Related papers: Generalized-hypergeometric solutions of the genera…
We present infinitely many solutions of the general Heun equation in terms of generalized hypergeometric functions. Each solution assumes that two restrictions are imposed on the involved parameters: a characteristic exponent of one of the…
We show that for every second order Fuchsian linear differential equation $E$ with $n$ singularities of which $n-3$ are apparent there exists a hypergeometric equation $H$ and a linear differential operator with polynomial coefficients…
A Fuchsian system of rank 8 in 3 variables with 4 parameters is presented. The singular locus consists of six planes and a cubic surface. The restriction of the system onto the intersection of two singular planes is an ordinary differential…
The reducible double confluent Heun equation (DCHE) is the only DCHE whose general symmetric unfolding leads to a Fuchsian equation. Contrary to general Heun equation the unfolded Fuchsian equation has 5 singular points :…
Fuchsian differential equations $H_j$ of order $j=3,\dots,6$ with three singular points and one accessory parameter are presented. The shift operators for $H_6$ are studied. They lead to assign the accessory parameter of $H_6$ a cubic…
We give the hypergeometric solutions of some algebraic equations including the general fifth degree equation.
We show that the Heun confluent equation admits infinitely many solutions in terms of the confluent generalized hypergeometric functions. For each of these solutions a characteristic exponent of a regular singularity of the Heun confluent…
In this paper, we introduce a new class of confluent hypergeometric functions of many variables, study their properties, and determine a system of partial differential equations that this function satisfies. It turns out that all the…
In our preceding paper, we have proposed an algorithm for obtaining finite-norm solutions of higher-order linear ordinary differential equations of the Fuchsian type [\sum_m p_m (x) (d/dx)^m] f(x) = 0 (where p_m is a polynomial with…
We present a multidomain spectral approach for Fuchsian ordinary differential equations in the particular case of the hypergeometric equation. Our hybrid approach uses Frobenius' method and Moebius transformations in the vicinity of each of…
The main result of the present paper is the construction of fundamental solutions for a class of multidimensional elliptic equations with several singular coefficients. These fundamental solutions are directly connected with multiple…
We show that in four particular cases the derivative of the solution of Heun general equation can be expressed in terms of a solution to another Heun equation. Starting from this property, we use the Gauss hypergeometric functions to…
We focus on Fuchsian equations with four accessory parameters and three singular points. We see that the Fuchsian equations admit a "degeneration scheme" in some sense, which is expected to give rise to a degeneration scheme of discrete…
The form of the coefficients of power series expressions corresponding to solutions of Fuchsian differential equations (or their associated degenerated confluent forms) with n regular singular points is determined by solving the…
A general formalism to solve nonlinear differential equations is given. Solutions are found and reduced to those of second order nonlinear differential equations in one variable. The approach is uniformized in the geometry and solves…
This article addresses linear hyperbolic partial differential equations and pseudodifferential equations with strongly singular coefficients and data, modelled as members of algebras of generalised functions. We employ the recently…
We construct solutions to nonlinear wave equations that are singular along a prescribed noncharacteristic hypersurface which is the graph of a function satisfying not the Eikonal but another partial differential equation of the first order.…
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…
We find a new class of the Fuchsian equations, which have an algebraic geometric solutions with the parameter belonging to a hyperelliptic curve. Methods of calculating the algebraic genus of the curve, and its branching points, are…
Differential systems with a Fuchsian linear part are studied in regions including all the singularities in the complex plane of these equations. Such systems are not necessarily analytically equivalent to their linear part (they are not…