Related papers: On Certain Solutions for Confluent and Double-Conf…
When studying boundary value problems for some partial differential equations arising in applied mathematics, we often have to study the solution of a system of partial differential equations satisfied by hypergeometric functions and find…
We review different methods of generating potentials such that the one-dimensional Schr\"{o}dinger equation (ODSE) can be transformed into the hypergeometric equation. We compare our results with previous studies, and complement the subject…
In this article the unique solution of the Cauchy problem is founded by the Riemann method. Some relations for given here confluent hypergeometric functions of two and three variables are used.
Uniformly regular equilibrium problems are natural generalizations of abstract equilibrium prob lems and they are defined over the uniformly prox-regular nonconvex sets. Some new efficient implicit methods for solving uniformly regular…
We report an alternative method to solve second order differential equations which have at most four singular points. This method is developed by changing the degrees of the polynomials in the basic equation of Nikiforov-Uvarov (NU) method.…
By means of topological methods, we provide new results on the existence, non-existence, localization and multiplicity of nontrivial solutions for systems of perturbed Hammerstein integral equations. In order to illustrate our theoretical…
A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. We present in a unified and explicit way all these systems of orthogonal polynomials, the associated…
The $q$-Heun equation is a $q$-difference analogue of Heun's differential equation. We review several solutions of Heun's differential equation and investigate polynomial-type solutions of $q$-Heun equation. The limit $q\to 1$ corresponding…
In this paper, we introduce a new function, the multiple confluent hypergeometric functions, and establish a functional equation for the $r$-variable Euler--Zagier multiple zeta functions using it. In the case when $r=2$, this functional…
We give a list of Heun equations which are Picard-Fuchs associated to families of algebraic varieties. Our list is based on the classification of families of elliptic curves with four singular fibers done by Herfurtner. We also show that…
We present some recent progresses on Heun functions, gathering results from classical analysis up to elliptic functions. We describe Picard's generalization of Floquet's theory for differential equations with doubly periodic coefficients…
In this article, we study about the solutions of second order linear differential equations by considering several conditions on the coefficients of homogenous linear differential equation and its associated non-homogenous linear…
We investigate the integral representations of solutions to the variant of $q$-hypergeometric equation of degree 2 obtained through $q$-middle convolution by using transformation formulas for $q$-hypergeometric series. We show the…
Modular and quasimodular solutions of specific second order differential equation in the upper-half plane which originates from a study of supersingular j-invariants are given explicitly. A characterization of the differential equation is…
We find confluent Heun solutions to the radial equations of two Halilsoy-Badawi metrics. For the first metric, we studied the radial part of the massless Dirac equation and for the second case, we studied the radial part of the massless…
In this paper, using the similarity method, we construct particular solutions with singularities for degenerate high-order equations. The considered equations have singularities of the first and second kind. Particular solutions are…
The motion of compressible, inviscid fluid under the constant pressure on a rotating sphere is studied. The hodograph equations for the corresponding Euler equation are presented. They provide us with the class of solutions of the Euler…
An analysis of the generalized confluent Heun equation $(\alpha_2r^2+\alpha_1r)\,y''+(\beta_2r^2+\beta_1r+\beta_0)\,y'-(\varepsilon_1r+\varepsilon_0)\,y=0$ in $d$-dimensional space, where $\{\alpha_i, \beta_i, \varepsilon_i\}$ are real…
This report addresses the solution of Riemann problems for hyperbolic equations when the nonlinear characteristic fields loose their genuine nonlinearity. In this context, exact solvers for nonconvex 1D Riemann problems are developed. First…
Streets and Tian introduced a parabolic flow of pluriclosed metrics. We classify the long time behavior of homogeneous solutions of this flow on closed complex surfaces including minimal Hopf, Inoue, Kodaira, and non-Kahler, properly…