Regular Singular Points Research Articles

Outgoing modal solutions for Galbrun's equation in helioseismology

We construct modal outgoing Green's kernels for the simplified Galbrun's equation under spherical symmetry, in the context of helioseismology. The coefficients of the equation are C2 functions representing the solar interior model S, complemented with an isothermal atmospheric model. We solve the equation in vectorial spherical harmonics basis to obtain modal equations for the different components of the unknown wave motions. These equations are then decoupled and written in Schrödinger form, whose coefficients are shown to be C2 apart from at most two regular singular points, and to decay like a Coulomb potential at infinity. These properties allow us to construct an outgoing Green's kernel for each spherical mode. We also compute asymptotic expansions of coefficients up to order r−3 as r tends to infinity, and show numerically that their accuracy is improved by including the contribution from the gravity although this term is of order r−3.

Equivariant quantum differential equation, Stokes bases, and K-theory for a projective space

We consider the equivariant quantum differential equation for the projective space $$P^{n-1}$$ and introduce a compatible system of difference equations. We prove an equivariant gamma theorem for $$P^{n-1}$$ , which describes the asymptotics of the differential equation at its regular singular point in terms of the equivariant characteristic gamma class of the tangent bundle of $$P^{n-1}$$ . We describe the Stokes bases of the differential equation at its irregular singular point in terms of the exceptional bases of the equivariant K-theory algebra of $$P^{n-1}$$ and a suitable braid group action on the set of exceptional bases. Our results are an equivariant version of the well-known results of Dubrovin and Guzzetti..

Further study on the conformable fractional Gauss hypergeometric function

<abstract> This article presents an exhaustive study on the conformable fractional Gauss hypergeometric function (CFGHF). We start by solving the conformable fractional Gauss hypergeometric differential equation (CFGHDE) about the fractional regular singular points $ x = 1 $ and $ x = \infty $. Next, various generating functions of the CFGHF are established. We also develop some differential forms for the CFGHF. Subsequently, differential operators and contiguous relations are reported. Furthermore, we introduce the conformable fractional integral representation and the fractional Laplace transform of CFGHF. As an application, and after making a suitable change of the independent variable, we provide general solutions of some known conformable fractional differential equations, which could be written by means of the CFGHF. </abstract>

Eigenfunctions of Ordinary Differential Euler Operators

Asymptotic solutions of the eigenvalue problem for an Euler operator in a neighborhood of a regular singular point are considered. We find a condition under which the asymptotic expansion is free of logarithms. Eigenvalues expressed in terms of elementary functions in the form of a finite sum of quasi-polynomials are obtained for third-order Euler operators and also for commuting Euler operators of sixth and ninth orders. The problem on common eigenfunctions for commuting Euler operators is examined. In the case of operators of rank 2 and 3, it can be reduced to second- and third-order Bessel equations by differential substitutions.

Logarithmic A-hypergeometric series

The method of Frobenius is a standard technique to construct series solutions of an ordinary linear differential equation around a regular singular point. In the classical case, when the roots of the indicial polynomial are separated by an integer, logarithmic solutions can be constructed by means of perturbation of a root. The method for a regular [Formula: see text]-hypergeometric system is a theme of the book by Saito, Sturmfels and Takayama. Whereas they perturbed a parameter vector to obtain logarithmic [Formula: see text]-hypergeometric series solutions, we adopt a different perturbation in this paper.

Electromagnetic waves in graded-index planar waveguides

The propagation of guided TE and TM modes through graded-index planar waveguides is reported in this paper. Both a real-valued and complex-valued position-dependent refractive index are supposed for a film layer. It is possible to set various refractive index profiles based on five distribution parameters. For the position-dependent refractive index, the governing equations are transformed to Heun’s differential equation, an exact local solution expressed in terms of local Heun functions. The general nature of these functions is demonstrated based on four degenerate cases of Heun’s equation. The calculation of guided modes requires evaluation of the general solution in the interval containing two regular singular points. For this purpose, the generalized Heun function is introduced and employed in general solutions to the governing equations. The applicability of the generalized solutions is demonstrated by the calculation of guided modes for both the real-valued and complex-valued refractive index.

A geometric-analytic study of linear differential equations of order two

&lt;p style='text-indent:20px;'&gt;We study second order linear differential equations with analytic coefficients. One important case is when the equation admits a so called regular singular point. In this case we address some untouched and some new aspects of Frobenius methods. For instance, we address the problem of finding formal solutions and studying their convergence. A characterization of regular singularities is given in terms of the space of solutions. An analytic-geometric classification of such linear polynomial homogeneous ODEs is obtained by the use of techniques from geometric theory of foliations means. This is done by associating to such an ODE a rational Riccati differential equation and therefore a &lt;i&gt;global holonomy group&lt;/i&gt;. This group is a computable group of Moebius maps. These techniques apply to classical equations as Bessel and Legendre equations. We also address the problem of deciding which such polynomial equations admit a Liouvillian solution. A normal form for such a solution is then obtained. Our results are concrete and (computationally) constructive and are aimed to shed a new light in this important subject.&lt;/p&gt;

On the fully anisotropic honeycomb lattice Green function, Bessel function integrals and Pearson random walks

The analytical properties of the lattice Green function for the fully anisotropic honeycomb lattice are studied, where (α 1, α 2, α 3) are anisotropy parameters with {α j ∈ (0, ∞): j = 1, 2, 3}, and w = u + iv is a complex variable in a (u, v) plane. This integral defines a single-valued analytic function G H(α 1, α 2, α 3; w) provided that a cut is made along the real axis from u = −(α 1 + α 2 + α 3) to u = (α 1 + α 2 + α 3). We show that G H(α 1, α 2, α 3; w) is a solution of a second-order linear differential equation with ten ordinary regular singular points and four apparent singular points. The apparent singularities are removed by constructing a particular differential equation of fourth order. Next, the series solution where |w| > (α 1 + α 2 + α 3), and is introduced. It is proved that, in general, satisfies a five-term linear recurrence relation. The asymptotic behaviour of as n → ∞ is also established. In order to determine the behaviour of G H(α 1, α 2, α 3; w) along the edges of the cut we define the limit function where u ∈ [−(α 1 + α 2 + α 3), (α 1 + α 2 + α 3)]. Integral representations are established for and . In particular, it is found that where J 0(z) and Y 0(z) denote Bessel functions of the first and second kind, respectively, and u ∈ (0, α 1 + α 2 + α 3). It is also demonstrated that the piecewise functions and can be sectionally evaluated exactly for all u ∈ (0, α 1 + α 2 + α 3), in terms of complete elliptic integrals of the first kind K(k), where k 2 ≡ k 2(α 1, α 2, α 3, u) is a rational function of (α 1, α 2, α 3) and u. Finally, applications of the results are made to the lattice Green function for the fully anisotropic simple cubic lattice, and to the theory of Pearson random walks in a plane. In particular, various Bessel function integrals are evaluated in order to derive a new exact formula for the mean end-to-end distance of a general three-step random walk.

Solving Second-Order Linear Differential Equations with Random Analytic Coefficients about Regular-Singular Points

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.

Exact Analytical and Numerical Solutions for Convective Heat Transfer in a Semi-Spherical Extended Surface with Regular Singular Points

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.

Scalar particle with the Darwin – Cox intrinsic structure in the external Coulomb field

The generalized Klein – Fock – Gordon equation for a particle with the Darwin–Cox structure allowing for a charge distribution of a particle over a sphere of finite radius is studied with regard to the external Coulomb field. The separation of variables is carried out, the obtained radial equation is significantly more complicated than the equation in the case of ordinary particles, it has essentially singular points r = 0 of rank 3, r = ∞ of rank 2 and 4 regular singular points. In the case of a minimum orbital momentum l = 0, the structure of singularities is simplified: there are essentially singular points r = 0, r = ∞ of rank 2 and 4 regular singular points. Frobenius solutions of this equation are constructed and the structure of the 7-term recurrence relations for the coefficients of the arising power series is investigated. As an analytical quantization condition, the generalized transcendence requirement of solutions is used; it allows one to obtain a fourth-degree algebraic equation for energy levels. The equation has 4 sets of roots depending on the orbital moment l and the main quantum number k = 1,2,3,… . The numerical analysis shows that one of the sets of the roots 0 &lt; εl,k &lt; mc2 can be interpreted as those corresponding to certain bound states of the particle in the Coulomb field.

The elliptic KZB connection and algebraic de Rham theory for unipotent fundamental groups of elliptic curves

In this paper, we develop an algebraic de Rham theory for unipotent fundamental groups of once punctured elliptic curves over a field of characteristic zero using the universal elliptic KZB connection of Calaque-Enriquez-Etingof and Levin-Racinet. We use it to give an explicit version of Tannaka duality for unipotent connections over an elliptic curve with a regular singular point at the identity.

Scalar particle with Darwin-Cox structure in external Coulomb field

Generalized Klein-Fock-Gordon equation for a scalar particle with Darwin-Cox structure, which takes into account distribution of electric charge of the particle inside a finite spherical region is studied in external Coulomb field. Corresponding radial equation has two irregular singular points, r = 0 of the rank 3, r = ∞ of the rank 2, and four regular singular points. In the case of minimal angular momentum, l = 0, the structure of singularities becomes simpler: the points r = 0, r = ∞ are both of the rank 2, and four regular points remain the same. There are constructed formally exact Frobenius type solutions of the derived equations, convergence of relevant power series, with 8-term and 7-term recurrent relations respectively, is studied. As analytical quantization rule is taken so-called transcendence conditions. It provides us with 4-th order algebraic equation with respect to energy values, which has four sets of roots. Only one set of roots, 0 < El,k < 1, depending on angular momentum l = 0, 1, 2, … and main quantum number n = 0, 1, 2, … may be interpreted as corresponding to some bound states of the particle in the Coulomb field. In the same manner, a generalized nonrelativistic Schrödinger equation for such a particle was studied, the final results are similar.

Fractional Gauss hypergeometric differential equation

In this paper, we use the new concept of fractional regular singular point and use the technique of fractional power series to solve the fractional Gauss hypergeometric differential equation. Also, we introduce the forms of the conformable fractional derivative and the integral representation of the fractional Guassian function.

Generalized-Hypergeometric Solutions of the General Fuchsian Linear ODE Having Five Regular Singularities

We show that a Fuchsian differential equation having five regular singular points admits solutions in terms of a single generalized hypergeometric function for infinitely many particular choices of equation parameters. Each solution assumes four restrictions imposed on the parameters: two of the singularities should have non-zero integer characteristic exponents and the accessory parameters should obey polynomial equations.

On the Legendre differential equation with uncertainties at the regular‐singular point 1: L p ( Ω ) random power series solution and approximation of its statistical moments

is the peer reviewed version of the following article: Calatayud, J, Cortes, J-;C, Jornet, M. On the Legendre differential equation with uncertainties at the regular-singular point 1: Lp random power series solution and approximation of its statistical moments. Comp and Math Methods. 2019; 1:e1045. https://doi.org/10.1002/cmm4.1045 , which has been published in final form at https://doi.org/10.1002/cmm4.1045. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.

Analytical study of Bragg resonance by singly periodic sinusoidal ripples based on the modified mild-slope equation

The wave propagation over singly periodic sinusoidal ripples is studied analytically based on the modified mild-slope equation (MMSE). Firstly, each ripple region is divided into four monotonic subintervals such that there is only one regular singular point of the MMSE within each subinterval which is located at one of the two endpoints. Secondly, in each subinterval, due to the monotonicity, the implicit MMSE can be transformed into an explicit MMSE (EMMSE). Thirdly, in each subinterval, due to the unique regular singularity, a general solution in terms of Frobenius series to the EMMSE can be constructed and the convergence condition of the series solution is analyzed and graphically demonstrated. Finally, by using the mass-conserving matching conditions at each common boundary between any two adjacent subintervals, an analytical formula of the reflection coefficient is established. The present solution is validated against various existing solutions, especially, it is identical to a numerical solution to the MMSE. It is shown that, as the number of ripples increases, the peak amplitudes of both the primary and subharmonic Bragg resonances increase while the resonance bandwidths decrease, and the zero reflection occurs more frequently. When the number becomes very large, the peak amplitudes of the primary and subharmonic Bragg resonances achieve unity and the resonance bandwidths approach to their limit resonance bands. As the height of ripples increases, the peak amplitudes of both the primary and subharmonic Bragg resonances and the bandwidth of the primary resonance increase accordingly, and the positions of both the primary and subharmonic Bragg resonances downward shift to low frequency more significantly, but the number of the zero reflection keeps the same.

Exact results for the anisotropic honeycomb lattice Green function with applications to three-step Pearson random walks

The mathematical properties of the lattice Green function for the anisotropic honeycomb lattice are studied, where is a complex variable which lies in a plane, and is a real anisotropy parameter with . This double integral defines a single-valued analytic function provided that a cut is made along the real axis from to . In order to analyse the behaviour of along the edges of the cut it is convenient to define the limit function where . It is proved that the limit functions and can be sectionally evaluated exactly for all , in terms of various elliptic integrals of the first kind , where is a rational function of and u.Next, it is demonstrated that is a solution of a second order linear differential equation with eight ordinary regular singular points and two apparent singular points. It is shown that the apparent singularities can be removed by constructing a particular differential equation of third order. The series solution where and is investigated. In particular, we show that, in general, satisfies a four-term linear recurrence relation. This result is used to determine the asymptotic behaviour of as .Integral representations are established for and . It is found that where J0(z) and Y0(z) denote Bessel functions of the first and second kind, respectively, and .Finally, the results are applied to the lattice Green function for the anisotropic simple cubic lattice, and to the theory of Pearson random walks in a plane.

The massive Dirac equation in the Kerr-Newman-de Sitter and Kerr-Newman black hole spacetimes

Exact solutions of the Dirac general relativistic equation that describe the dynamics of a massive, electrically charged particle with half-integer spin in the curved spacetime geometry of an electrically charged, rotating Kerr-Newman-(anti) de Sitter black hole are investigated. We first, derive the Dirac equation in the Kerr-Newman-de Sitter (KNdS) black hole background using a generalised Kinnersley null tetrad in the Newman-Penrose formalism. Subsequently in this frame and in the KNdS black hole spacetime, we prove the separation of the Dirac equation into ordinary differential equations for the radial and angular parts. Under specific transformations of the independent and dependent variables we prove that the transformed radial equation for a massive charged spin fermion in the background KNdS black hole constitutes a highly non-trivial generalisation of Heun’s equation since it possess five regular finite singular points. Using a Regge-Wheeler-like independent variable we transform the radial equation in the KNdS background into a Schrödinger like differential equation and investigate its asymptotic behaviour near the event and cosmological horizons. For the case of a massive fermion in the background of a Kerr-Newman (KN) black hole we first prove that the radial and angular equations that result from the separation of Dirac’s equation reduce to the generalised Heun differential equation (GHE). The local solutions of such GHE are derived and can be described by holomorphic functions whose power series coefficients are determined by a four-term recurrence relation. In addition using asymptotic analysis we derive the solutions for the massive fermion far away from the KN black hole and the solutions near the event horizon . The determination of the separation constant as an eigenvalue problem in the KN background is investigated. Using the aforementioned four-term recursion formula we prove that in the non-extreme KN geometry there are no bound states with , where ω and μ are the energy and mass of the fermion respectively.

АНАЛІЗ ЗАГАЛЬНОГО РОЗВ’ЯЗКУ ОДНОГО ЛІНІЙНОГО ОДНОРІДНОГО ДИФЕРЕНЦІАЛЬНОГО РІВНЯННЯ З ПРАВИЛЬНОЮ ОСОБЛИВОЮ ТОЧКОЮ

У роботі виявлено, що обране для дослідження лінійне однорідне диференціальне рівняння другого порядку, яке має правильну особливу точку, за допомогою перетворення Куммера-Ліувілля зводиться до виродженого гіпергеометричного рівняння, яке подане в формі Уіттекера. Встановлені співвідношення між коефіцієнтами заданого рівняння, виродженого гіпергеометричного рівняння у загальному вигляді та у формі Уіттекера. На основі цих співвідношень показано, що досліджуване рівняння має лише один незалежний фундаментальний розв’язок у вигляді виродженої гіпергеометричної функції. Другий фундаментальний розв’язок може бути знайдений або за допомогою метода Фробеніуса у вигляді узагальненого степеневого ряду, або виражений через функцію Трікомі. В останньому випадку загальний розв’язок досліджуваного рівняння є лінійною комбінацією функцій Уіттекера.