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Generalized trigonometric functions are applied to the Legendre-Jacobi standard form of complete elliptic integrals, and a new form of the generalized complete elliptic integrals of the Borweins is presented. According to the form, it can…
A method of finding general solutions of second-order nonlinear ordinary differential equations by extending the Prelle-Singer (PS) method is briefly discussed. We explore integrating factors, integrals of motion and the general solution…
In this paper we present an iterative method, inspired by the inverse iteration with shift technique of finite linear algebra, designed to find the eigenvalues and eigenfunctions of the Laplacian with homogeneous Dirichlet boundary…
We deal with the higher-order fractional Laplacians by two methods: the integral method and the system method. The former depends on the integral equation equivalent to the differential equation. The latter works directly on the…
It is well known that second order homogeneous linear ordinary differential equations with slowly varying coefficients admit slowly varying phase functions. This observation underlies the Liouville-Green method and many other techniques for…
The exactness equation for Lepage 2-forms, associated with variational systems of ordinary differential equations on smooth manifolds, is analyzed with the aim to construct a concrete global variational principle. It is shown that locally…
Using the theory of orthogonal polynomials, their associated recursion relations and differential formulas we develop a method for evaluating new integrals. The method is illustrated by obtaining a closed-form expression for the value of an…
A general method for solving linear differential equations of arbitrary order, is used to arrive at new representations for the solutions of the known differential equations, both without and with a source term. A new quasi-solvable…
Functional integrals are central to modern theories ranging from quantum mechanics and statistical thermodynamics to biology, chemistry, and finance. In this work we present a new method for calculating functional integrals based on a…
A solution is proposed for the problem of composition of ordinary generating functions. A new class of functions that provides a composition of ordinary generating functions is introduced; main theorems are presented; compositae are written…
Finite differences have been widely used in mathematical theory as well as in scientific and engineering computations. These concepts are constantly mentioned in calculus. Most frequently-used difference formulas provide excellent…
The main aim of this paper is to provide a novel approach to deriving identities for the Bernstein polynomials using functional equations. We derive various functional equations and differential equations using generating functions.…
In the article, a general solution of an equation with a generalized Hilfer derivative, which has a degeneration, is constructed. Particular solutions are presented through the Kilbas-Saigo function. A representation of the solution of the…
Several integrals involving powers and ordinary hypergeometric functions are rederived by means of a generalized hypergeometric function of two variables (Appell's function) recovering some well-known expressions as particular cases. Simple…
We study a natural extension to complex numbers of the standard continued fractions. The basic algorithm is due to Lagrange and Gauss, though it seems to have gone mostly unnoticed as a way to create continued fractions. The new…
We review properties of confluent functions and the closely related Laguerre polynomials, and determine their bilinear integrals. As is well-known, these integrals are convergent only for a limited range of parameters. However, when one…
We consider an ordinary nonlinear differential equation with generalized coefficients as an equation in differentials in algebra of new generalized functions. Then the solution of such equation will be a new generalized function. In the…
We study incommensurate fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives and generalized fractional integrals and derivatives. We obtain necessary optimality…
We present a common ground for infinite sums, unordered sums, Riemann/Lebesgue integrals, arc length and some generalized means. It is based on extending functions on finite sets using Hausdorff metric in a natural way.
In this study, new master theorems and general formulas of integrals are presented and implemented to solve some complicated applications in different fields of science. The proposed theorems are considered to be generators of new problems,…