Related papers: On polynomial solutions of Heun equation
Based the homogeneous balance method, a general method is suggested to obtain several kinds of exact solutions for some kinds of nonlinear equations. The validity and reliability of the method are tested by applying it to the Bousseneq…
We present exercises with solutions related to A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode.
We consider methods for constructing explicit solutions of the non-stationary Lam\'e equation, which is a generalization of the classical Lam\'e equation, that has appeared in works on integrable models, conformal field theory, high energy…
We consider the extension of the method of Gauss-Newton from complex floating-point arithmetic to the field of truncated power series with complex floating-point coefficients. With linearization we formulate a linear system where the…
In this paper, we study the existence and uniqueness of periodic solutions of the differential equation of the form . Here, we obtain some sufficient conditions which guarantee the existence of periodic solutions. This equation is a quite…
Necessary conditions are obtained for certain types of rational delay differential equations to admit a non-rational meromorphic solution of hyper-order less than one. The equations obtained include delay Painlev\'e equations and equations…
Our aim in this paper is to prove, under some growth conditions on the datas, the solvability in a Gevrey class of a polynomially nonlinear functional differential equation.
The paper represents the method for construction of the families of particular solutions to some new classes of $(n+1)$ dimensional nonlinear Partial Differential Equations (PDE). Method is based on the specific link between algebraic…
We give improved lower bounds for the number of solutions of some $S$-unit equations over the integers, by counting the solutions of some associated linear equations as the coefficients in those equations vary over sparse sets. This method…
Tropical differential equations are introduced and an algorithm is designed which tests solvability of a system of tropical linear differential equations within the complexity polynomial in the size of the system and in its coefficients.…
We solve the difference equation with linear coefficients by the Momentenansatz to obtain explicit formulas for orthogonal polynomials.
The characterization of systems of differential equations admitting a superposition function allowing us to write the general solution in terms of any fundamental set of particular solutions is discussed. These systems are shown to be…
Some properties and relations satisfied by the polynomial solutions of the bispectral problem are studied. Given a differential operator, under certain restrictions its polynomial eigenfunctions are explicitly obtained, as well as the…
An approach is proposed for bounding the number of zeros that solutions of linear differential systems with polynomial coefficients may have. A bound is obtained in a special case which improves upon currently existing.
We reexamine and extend a group of solutions in series of Bessel functions for a limiting case of the confluent Heun equation and, then, apply such solutions to the one-dimensional Schr\"odinger equation with an inverted quasi-exactly…
We develop a new method that improves the efficiency of equation-by-equation algorithms for solving polynomial systems. Our method is based on a novel geometric construction, and reduces the total number of homotopy paths that must be…
Meromorphic solutions of autonomous nonlinear ordinary differential equations are studied. An algorithm for constructing meromorphic solutions in explicit form is presented. General expressions for meromorphic solutions (including rational,…
Several results on Heun's equation are generalized to a certain class of Fuchsian differential equations. Namely, we obtain integral representations of solutions and develop Hermite-Krichever Ansatz on them. In particular, we investigate…
In this article we present solutions of certain polynomial equations in periodic nested radicals. We also present a new way to solve the general tetranomial equation with new functions. As application of these new functions we solve the…
The object of the present work is to present the new classes of third-order and fourth-order iterative methods for solving nonlinear equations. Our third-order method includes methods of Weerakoon \cite{Weerakoon}, Homeier \cite{Homeier2},…