Euler integral symmetries for the confluent Heun equation and symmetries of the Painlevé equation PV

Euler integral symmetries relate solutions of ordinary linear differential equations and generate integral representations of the solutions in several cases or relations between solutions of constrained equations. These relations lead to the corresponding symmetries of the monodromy matrices for the differential equations. We discuss Euler symmetries in the case of the deformed confluent Heun equation, which is in turn related to the Painlevé equation PV. The existence of symmetries of the linear equations leads to the corresponding symmetries of the Painlevé equation of the Okamoto type. The choice of the system of linear equations that reduces to the deformed confluent Heun equation is the starting point for the constructions. The basic technical problem is to choose the bijective relation between the system parameters and the parameters of the deformed confluent Heun equation. The solution of this problem is quite large, and we use the algebraic computing system Maple for this.     

Euler integral symmetries for a deformed Heun equation and symmetries of the Painlevé PVI equation

Euler integral transformations relate solutions of ordinary linear differential equations and generate integral representations of the solutions in a number of cases or relations between solutions of constrained equations (Euler symmetries) in some other cases. These relations lead to the corresponding symmetries of the monodromy matrices. We discuss Euler symmetries in the case of the simplest Fuchsian system that is equivalent to a deformed Heun equation, which is in turn related to the Painlevé PVI equation. The existence of integral symmetries of the deformed Heun equation leads to the corresponding symmetries of the PVI equation. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 2, pp. 252–264, May, 2008.     

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.     

On the Central Connection Problem for the Double Confluent Heun Equation

The relation between the different local solutions of the double confluent Heun equation is called the central connection problem for this equation. We give limit formulae for the connection coefficients of the matrix which represents the relation between the fundamental sets of asymptotic solutions at the irregular singularities 0 and ∞, respectively. The problem is closely related to the connection problem of a special confluent Heun equation. By Laplace transform, a transfer between the connection problems of the corresponding differential equations takes place.     

Automorphisms of the solution spaces of special double-confluent Heun equations

Two new linear operators determining automorphisms of the solution space of a special double-confluent Heun equation in the general case are obtained. This equation has two singular points, both of which are irregular. The obtained result is applied to solve the nonlinear equation of the resistively shunted junction model for an overdamped Josephson junction in superconductors. The new operators are explicitly expressed in terms of structural polynomials, for which recursive computational algorithms are constructed. Two functional equations for the solutions of the special double-confluent Heun equation are found.     

On a class of bivariate second-order linear partial difference equations and their monic orthogonal polynomial solutions

In this paper we classify the bivariate second-order linear partial difference equations, which are admissible, potentially self-adjoint, and of hypergeometric type. Using vector matrix notation, explicit expressions for the coefficients of the three-term recurrence relations satisfied by monic orthogonal polynomial solutions are obtained in terms of the coefficients of the partial difference equation. Finally, we make a compilation of the examples existing in the literature belonging to the class analyzed in this paper, namely bivariate Charlier, Meixner, Kravchuk and Hahn orthogonal polynomials.     

An algorithm to obtain global solutions of the double confluent Heun equation

A procedure is proposed to construct solutions of the double confluent Heun equation with a determinate behaviour at the singular points. The connection factors are expressed as quotients of Wronskians of the involved solutions. Asymptotic expansions are used in the computation of those Wronskians. The feasibility of the method is shown in an example, namely, the Schrödinger equation with a quasi-exactly-solvable potential.     

Determinant Forms for Extended Heat Functions and Expansions of Solutions of Associated Cauchy Problems

We briefly review series solutions of differential equations problems of the second order that lead to coefficients expressed in terms of determinants. Derivative type formulas involving a generating function with several parameters are developed for these determinant coefficients in first order problems. These permit constructing determinant forms for the heat polynomials and their Appell transforms. Hadamard's theorem for bounding determinants and conical regions are used to deduce simplified versions of expansion theorems involving these polynomials and associated Appell transforms. Extended versions of the heat equation are also considered.     

Difference Equations and Divisibility Properties of Sequences

There are many different ways of defining a sequence in terms of solutions to difference equations. In fact, if a sequence satisfies one recurrence then it satisfies an infinite number of recurrences. Arithmetic properties of an integral sequence are often studied by direct methods based on the combinatorial or algebraic definition of the numbers or using their generating function. The rational generating function is the main tool in obtaining various difference equations with coefficients and initial values exhibiting divisibility patterns that can imply particular arithmetic properties of the solutions. In this process, we face the challenging task of finding difference equations that are relevant to the divisibility properties by transforming the original rational generating function. As a matter of fact, it is not necessarily the simple difference equation that helps the most in proving the properties. We illustrate this process on several examples and a sequence involving a p -sected binomial sum of the form y n = y n ( p , a )= ~ k =0 X n kp a k where p is an arbitrary prime. Let 𝜌 p ( m ) denote the exponent of the highest power of a prime p which divides m . Recently, the author obtained lower bounds for 𝜌 p ( y n ) based on recurrence relations of order p and p m 1. The cases with tight bounds have also been characterized. In this paper, we prove that 𝜌 p ( y np ( p , a ))= n for 𝜌 p ( a +1)=1, p S 3. We obtain alternative difference equations of order p 2 for y n and order p for the p -sected sequence y np by a generating function based method. We also extend general divisibility results relying on the arithmetic properties of the coefficients and initial values.     

Generalized Abel type integral equations with localized fractional integrals and derivatives

Generalized Abel type integral equations with Gauss, Kummer's and Humbert's confluent hypergeometric functions in the kernel and generalized Abel type integral equations with localized fractional integrals are considered. The left-hand sides of these equations are inversed by using generalized fractional derivatives. Explicit solutions of the equations in the class of locally summable functions are obtained. They are represented in terms of hypergeometric functions. Asymptotic power exponential type expansions of the generalized and localized fractional integrals are obtained. The base solutions of the generalized Abel type integral equation are given in the form of asymptotic series.     

An approximate method via Taylor series for stochastic functional differential equations

The subject of this paper is an analytic approximate method for stochastic functional differential equations whose coefficients are functionals, sufficiently smooth in the sense of Fréchet derivatives. The approximate equations are defined on equidistant partitions of the time interval, and their coefficients are general Taylor expansions of the coefficients of the initial equation. It will be shown that the approximate solutions converge in the Lp-norm and with probability one to the solution of the initial equation, and also that the rate of convergence increases when degrees in Taylor expansions increase, analogously to real analysis.     

Full low-frequency asymptotic expansion for second-order elliptic equations in two dimensions

The present paper contains the low-frequency expansions of solutions of a large class of exterior boundary value problems involving second-order elliptic equations in two dimensions. The differential equations must coincide with the Helmholtz equation in a neighbourhood of infinity, however, they may depart radically from the Helmholtz equation in any bounded region provided they retain ellipticity. In some cases the asymptotic expansion has the form of a power series with respect to k² and k² (ln k + a)^?1, where k is the wave number and a is a constant. In other cases it has the form of a power series with respect to k², coefficients of which depend polynomially on In k. The procedure for determining the full low-frequency expansion of solutions of the exterior Dirichlet and Neumann problems for the Helmholtz equation is included as a special case of the results presented here.     

Composite Approximations to the Solutions of the Orr-Sommerfeld Equation

The method of matched asymptotic expansions is used to derive composite approximations to the solutions of the Orr-Sommerfeld equation which satisfy Olver's completeness requirement. It is shown that the inner expansions can be obtained to all orders in terms of a certain class of generalized Airy functions, and these expansions are then used to derive approximations to the connection formulae. Because of the linearity of the problem it is possible and convenient to fix the normalization of the inner and outer expansions separately and then to relate them through the central matching coefficients. The Stokes multipliers can then be expressed in terms of the central matching coefficients and the coefficients which appear in the connection formulae. Once the inner and outer expansions have been matched they can be combined, if desired, to form composite approximations of either the additive or multiplicative type. For example, the ‘modified’ viscous solutions of Tollmien emerge in a natural way as first-order composite approximations obtained by multiplicative composition; similarly, the form of the ‘viscous correction’ to the singular inviscid solutions which I conjectured some years ago emerges as a first-order additive composite approximation. Because of the completeness requirement, however, these composite approximations are valid only in certain wedge shaped domains; approximations which are valid in the complementary sectors can then be obtained by the use of the connections formulae. The theory thus provides a relatively simple and explicit method of obtaining higher approximations, and its structure permits a direct comparison of the present results not only with the older heuristic theories but also with the comparison equation method.     

On Complicated Expansions of Solutions to ODES

Polynomial ordinary differential equations are studied by asymptotic methods. The truncated equation associated with a vertex or a nonhorizontal edge of their polygon of the initial equation is assumed to have a solution containing the logarithm of the independent variable. It is shown that, under very weak constraints, this nonpower asymptotic form of solutions to the original equation can be extended to an asymptotic expansion of these solutions. This is an expansion in powers of the independent variable with coefficients being Laurent series in decreasing powers of the logarithm. Such expansions are sometimes called psi-series. Algorithms for such computations are described. Six examples are given. Four of them are concern with Painlevé equations. An unexpected property of these expansions is revealed.     

An observation on the periodic solutions to nonlinear physical models by means of the auxiliary equation with a sixth-degree nonlinear term

It is a fact that in auxiliary equation methods, the exact solutions of different types of auxiliary equations may produce new types of exact travelling wave solutions to nonlinear equations. In this manner, various auxiliary equations of first-order nonlinear ordinary differential equation with distinct-degree nonlinear terms are examined and, by means of symbolic computation, the new solutions of original auxiliary equation of first-order nonlinear ordinary differential equation with sixth-degree nonlinear term are presented. Consequently, the novel exact solutions of the generalized Klein–Gordon equation and the active-dissipative dispersive media equation are found out for illustration purposes. They are also applicable, where conventional perturbation method fails to provide any solution of the nonlinear problems under study.     

Modified equations of the first kind for the Helmholtz equation

Integral equations of the first kind for exterior problems arising in the study of the three‐dimensional Helmholtz equation are considered. These equations are derived by seeking solutions in the form of layer potentials with modified fundamental solutions. For each first kind equation, existence and uniqueness of solution are proved with the aid of composition relations involving associated modified boundary integral operators. For the Dirichlet problem, an optimal choice of the modification coefficients is considered in order to minimize the condition number of the resulting integral operator. Copyright © 2014 John Wiley & Sons, Ltd.     

Asymptotic analysis of singularly perturbed Hamilton-Jacobi equations

Unlike the previous investigation of the sufficient conditions for the convergence of minimax solutions of singularly perturbed Hamilton-Jacobi (H-J) equations, a typical example of which would be the Bellman-Isaacs (B-I) equations, convergence conditions are formulated not in terms of auxiliary constructs [1], but in terms of the Hamiltonian, the boundary function, assumptions regarding their continuity, Lipschitz continuity, etc. In addition, an asymptotic equation is derived, that is, a H-J equation whose minimax solution is the limit of solutions of H-J equations in which some of the momentum variables have coefficients whose denominators contain a small parameter which is made to approach zero.     

构造辅助函数计算准Vandermonde行列式

通过构造线性方程组和一元高次方程,利用线性方程组的解与一元高次方程根与系数的关系推导出第一类准Vandermonde行列式的值.通过构造辅助函数计算一个特殊的第一类准Vandermonde行列式,并把这种方法推广于两类特殊第二类准Vandermonde行列式的计算.