Expansion of solutions to the fifth Painlevé equation in a neighborhood of a nonsingular point

For the fifth Painlevé equation, the asymptotic expansions of solutions are obtained in a neighborhood of a nonsingular point of the independent variable by using methods of power geometry. Four families of expansions are found. All of these expansions are convergent series in integral powers with complex coefficients. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 36, Suzdal Conference-2004, Part 2, 2005.     

Expansion of solutions of the sixth Painlevé equation near a regular point

For the sixth Painlevé equation, by using power-geometry methods, all asymptotic expansions of solutions are obtained in a neighborhood of a regular point of the independent variable. All of these expansions are convergent series in integer powers with constant complex coefficients. Five families of expansions are found. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 36, Suzdal Conference-2004, Part 2, 2005.     

Asymptotic expansion of differential equations with an irregular singular point of finite rank

An asymptotic expansion is constructed for solutions of second-order differential equations with an irregular singular point of finite rank at infinity. Formal solutions are obtained. By reduction of the differential equation to a Volterra integral equation we prove that the formal solutions are asymptotic expansions of the solutions of the differential equation.Simferopol' University. Translated from Dinamicheskie Sistemy, No. 10, pp. 78–83, 1992.     

Asymptotic expansions for partial solutions of the sixth Painlevé equation

A formalism for an averaging method for the Painlevé equations, in particular, the sixth equation, is developed. The problem is to describe the asymptotic behavior of the sixth Painlevé transcendental in the case where the module of the independent variable tends to infinity. The corresponding expansions contain an elliptic function (ansatz) in the principal term. The parameters of this function depend on the variable because of the modulation equation. The elliptic ansatz and the modulation equation for the sixth Painlevé equation are obtained in their explicit form. A partial solution of the modulation equation leading to a previously unknown asymptotic expansion for the partial solution of the sixth Painlevé equation is obtained.     

Asymptotic solvers for ordinary differential equations with multiple frequencies

We construct asymptotic expansions for ordinary differential equations with highly oscillatory forcing terms,focusing on the case of multiple,non-commensurate frequencies.We derive an asymptotic expansion in inverse powers of the oscillatory parameter and use its truncation as an exceedingly effective means to discretize the differential equation in question.Numerical examples illustrate the effectiveness of the method.     

Asymptotic Expansions of Generalized Stieltjes Transforms of Algebraically Decaying Functions

Asymptotic expansions of Stieltjes and generalized Stieltjes transforms of functions having an asymptotic expansion in negative integer powers of their variable have been exhaustively investigated by R. Wong. In this article, we extend this analysis to Stieltjes and generalized Stieltjes transforms of functions having an asymptotic expansion in negative rational powers of their variable. Distributional approach is used to derive asymptotic expansions of the Stieltjes and generalized Stieltjes transforms of this kind of functions for large values of the parameter(s) of the transformation. Error bounds are obtained at any order of the approximation for a large family of integrands. The asymptotic approximation of an integral involved in the calculation of the mass renormalization of the quantum scalar field and of the third symmetric elliptic integral are given as illustrations.     

Asymptotic Expansions of Generalized Stieltjes Transforms of Algebraically Decaying Functions

Asymptotic expansions of Stieltjes and generalized Stieltjes transforms of functions having an asymptotic expansion in negative integer powers of their variable have been exhaustively investigated by R. Wong. In this article, we extend this analysis to Stieltjes and generalized Stieltjes transforms of functions having an asymptotic expansion in negative rational powers of their variable. Distributional approach is used to derive asymptotic expansions of the Stieltjes and generalized Stieltjes transforms of this kind of functions for large values of the parameter(s) of the transformation. Error bounds are obtained at any order of the approximation for a large family of integrands. The asymptotic approximation of an integral involved in the calculation of the mass renormalization of the quantum scalar field and of the third symmetric elliptic integral are given as illustrations.     

Singularly Perturbed Markov Chains with Two Small Parameters: A Matched Asymptotic Expansion

This work is concerned with asymptotic properties of solutions to forward equations for singularly perturbed Markov chains with two small parameters. It is motivated by the model of a cost-minimizing firm involving production planning and capacity expansion and a two-level hierarchical decomposition. Our effort focuses on obtaining asymptotic expansions of the solutions to the forward equation. Different from previous work on singularly perturbed Markov chains, the inner expansion terms are constructed by solving certain partial differential equations. The methods of undetermined coefficients are used. The error bound is obtained.     

The Asymptotic Expansion and Numerical Verification of Van der Pol's Equation

A technique is presented in this paper to verify the order of accuracy of asymptotic expansion of Van der Pol's equation. The technique is focused on using numerical solutions as an independent means of verifying the validity of asymptotic expansions.     

Asymptotic Solutions of a Fourth Order Differential Equation

In this paper, we derive uniform asymptotic expansions of solutions to the fourth order differential equation where x is a real variable and λ is a large positive parameter. The solutions of this differential equation can be expressed in the form of contour integrals, and uniform asymptotic expansions are derived by using the cubic transformation introduced by Chester, Friedman, and Ursell in 1957 and the integration-by-part technique suggested by Bleistein in 1966. There are two advantages to this approach: (i) the coefficients in the expansion are defined recursively, and (ii) the remainder is given explicitly. Moreover, by using a recent method of Olde Daalhuis and Temme, we extend the validity of the uniform asymptotic expansions to include all real values of x .     

Uniform asymptotic expansions of solutions of the Mathieu equation and the modified Mathieu equation

By the method of the model equation, uniform asymptotic expansions of the Floquet solutions of the Mathieu equation and two linearly independent solutions of the modified Mathieu equation are obtained for any real values of the separation parameter contained in these equations.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 62, pp. 60–91, 1976.     

非线性积分方程迭代配置法的渐近展开及其外推

非线性积分方程迭代配置法的渐近展开及其外推韩国强（华南理工大学计算机工程与科学系）ＡＳＹＭＰＴＯＴＩＣＥＲＲＯＲＥＸＭＮＳＩＯＮＳＡＮＤＥＸＴＲＡＰＯＬＡＴＩＯＮＦＯＲＴＨＥＩＴＥＲＡＴＥＤＣＯＬＬＯＣＡＴＩＯＮＭＥＴＨＯＤＳＯＦＮＯＮＬＩＮＥＡＲＩ...     

Series solutions of confluent Heun equations in terms of incomplete Gamma-functions

We present a simple systematic algorithm for construction of expansions of the solutions of ordinary differential equations with rational coefficients in terms of mathematical functions having indefinite integral representation. The approach employs an auxiliary equation involving only the derivatives of a solution of the equation under consideration. Using power-series expansions of the solutions of this auxiliary equation, we construct several expansions of the four confluent Heun equations'' solutions in terms of the incomplete Gamma-functions. In the cases of single- and double-confluent Heun equations the coefficients of the expansions obey four-term recurrence relations, while for the bi- and tri-confluent Heun equations the recurrence relations in general involve five terms. Other expansions for which the expansion coefficients obey recurrence relations involving more terms are also possible. The particular cases when these relations reduce to ones involving less number of terms are identified. The conditions for deriving closed-form finite-sum solutions via right-hand side termination of the constructed series are discussed.     

Applications of expansions in powers of a small parameter

During the last 35 years there has been an explosive interest in the problem of solving nonlinear differential equations. The purpose of this paper is to provide an historically oriented expository introduction to an analytical method of solving certain nonlinear differential equations in which a series expansion in powers of a small parameter is used. These series expansions turn out frequently to be asymptotic series.     

Uniform Asymptotic Expansions of Symmetric Elliptic Integrals

Symmetric standard elliptic integrals are considered when two or more parameters are larger than the others. The distributional approach is used to derive seven expansions of these integrals in inverse powers of the asymptotic parameters. Some of these expansions also involve logarithmic terms in the asymptotic variables. These expansions are uniformly convergent when the asymptotic parameters are greater than the remaining ones. The coefficients of six of these expansions involve hypergeometric functions with less parameters than the original integrals. The coefficients of the seventh expansion again involve elliptic integrals, but with less parameters than the original integrals. The convergence speed of any of these expansions increases for an increasing difference between the asymptotic variables and the remaining ones. All the expansions are accompanied by an error bound at any order of the approximation. January 31, 2000. Date revised: May 18, 2000. Date accepted: August 4, 2000.     

Approximate methods based on order reduction for the periodic solutions of nonlinear third-order ordinary differential equations

Four approximate methods based on order reduction, the introduction of a book-keeping parameter and power series expansions for the solution and the frequency of oscillation are used to analyze three autonomous, nonlinear, third-order ordinary differential equations which have analytical periodic solutions. The first method introduces the velocity in both sides of the equation if this (linear) term is not present. The second one is based on the first one but employs a new independent variable, whereas, in the third and fourth techniques, a term equal to the velocity times the square of the unknown frequency of oscillation is introduced in both sides of the equation. The third method uses the original independent variable, whereas the fourth one employs a new independent variable which depends linearly on the unknown frequency of oscillation. It is shown that the second method provides accurate solutions only for initial velocities close to unity, whereas the third one is found to yield very accurate results for the first and second equations considered here and only for large initial velocities for the third one. The fourth technique provides as accurate results as or more accurate results than parameter-perturbation techniques which deal with the third-order equations directly and are based on the expansion of certain constants that appear in the differential equations in terms of a book-keeping parameter.     

Asymptotic analysis of certain systems of linear differential equations with a large parameter

Two classes of systems of linear ordinary differential equations containing high-frequency terms proportional to certain positive powers of the oscillation frequency are considered. Complete asymptotic expansions of fundamental systems of solutions are constructed and substantiated.     

Asymptotics at infinity of solutions of equations close to Hamiltonian equations

A nonlinear non-autonomous system of two ordinary differential equations is considered. It is assumed that the equations corresponding to the principal part in the asymptotics at infinity with respect to the independent variable are integrable and written in the action-angle variables. In the case where the lower terms in the equation periodically depend on the angle, the asymptotic expansion at infinity of the two-parametric family of solutions is constructed. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 53, Suzdal Conference-2006, Part 1, 2008.     

Asymptotics of the eigenvalues of the rotating harmonic oscillator

The eigenenergies λ of a radial Schrödinger equation associated with the problem of a rotating harmonic oscillator are studied, these being values which admit eigensolutions which vanish at both the origin (a regular singularity of the equation) and at infinity. Asymptotic expansions, for the case where a coupling parameter α is small, are derived for λ. The approximation for λ consists of two components, an asymptotic expansion in powers of α, and a single term which is exponentially small (which can be associated with tunneling effects). The method of proof is rigorous, and utilizes three separate asymptotic approximations for the eigenfunction in the complex radial plane, involving elementary functions (WKB or Liouville-Green approximations), a modified Bessel function and a parabolic cylinder function.     

Complete Low Frequency Analysis for the Reduced Wave Equation with Variable Coefficients in Three Dimensions *

We study the low frequency asymptotics of the solutions u_w, to some exterior boundary value problems for reduced wave equations in three dimensions with frequencies w. The coefficients of the underlying differential operator and of the right hand sides are allowed to be variable and tend to those of the Laplacian and to 0 respectively at infinity with a certain order. As far as allowed by this order an asymptotic expansion in terms of powers of w is determined.