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Journal articles on the topic 'Asymptotic expansions of solutions'

Author: Grafiati

Published: 4 June 2021

Last updated: 3 February 2022

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1

MAJDALANI, JOSEPH. "Multiple asymptotic solutions for axially travelling waves in porous channels." Journal of Fluid Mechanics 636 (September 25, 2009): 59–89. http://dx.doi.org/10.1017/s0022112009007939.

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Travelling waves in confined enclosures, such as porous channels, develop boundary layers that evolve over varying spatial scales. The present analysis employs a technique that circumvents guessing of the inner coordinate transformations at the forefront of a multiple-scales expansion. The work extends a former study in which a two-dimensional oscillatory solution was derived for the rotational travelling wave in a porous channel. This asymptotic solution was based on a free coordinate that could be evaluated using Prandtl's principle of matching with supplementary expansions. Its derivation required matching the dominant term in the multiple-scales expansion to an available Wentzel-Kramers-Brillouin (WKB) solution. Presently, the principle of least singular behaviour is used. This approach leads to a multiple-scales approximation that can be obtained independently of supplementary expansions. Furthermore, a procedure that yields different types of WKB solutions is described and extended to arbitrary order in the viscous perturbation parameter. Among those, the WKB expansion of type I is shown to exhibit an alternating singularity at odd orders in the perturbation parameter. This singularity is identified and suppressed using matched asymptotic tools. In contrast, the WKB expansion of type II is found to be uniformly valid at any order. Additionally, matched asymptotic, WKB and multiple-scales expansions are developed for several test cases. These enable us to characterize the essential vortico-acoustic features of the axially travelling waves in a porous channel. All solutions are numerically verified, compared and discussed.

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2

Fahrenwaldt, Matthias A. "Short-time asymptotic expansions of semilinear evolution equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 146, no. 1 (January 7, 2016): 141–67. http://dx.doi.org/10.1017/s0308210515000372.

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We develop an algebraic approach to constructing short-time asymptotic expansions of solutions of a class of abstract semilinear evolution equations. The expansions are typically valid for both the solution of the equation and its gradient. We apply a perturbation approach based on the symbolic calculus of pseudo-differential operators and heat kernel methods. The construction is explicit and can be done to arbitrary order. All results are rigorously formulated in terms of Banach algebras. As an application we obtain a novel approach to finding approximate solutions of Markovian backward stochastic differential equations.

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3

Meftah, Safia. "A New Approach to Approximate Solutions for Nonlinear Differential Equation." International Journal of Mathematics and Mathematical Sciences 2018 (July 16, 2018): 1–8. http://dx.doi.org/10.1155/2018/5129502.

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The question discussed in this study concerns one of the most helpful approximation methods, namely, the expansion of a solution of a differential equation in a series in powers of a small parameter. We used the Lindstedt-Poincaré perturbation method to construct a solution closer to uniformly valid asymptotic expansions for periodic solutions of second-order nonlinear differential equations.

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4

Steinrück, Herbert. "Mixed convection over a cooled horizontal plate: non-uniqueness and numerical instabilities of the boundary-layer equations." Journal of Fluid Mechanics 278 (November 10, 1994): 251–65. http://dx.doi.org/10.1017/s0022112094003691.

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The boundary-layer flow over a cooled horizontal plate is considered. It is shown that the real part of the spectrum of the evolution operator of the linearized equations is not bounded uniformly from above which explains the difficulties encounterd by a numerical solution. Furthermore it is shown that near the leading edge an asymptotic expansion of the solution is not unique. A one-parametric family of asymptotic expansions of solutions can be constructed.

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5

Archibasov, A. A. "ASYMPTOTIC EXPANSIONS OF SOLUTIONS FOR HIV EVOLUTION MODEL." Vestnik of Samara University. Natural Science Series 19, no. 3 (June 1, 2017): 5–11. http://dx.doi.org/10.18287/2541-7525-2013-19-3-5-11.

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In the paper the mathematical model of HIV evolution is considered. This model is a singularly perturbed partial integro-differential equations system. Based on the Tikhonov—Vasilieva method of boundary function the first approximation of the system solutions is realized.

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6

Geng, Jun, and Zhongwei Shen. "Asymptotic expansions of fundamental solutions in parabolic homogenization." Analysis & PDE 13, no. 1 (January 6, 2020): 147–70. http://dx.doi.org/10.2140/apde.2020.13.147.

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7

Samovol, V. S. "Asymptotic Expansions of Solutions to the Riccati Equation." Doklady Mathematics 101, no. 1 (January 2020): 49–52. http://dx.doi.org/10.1134/s1064562420010196.

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8

Ishige, Kazuhiro, Tatsuki Kawakami, and Hironori Michihisa. "Asymptotic Expansions of Solutions of Fractional Diffusion Equations." SIAM Journal on Mathematical Analysis 49, no. 3 (January 2017): 2167–90. http://dx.doi.org/10.1137/16m1101428.

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9

Schulz, Friedmar. "Asymptotic expansions for solutions of elliptic differential inequalities." Analysis 14, no. 2-3 (September 1994): 139–46. http://dx.doi.org/10.1524/anly.1994.14.23.139.

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10

Caflisch, R. E. "Asymptotic expansions of solutions for the Boltzmann equation." Transport Theory and Statistical Physics 16, no. 4-6 (June 1987): 701–25. http://dx.doi.org/10.1080/00411458708204310.

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11

Jha, Shing-Whu, Attila Máté, and Paul Nevai. "Asymptotic expansions for solutions of smooth recurrence equations." Proceedings of the American Mathematical Society 110, no. 2 (February 1, 1990): 365. http://dx.doi.org/10.1090/s0002-9939-1990-1014646-7.

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12

Shchitov, I. N. "Asymptotic expansions of solutions to singularly perturbed systems." Ukrainian Mathematical Journal 45, no. 4 (April 1993): 598–608. http://dx.doi.org/10.1007/bf01062954.

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13

Lastra, Alberto, and Stéphane Malek. "On a q-Analog of a Singularly Perturbed Problem of Irregular Type with Two Complex Time Variables." Mathematics 7, no. 10 (October 3, 2019): 924. http://dx.doi.org/10.3390/math7100924.

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The analytic solutions of a family of singularly perturbed q-difference-differential equations in the complex domain are constructed and studied from an asymptotic point of view with respect to the perturbation parameter. Two types of holomorphic solutions, the so-called inner and outer solutions, are considered. Each of them holds a particular asymptotic relation with the formal ones in terms of asymptotic expansions in the perturbation parameter. The growth rate in the asymptotics leans on the - 1 -branch of Lambert W function, which turns out to be crucial.

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14

Ferreira, Chelo, José L. López, and Ester Pérez Sinusía. "Convergent and asymptotic expansions of solutions of second-order differential equations with a large parameter." Analysis and Applications 12, no. 05 (August 28, 2014): 523–36. http://dx.doi.org/10.1142/s0219530514500328.

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We consider the second-order linear differential equation [Formula: see text] where x ∈ [0, X], X > 0, α ∈ (-∞, 2), Λ is a large complex parameter and g is a continuous function. The asymptotic method designed by Olver [Asymptotics and Special Functions (Academic Press, New York, 1974)] gives the Poincaré-type asymptotic expansion of two independent solutions of the equation in inverse powers of Λ. We add initial conditions to the differential equation and consider the corresponding initial value problem. By using the Green's function of an auxiliary problem, we transform the initial value problem into a Volterra integral equation of the second kind. Then using a fixed point theorem, we construct a sequence of functions that converges to the unique solution of the problem. This sequence has also the property of being an asymptotic expansion for large Λ (not of Poincaré-type) of the solution of the problem. Moreover, we show that the idea may be applied also to nonlinear differential equations with a large parameter.

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15

Gaulter, Simon N., and Nicholas R. T. Biggs. "Acoustic trapped modes in a three-dimensional waveguide of slowly varying cross section." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 469, no. 2149 (January 8, 2013): 20120384. http://dx.doi.org/10.1098/rspa.2012.0384.

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In this paper, we develop an asymptotic scheme to approximate the trapped mode solutions to the time harmonic wave equation in a three-dimensional waveguide with a smooth but otherwise arbitrarily shaped cross section and a single, slowly varying ‘bulge’, symmetric in the longitudinal direction. Extending previous research carried out in the two-dimensional case, we first use a WKBJ-type ansatz to identify the possible quasi-mode solutions that propagate only in the thicker region, and hence find a finite cut-on region of oscillatory behaviour and asymptotic decay elsewhere. The WKBJ expansions are used to identify a turning point between the cut-on and cut-off regions. We note that the expansions are non-uniform in an interior layer centred on this point, and we use the method of matched asymptotic expansions to connect the cut-on and cut-off regions within this layer. The behaviour of the expansions within the interior layer then motivates the construction of a uniformly valid asymptotic expansion. Finally, we use this expansion and the symmetry of the waveguide around the longitudinal centre, x =0, to extract trapped mode wavenumbers, which are compared with those found using a numerical scheme and seen to be extremely accurate, even to relatively large values of the small parameter.

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16

Gie, Gung-Min, Chang-Yeol Jung, and Roger Temam. "Analysis of Mixed Elliptic and Parabolic Boundary Layers with Corners." International Journal of Differential Equations 2013 (2013): 1–13. http://dx.doi.org/10.1155/2013/532987.

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We study the asymptotic behavior at small diffusivity of the solutions,uε, to a convection-diffusion equation in a rectangular domainΩ. The diffusive equation is supplemented with a Dirichlet boundary condition, which is smooth along the edges and continuous at the corners. To resolve the discrepancy, on∂Ω, betweenuεand the corresponding limit solution,u0, we propose asymptotic expansions ofuεat any arbitrary, but fixed, order. In order to manage some singular effects near the four corners ofΩ, the so-called elliptic and ordinary corner correctors are added in the asymptotic expansions as well as the parabolic and classical boundary layer functions. Then, performing the energy estimates on the difference ofuεand the proposed expansions, the validity of our asymptotic expansions is established in suitable Sobolev spaces.

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17

Olde Daalhuis, A. B. "INVERSE FACTORIAL-SERIES SOLUTIONS OF DIFFERENCE EQUATIONS." Proceedings of the Edinburgh Mathematical Society 47, no. 2 (June 2004): 421–48. http://dx.doi.org/10.1017/s0013091503000609.

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AbstractWe obtain inverse factorial-series solutions of second-order linear difference equations with a singularity of rank one at infinity. It is shown that the Borel plane of these series is relatively simple, and that in certain cases the asymptotic expansions incorporate simple resurgence properties. Two examples are included. The second example is the large $a$ asymptotics of the hypergeometric function ${}_2F_1(a,b;c;x)$.AMS 2000 Mathematics subject classification: Primary 34E05; 39A11. Secondary 33C05

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18

Bruno, A. D., and I. V. Goryuchkina. "Asymptotic expansions of solutions of the sixth Painlevé equation." Transactions of the Moscow Mathematical Society 71 (2010): 1. http://dx.doi.org/10.1090/s0077-1554-2010-00186-0.

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19

MCLEAN, W. "Asymptotic Error Expansions for Numerical Solutions of Integral Equations." IMA Journal of Numerical Analysis 9, no. 3 (1989): 373–84. http://dx.doi.org/10.1093/imanum/9.3.373.

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20

Chen, Xinfu, and Susmita Sadhu. "Uniform asymptotic expansions of solutions of an inhomogeneous equation." Journal of Differential Equations 253, no. 3 (August 2012): 951–76. http://dx.doi.org/10.1016/j.jde.2012.04.018.

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21

CAO, LI-HUA, and YU-TIAN LI. "LINEAR DIFFERENCE EQUATIONS WITH A TRANSITION POINT AT THE ORIGIN." Analysis and Applications 12, no. 01 (December 13, 2013): 75–106. http://dx.doi.org/10.1142/s0219530513500371.

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A pair of linearly independent asymptotic solutions are constructed for the second-order linear difference equation [Formula: see text] where An and Bn have asymptotic expansions of the form [Formula: see text] with θ ≠ 0 and α0 ≠ 0 being real numbers, and β0 = ±2. Our result holds uniformly for the scaled variable t in an infinite interval containing the transition point t1 = 0, where t = (n + τ0)-θx and τ0 is a small shift. In particular, it is shown how the Bessel functions Jν and Yν get involved in the uniform asymptotic expansions of the solutions to the above linear difference equation. As an illustration of the main result, we derive a uniform asymptotic expansion for the orthogonal polynomials associated with the Laguerre-type weight xα exp (-qmxm), x > 0, where m is a positive integer, α > -1 and qm > 0.

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22

FRIDMAN, G. M. "Matched asymptotics for two-dimensional planing hydrofoils with spoilers." Journal of Fluid Mechanics 358 (March 10, 1998): 259–81. http://dx.doi.org/10.1017/s0022112097008215.

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The purpose of the paper is to demonstrate the effectiveness of the matched asymptotic expansions (MAE) method for the planing flow problem. The matched asymptotics, taking into account the flow nonlinearities in those regions where they are most pronounced (i.e. in the vicinity of the edges), are shown to significantly extend the range where the linear theory gives good results. Two model problems are used: the planing flat plate with a spoiler on the trailing edge and the curved planing foil. Asymptotic solutions obtained by the MAE method are compared with those obtained using linear and exact nonlinear theories. Based on the results, the asymptotic solution to the planing problem under the gravity is proposed.

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23

Olver, F. W. J. "Asymptotic expansions of the coefficients in asymptotic series solutions of linear differential equations." Methods and Applications of Analysis 1, no. 1 (1994): 1–13. http://dx.doi.org/10.4310/maa.1994.v1.n1.a1.

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24

CHRUŚCIEL, PIOTR T., and SZYMON ŁȨSKI. "POLYHOMOGENEOUS SOLUTIONS OF NONLINEAR WAVE EQUATIONS WITHOUT CORNER CONDITIONS." Journal of Hyperbolic Differential Equations 03, no. 01 (March 2006): 81–141. http://dx.doi.org/10.1142/s0219891606000732.

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The study of Einstein equations leads naturally to Cauchy problems with initial data on hypersurfaces which closely resemble hyperboloids in Minkowski space-time, and with initial data with polyhomogeneous asymptotics, that is, with asymptotic expansions in terms of powers of ln r and inverse powers of r. Such expansions also arise in the conformal method for analysing wave equations in odd space-time dimension. In recent work it has been shown that for non-linear wave equations, or for wave maps, polyhomogeneous initial data lead to solutions which are also polyhomogeneous provided that an infinite hierarchy of corner conditions holds. In this paper we show that the result is true regardless of corner conditions.

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25

CALO, VICTOR M., YALCHIN EFENDIEV, and JUAN GALVIS. "ASYMPTOTIC EXPANSIONS FOR HIGH-CONTRAST ELLIPTIC EQUATIONS." Mathematical Models and Methods in Applied Sciences 24, no. 03 (December 29, 2013): 465–94. http://dx.doi.org/10.1142/s0218202513500565.

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In this paper, we present a high-order expansion for elliptic equations in high-contrast media. The background conductivity is taken to be one and we assume the medium contains high (or low) conductivity inclusions. We derive an asymptotic expansion with respect to the contrast and provide a procedure to compute the terms in the expansion. The computation of the expansion does not depend on the contrast which is important for simulations. The latter allows avoiding increased mesh resolution around high conductivity features. This work is partly motivated by our earlier work in [Domain decomposition preconditioners for multiscale flows in high-contrast media, Multiscale Model Simul.8 (2010) 1461–1483] where we design efficient numerical procedures for solving high-contrast problems. These multiscale approaches require local solutions and our proposed high-order expansion can be used to approximate these local solutions inexpensively. In the case of a large-number of inclusions, the proposed analysis can help to design localization techniques for computing the terms in the expansion. In the paper, we present a rigorous analysis of the proposed high-order expansion and estimate the remainder of it. We consider both high- and low-conductivity inclusions.

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26

Korovina, Maria. "Asymptotics of Solutions of Linear Differential Equations with Holomorphic Coefficients in the Neighborhood of an Infinitely Distant Point." Mathematics 8, no. 12 (December 20, 2020): 2249. http://dx.doi.org/10.3390/math8122249.

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This study is devoted to the description of the asymptotic expansions of solutions of linear ordinary differential equations with holomorphic coefficients in the neighborhood of an infinitely distant singular point. This is a classical problem of analytical theory of differential equations and an important particular case of the general Poincare problem on constructing the asymptotics of solutions of linear ordinary differential equations with holomorphic coefficients in the neighborhoods of irregular singular points. In this study we consider such equations for which the principal symbol of the differential operator has multiple roots. The asymptotics of a solution for the case of equations with simple roots of the principal symbol were constructed earlier.

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27

ALLEN, PAUL T., and ALAN D. RENDALL. "ASYMPTOTICS OF LINEARIZED COSMOLOGICAL PERTURBATIONS." Journal of Hyperbolic Differential Equations 07, no. 02 (June 2010): 255–77. http://dx.doi.org/10.1142/s0219891610002141.

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In cosmology an important role is played by homogeneous and isotropic solutions of the Einstein–Euler equations and linearized perturbations of these. This paper proves results on the asymptotic behavior of scalar perturbations both in the approach to the initial singularity of the background model and at late times. The main equation of interest is a linear hyperbolic equation whose coefficients depend only on time. Expansions for the solutions are obtained in both asymptotic regimes. In both cases, it is shown how general solutions with a linear equation of state can be parametrized by certain functions which are coefficients in the asymptotic expansion. For some nonlinear equations of state, it is found that the late-time asymptotic behavior is qualitatively different from that in the linear case.

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28

Olde Daalhuis, A. B. "Hyperasymptotics for nonlinear ODEs I. A Riccati equation." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 461, no. 2060 (June 24, 2005): 2503–20. http://dx.doi.org/10.1098/rspa.2005.1462.

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We illustrate how one can obtain hyperasymptotic expansions for solutions of nonlinear ordinary differential equations. The example is a Riccati equation. The main tools that we need are transseries expansions and the Riemann sheet structure of the Borel transform of the divergent asymptotic expansions. Hyperasymptotic expansions determine the solutions uniquely. A numerical illustration is included.

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29

LLEWELLYN SMITH, STEFAN G. "The asymptotic behaviour of Ramanujan's integral and its application to two-dimensional diffusion-like equations." European Journal of Applied Mathematics 11, no. 1 (February 2000): 13–28. http://dx.doi.org/10.1017/s0956792599004039.

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The large-time behaviour of a large class of solutions to the two-dimensional linear diffusion equation in situations with radial symmetry is governed by the function known as Ramanujan's integral. This is also true when the diffusion coefficient is complex, which corresponds to Schrödinger's equation. We examine the asymptotic expansion of Ramanujan's integral for large values of its argument over the whole complex plane by considering the analytic continuation of Ramanujan's integral to the left half-plane. The resulting expansions are compared to accurate numerical computations of the integral. The large-time behaviour derived from Ramanujan's integral of the solution to the diffusion equation outside a cylinder is not valid far from the domain boundary. A simple method based on matched asymptotic expansions is outlined to calculate the solution at large times and distances: the resulting form of the solution combines the inverse logarithmic decay in time typical of Ramanujan's integral with spatial dependence on the usual similarity variable for the diffusion equation.

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30

Cao, Dat, and Luan Hoang. "Long-time asymptotic expansions for Navier-Stokes equations with power-decaying forces." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 150, no. 2 (January 22, 2019): 569–606. http://dx.doi.org/10.1017/prm.2018.154.

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AbstractThe Navier-Stokes equations for viscous, incompressible fluids are studied in the three-dimensional periodic domains, with the body force having an asymptotic expansion, when time goes to infinity, in terms of power-decaying functions in a Sobolev-Gevrey space. Any Leray-Hopf weak solution is proved to have an asymptotic expansion of the same type in the same space, which is uniquely determined by the force, and independent of the individual solutions. In case the expansion is convergent, we show that the next asymptotic approximation for the solution must be an exponential decay. Furthermore, the convergence of the expansion and the range of its coefficients, as the force varies are investigated.

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31

Parenti, Cesare, and Hidetoshi Tahara. "Asymptotic expansions of distribution solutions of some Fuchsian hyperbolic equations." Publications of the Research Institute for Mathematical Sciences 23, no. 6 (1987): 909–22. http://dx.doi.org/10.2977/prims/1195175863.

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32

Leppington, F. G. "MATCHING OF ASYMPTOTIC EXPANSIONS OF SOLUTIONS OF BOUNDARY VALUE PROBLEMS." Bulletin of the London Mathematical Society 26, no. 2 (March 1994): 198–200. http://dx.doi.org/10.1112/blms/26.2.198.

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33

Sgibnev, M. S. "Exact asymptotic expansions for solutions of multi-dimensional renewal equations." Izvestiya: Mathematics 70, no. 2 (April 30, 2006): 363–83. http://dx.doi.org/10.1070/im2006v070n02abeh002315.

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34

Bruno, A. D., and I. V. Goryuchkina. "All asymptotic expansions of solutions to the sixth Painlevé equation." Doklady Mathematics 76, no. 3 (December 2007): 851–55. http://dx.doi.org/10.1134/s1064562407060129.

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35

Goryuchkina, I. V. "Basic asymptotic expansions of solutions to the sixth Painlevé equation." Journal of Mathematical Sciences 160, no. 1 (June 11, 2009): 1–9. http://dx.doi.org/10.1007/s10958-009-9489-9.

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36

Samovol, V. S. "On Asymptotic Series Expansions of Solutions to the Riccati Equation." Mathematical Notes 110, no. 1-2 (July 2021): 135–44. http://dx.doi.org/10.1134/s0001434621070142.

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37

Arnold, Douglas N., and Alexandre L. Madureira. "Asymptotic Estimates of Hierarchical Modeling." Mathematical Models and Methods in Applied Sciences 13, no. 09 (September 2003): 1325–50. http://dx.doi.org/10.1142/s0218202503002933.

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In this paper we propose a way to analyze certain classes of dimension reduction models for elliptic problems in thin domains. We develop asymptotic expansions for the exact and model solutions, having the thickness as small parameter. The modeling error is then estimated by comparing the respective expansions, and the upper bounds obtained make clear the influence of the order of the model and the thickness on the convergence rates. The techniques developed here allows for estimates in several norms and semi-norms, and also interior estimates (which disregards boundary layers).

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38

Bruno, Alexander D. "Power Geometry and Elliptic Expansions of Solutions to the Painlevé Equations." International Journal of Differential Equations 2015 (2015): 1–13. http://dx.doi.org/10.1155/2015/340715.

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We consider an ordinary differential equation (ODE) which can be written as a polynomial in variables and derivatives. Several types of asymptotic expansions of its solutions can be found by algorithms of 2D Power Geometry. They are power, power-logarithmic, exotic, and complicated expansions. Here we develop 3D Power Geometry and apply it for calculation power-elliptic expansions of solutions to an ODE. Among them we select regular power-elliptic expansions and give a survey of all such expansions in solutions of the Painlevé equationsP1,…,P6.

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39

Ablowitz, M. J., and T. S. Haut. "Asymptotic expansions for solitary gravity-capillary waves in two and three dimensions." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, no. 2109 (June 24, 2009): 2725–49. http://dx.doi.org/10.1098/rspa.2009.0112.

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High-order asymptotic series are obtained for two- and three-dimensional gravity-capillary solitary waves. In two dimensions, the first term in the asymptotic series is the well-known sech 2 solution of the Korteweg–de Vries equation; in three dimensions, the first term is the rational lump solution of the Kadomtsev–Petviashvili equation I. The two-dimensional series is used (with nine terms included) to investigate how small surface tension affects the height and energy of large-amplitude waves and waves close to the solitary version of Stokes’ extreme wave. In particular, for small surface tension, the solitary wave with the maximum energy is obtained. For large surface tension, the two-dimensional series is also used to study the energy of depression solitary waves. Energy considerations suggest that, for large enough surface tension, there are solitary waves that can get close to the fluid bottom. In three dimensions, analytic solutions for the high-order perturbation terms are computed numerically, and the resulting asymptotic series (to three terms) is used to obtain the speed versus maximum amplitude curve for solitary waves subject to sufficiently large surface tension. Finally, the above asymptotic method is applied to the Benney–Luke (BL) equation, and the resulting asymptotic series (to three terms) is verified to agree with the solitary-wave solution of the BL equation.

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40

Chernyshov, K. I. "Asymptotic expansions of the solutions of linear, singularly perturbed differential equations." Journal of Soviet Mathematics 62, no. 6 (December 1992): 3153–64. http://dx.doi.org/10.1007/bf01095689.

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41

Bruno, A. D. "Asymptotic behaviour and expansions of solutions of an ordinary differential equation." Russian Mathematical Surveys 59, no. 3 (June 30, 2004): 429–80. http://dx.doi.org/10.1070/rm2004v059n03abeh000736.

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42

Bruno, Alexander Dmitrievich. "Calculation of complicated asymptotic expansions of solutions to the Painlevé equations." Keldysh Institute Preprints, no. 55 (2017): 1–27. http://dx.doi.org/10.20948/prepr-2017-55.

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43

Tulyakov, D. N. "A procedure for finding asymptotic expansions for solutions of difference equations." Proceedings of the Steklov Institute of Mathematics 272, S2 (April 2011): 162–67. http://dx.doi.org/10.1134/s0081543811030114.

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44

Novaga, Matteo, and Enrico Valdinoci. "Multibump solutions and asymptotic expansions for mesoscopic Allen-Cahn type equations." ESAIM: Control, Optimisation and Calculus of Variations 15, no. 4 (August 20, 2008): 914–33. http://dx.doi.org/10.1051/cocv:2008058.

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45

Popivanov, Nedyu, Todor Popov, and Rudolf Scherer. "Asymptotic expansions of singular solutions for (3+1)-D Protter problems." Journal of Mathematical Analysis and Applications 331, no. 2 (July 2007): 1093–112. http://dx.doi.org/10.1016/j.jmaa.2006.09.036.

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46

Pop, I., D. B. Ingham, and P. Cheng. "Transient Natural Convection in a Horizontal Concentric Annulus Filled With a Porous Medium." Journal of Heat Transfer 114, no. 4 (November 1, 1992): 990–97. http://dx.doi.org/10.1115/1.2911911.

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The method of matched asymptotic expansions is applied to study the problem of transient natural convection in a horizontal concentric porous annulus with inner and outer cylinders maintained at uniform temperatures. Asymptotic solutions for the inner layer, the outer layer, and the core are obtained for small times. A uniformly valid solution for stream function, tangential velocity, and temperature, which is valid for the whole domain, is constructed.

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47

Taber, L. A. "Nonlinear Asymptotic Solution of the Reissner Plate Equations." Journal of Applied Mechanics 52, no. 4 (December 1, 1985): 907–12. http://dx.doi.org/10.1115/1.3169167.

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Based on the Reissner plate equations for large displacements and rotations within the limits of small strain, asymptotic solutions are developed for circular plates under uniform pressure. With the boundary layer solution assumed in exponential form, the boundary conditions are applied directly at the plate edge without the need for matched asymptotic expansions. Results are presented for plates with clamped edges. When compared to the solution for the special case of von Ka´rma´n plate theory, stresses generally deviate by less than 10 percent for rotation angles up to about 30 deg.

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48

Dunster, T. M. "Olver's error bound methods applied to linear ordinary differential equations having a simple turning point." Analysis and Applications 12, no. 04 (June 17, 2014): 385–402. http://dx.doi.org/10.1142/s0219530514500298.

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Uniform asymptotic solutions of linear ordinary differential equations having a large parameter and a simple turning point are well known. Classical expansions involve Airy functions and their derivatives, and one of Frank Olver's major achievements was obtaining explicit and realistic error bounds. Here alternative expansions are considered, which involve the Airy function alone (and not its derivative). This is based on the early work of Cherry, and using Olver's techniques explicit error bounds are derived. The derivative of asymptotic solutions of turning point problems is also considered, and again using Olver's techniques, sharper error bounds are derived via the differential equation satisfied by such solutions.

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Cengizci, Süleyman. "A comparison between MMAE and SCEM for solving singularly perturbed linear boundary layer problems." Filomat 33, no. 7 (2019): 2135–48. http://dx.doi.org/10.2298/fil1907135c.

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In this study, we propose an efficient method so-called Successive Complementary Expansion Method (SCEM), that is based on generalized asymptotic expansions, for approximating to the solutions of singularly perturbed two-point boundary value problems. In this easy-applicable method, in contrast to the well-known method the Method of Matched Asymptotic Expansions (MMAE), any matching process is not required to obtain uniformly valid approximations. The key point: A uniformly valid approximation is adopted first, and complementary functions are obtained imposing the corresponding boundary conditions. An illustrative and two numerical experiments are provided to show the implementation and numerical properties of the present method. Furthermore, MMAE results are also obtained in order to compare the numerical robustnesses of the methods.

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Makarenko, Andrey N., and Alexander N. Myagky. "The asymptotic behavior of bouncing cosmological models in F(𝒢) gravity theory." International Journal of Geometric Methods in Modern Physics 14, no. 10 (September 13, 2017): 1750148. http://dx.doi.org/10.1142/s0219887817501481.

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We reconstruct [Formula: see text] gravity theory with an exponential scale factor to realize the bouncing behavior in the early universe and examine the asymptotic behavior of late-time solutions in this model. We propose an approach for the construction of asymptotic expansions of solutions of the Friedmann equations on the basis of Puiseux series.

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