Riemann-Hilbert Problem

Stokes curves of second order Fuchsian dfferential equations on the Riemann sphere with a large parameter form sphere graphs, which are called Stokes graphs. Topological classification of Stokes graphs are given for the case where equations have five regular singular points. It is proved that there are exactly 25 degree sequences of sphere triangulations associated with Stokes graphs under suitable generic conditions.

Chapter 7 - Ordinary Differential Equations

In the paper we deal with the Heun functions --- solutions of the Heun\nequation, which is the most general Fuchsian equation of second order with four\nregular singular points. Despite the increasing interest to the equation and\nnumerous applications of the functions in a wide variety of physical problems,\nit is only Maple amidst known software packages which is able to evaluate the\nHeun functions numerically. But the Maple routine is known to be imperfect:\neven at regular points it may return infinities or end up with no result.\nImproving the situation is difficult because the code is not publicly\navailable. The purpose of the work is to suggest and develop alternative\nalgorithms for numerical evaluation of the Heun functions. A procedure based on\npower series expansions and analytic continuation is suggested which allows us\nto avoid numerical integration of the differential equation and to achieve\nreasonable efficiency and accuracy. Results of numerical tests are given.\n

In this contribution, we construct approximations for the density associated with the solution of second-order linear differential equations whose coefficients are analytic stochastic processes about regular-singular points. Our analysis is based on the combination of a random Fröbenius technique together with the random variable transformation technique assuming mild probabilistic conditions on the initial conditions and coefficients. The new results complete the ones recently established by the authors for the same class of stochastic differential equations, but about regular points. In this way, this new contribution allows us to study, for example, the important randomized Bessel differential equation.

Isomonodromic Deformations of sl(2) Fuchsian Systems on the Riemann Sphere

We study the integrability properties of nonlinearly perturbed Euler equations (linear ordinary differential equations with one regular singular point in the complex plane plus a nonlinear perturbation) near the singular point. We allow for first integrals with essential singularities and give sufficient conditions for the nonintegrability of the equations in the complex domain. We extend normal form theorems for singular equations and argue that equivalence to normal forms captures the spirit of the poly-Painlevé test and is a powerful tool for a rigorous approach to nonintegrability.

We consider the existence of cohomogeneity one solitons for the isometric flow of G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-structures on the following classes of torsion-free G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-manifolds: the Euclidean R7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^7$$\\end{document} with its standard G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-structure, metric cylinders over Calabi–Yau 3-folds, metric cones over nearly Kähler 6-manifolds, and the Bryant–Salamon G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document}-manifolds. In all cases we establish existence of global solutions to the isometric soliton equations, and determine the asymptotic behaviour of the torsion. In particular, existence of shrinking isometric solitons on R7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^7$$\\end{document} is proved, giving support to the likely existence of type I singularities for the isometric flow. In each case, the study of the soliton equation reduces to a particular nonlinear ODE with a regular singular point, for which we provide a careful analysis. Finally, to simplify the derivation of the relevant equations in each case, we first establish several useful Riemannian geometric formulas for a general class of cohomogeneity one metrics on total spaces of vector bundles which should have much wider application, as such metrics arise often as explicit examples of special holonomy metrics.

In this paper we consider families of differential equations on some real interval that depend on a parameter $$\varepsilon $$ ε , are regular for $$\varepsilon \ne 0$$ ε ≠ 0 and have exactly one regular singular point for $$\varepsilon =0$$ ε = 0 . Our aim is to give a formula which describes the asymptotic behavior of the solutions for $$\varepsilon \rightarrow 0$$ ε → 0 . To this end, we use a method that combines matched asymptotic analysis with uniform asymptotic integration. We then apply our results to some typical examples, such as a differential equation with two coalescing regular singular points as well as a singular perturbation of the Bessel equation.

We determine closed and compact expressions for the ϵ-expansion of certain Gaussian hypergeometric functions expanded around half-integer values by explicitly solving for their recurrence relations. This ϵ-expansion is identified with the normalized solution of the underlying Fuchs system of four regular singular points. We compute its regularized zeta series (giving rise to two independent associators) whose ratio gives the ϵ-expansion at a specific value. Furthermore, we use the well known one-loop massive bubble integral as an example to demonstrate how to obtain all-order ϵ-expansions for Feynman integrals and how to construct representations for Feynman integrals in terms of generalized hypergeometric functions. We use the method of differential equations in combination with the recently established general solution for recurrence relations with non-commutative coefficients.

In this study, an exact analytical solution for the convective heat transfer equation from a semi-spherical fin was presented. To obtain a mathematical model, the system was assumed to be a lump in the vertical direction and the governing equation in the Cartesian coordinate was transferred to the Mathieu equation. The exact solution was compared with numerical results such as the finite difference method and midpoint method with Richardson extrapolation (Midrich). Not surprisingly, the exact solution prevailed over the numerical solutions in terms of accuracy and ease of use. Furthermore, the effect of Biot number on the heat transfer of the fin and the fine performance was investigated. The relative error of the results obtained from the analytical and numerical solutions at the base, center, and tip of the fin was 0, 7.72, and 40.25 percent, respectively. The results showed that the relative error between the analytical and numerical solutions depends on the Biot number and varies as a function of the fin length. The obtained analytical solution could be encouraging from different mathematical and industrial applications' points of view.

One-dimensional propagation of longitudinal elastic waves through functionally graded materials

We have previously formulated a program for deducing the intervals of oscillations in the solutions of ordinary second-order linear homogeneous differential equations. In this work, we demonstrate how the oscillation-detection program can be carried out around the regular singular points $x=\pm1$ of the Legendre differential equations. The solutions $y_{n}(x)$ of the Legendre equation are predicted to be oscillatory in $|x| < 1$ for $n\geq3$ and nonoscillatory outside of that interval for all values of n. In contrast, the solutions $y_{n}^{m}(x)$ of the associated Legendre equation are predicted to be oscillatory for $n\geq3$ and $m\leq n-2$ only in smaller subintervals $|x| < x_{*} < 1$ , the sizes of which are determined by n and m. Numerical integrations confirm that such subintervals are distinctly smaller than $(-1, +1)$ .

The Solution of a Second Order Linear Differential Equation Near a Regular Singular Point

Starting from a second-order Fuchsian differential equation having five regular singular points, an equation obeyed by a function proportional to the first derivative of the solution of the Heun equation, we construct several expansions of the solutions of the general Heun equation in terms of Appell generalized hypergeometric functions of two variables of the first kind. Several cases when the expansions reduce to those written in terms of simpler mathematical functions such as the incomplete Beta function or the Gauss hypergeometric function are identified. The conditions for deriving finite-sum solutions via termination of the series are discussed. In general, the coefficients of the expansions obey four-term recurrence relations; however, there exist certain choices of parameters for which the recurrence relations involve only two terms, though not necessarily successive. For such cases, the coefficients of the expansions are explicitly calculated and the general solution of the Heun equation is constructed in terms of the Gauss hypergeometric functions.

A precise analysis of partial match retrieval of multidimensional data is presented. The structures considered here are multidimensional search trees ( k -d-trees) and digital tries ( k -d-tries), as well as structures designed for efficient retrieval of information stored on external devices. The methods used include a detailed study of a differential system around a regular singular point in conjunction with suitable contour integration techniques for the analysis of k -d-trees, and properties of the Mellin integral transform for k -d-tries and extendible cell algorithms.