family of methods and it would obviously take a book [5] to describe them all.…”

Section: F = 4/(1 + Xmentioning

confidence: 99%