Related papers: Comments on Lagrange Partial Differential Equation
This work is divided into three parts. The first part concerns polynomials in one variable with all real roots. We consider linear transformations that preserve real rootedness, as well as matrices that preserve interlacing. The second part…
We consider a system of differential equations and obtain its solutions with exponential asymptotics and analyticity with respect to the spectral parameter. Solutions of such type have importance in studying spectral properties of…
In this work we study the solutions to some fractional higher-order equations. Special cases in which time-fractional derivatives take integer values are also examined and the explicit solutions are presented. Such solutions can be…
The existence of sufficiently many finite order meromorphic solutions of a differential equation, or difference equation, or differential-difference equation, appears to be a good indicator of integrability. In this paper, we investigate…
Often a non-linear mechanical problem is formulated as a non-linear differential equation. A new method is introduced to find out new solutions of non-linear differential equations if one of the solutions of a given non-linear differential…
We introduce two ordinary second-order linear differential equations of the Laguerre- and Jacobi-type. Solutions are written as infinite series of square integrable functions in terms of the Laguerre and Jacobi polynomials, respectively.…
We present a method of deriving linearizing transformations for a class of second order nonlinear ordinary differential equations. We construct a general form of a nonlinear ordinary differential equation that admits Bernoulli equation as…
This work investigates diagonalization-based methods for efficiently solving linear evolution problems, with a particular focus on the heat equation. The plain diagonalization of the differential operator, though effective for elliptic…
We present a stable and convergent method for solving initial value problems based on the use of differentiation matrices obtained by Lagrange interpolation. This implicit multistep-like method is easy-to-use and performs pretty well in the…
It is well known that for a first order system of linear difference equations with rational function coefficients, a solution that is holomorphic in some left half plane can be analytically continued to a meromorphic solution in the whole…
It is well-known that the equations for a simple fluid can be cast into what is called their Lagrange formulation. We introduce a notion of a generalized Lagrange formulation, which is applicable to a wide variety of systems of partial…
We develop a systematic way to solve linear equations involving tensors of arbitrary rank. We start off with the case of a rank $3$ tensor, which appears in many applications, and after finding the condition for a unique solution we derive…
In this paper we use different techniques from the fractional and pseudo-operators calculus to solve partial differential equations involving operators with non integer exponents. We apply the method to equations resembling generalizations…
A functional differential equation related to the logistic equation is studied by a combination of numerical and perturbation methods. Parameter regions are identified where the solution to the nonlinear problem is approximated well by…
In this paper, the solution of the multi-order differential equations, by using Mellin Transform, is proposed. It is shown that the problem related to the shift of the real part of the argument of the transformed function, arising when the…
We present a method for reducing the order of ordinary differential equations satisfying a given scaling relation (Majorana scale-invariant equations). We also develop a variant of this method, aimed to reduce the degree of non-linearity of…
In this article, the order of some classes of fractional linear differential equations is determined, based on asymptotic behavior of the solution as time tends to infinity. The order of fractional derivative has been proved to be of great…
As a first step towards a theory of differential equations involving para-Grassmann variables the linear equations with constant coefficients are discussed and solutions for equations of low order are given explicitly. A connection to…
We study solutions of the Yang-Baxter equation on a tensor product of an arbitrary finite-dimensional and an arbitrary infinite-dimensional representations of the rank one symmetry algebra. We consider the cases of the Lie algebra sl_2, the…
A method to calculate the adjoint solution for a large class of partial differential equations is discussed. It differs from the known continuous and discrete adjoint, including automatic differentiation. Thus, it represents an alternative,…