Related papers: Solving Third Order Linear Difference Equations in…
In this paper, we present a new numerical method to solve fractional differential equations. Given a fractional derivative of arbitrary real order, we present an approximation formula for the fractional operator that involves integer-order…
Quantum algorithms offer an exponential advantage with respect to the number of dependent variables for solving certain nonlinear ordinary differential equations (ODEs). These algorithms typically begin by transforming the original…
We present new algorithms to detect and correct errors in the lower-upper factorization of a matrix, or the triangular linear system solution, over an arbitrary field. Our main algorithms do not require any additional information or…
We report a detailed analysis of the optical realization [1, 3, 2, 4] of the analogue algorithm described in the first paper of this series [5] for the simultaneous factorization of an exponential number of integers. Such an analogue…
We have been working in many aspects of the problem of analyzing, understanding and solving ordinary differential equations (first and second order). As we have extensively mentioned, while working in the Darboux type methods, the most…
We discuss a general method by which a higher order difference equation on a group is transformed into an equivalent triangular system of two difference equations of lower orders. This breakdown into lower order equations is based on the…
We study the algorithmic complexity of fair division problems with a focus on minimizing the number of queries needed to find an approximate solution with desired accuracy. We show for several classes of fair division problems that under…
Linearization of coupled second order nonlinear ordinary differential equations (SNODEs) is one of the open and challenging problems in the theory of differential equations. In this paper we describe a simple and straightforward method to…
Linear differential equations are ubiquitous in science and engineering. Quantum computers can simulate quantum systems, which are described by a restricted type of linear differential equations. Here we extend quantum simulation algorithms…
Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…
For a nonlinear ordinary differential equation with time delay, the differentiation of the solution with respect to the delay is investigated. Special emphasis is laid on the second-order derivative. The results are applied to an associated…
A systematic algorithm for building integrating factors of the form mu(x,y), mu(x,y') or mu(y,y') for second order ODEs is presented. The algorithm can determine the existence and explicit form of the integrating factors themselves without…
We present two algorithms for computing hypergeometric solutions of second order linear differential operators with rational function coefficients. Our first algorithm searches for solutions of the form \[ \exp(\int r \,…
A characterization of the symmetry algebra of the $n$th order ordinary differential equations (ODEs) with maximal symmetry and all third order linearizable ODEs is given. This is used to show that such an algebra $\mathfrak{g}$ determines…
A new method of solving third-order ordinary complex differential equations (OCDEs) by generalizing Prelle-Singer. The idea which is a procedure for finding the solution for second-order differential equations in the real domain. We have…
A typical system of k difference (or differential) equations can be compressed, or folded into a difference (or ordinary differential) equation of order k. Such foldings appear in control theory as the canonical forms of the controllability…
In this article we study solutions to second order linear difference equations with variable coefficients. Under mild conditions we provide closed form solutions using finite continued fraction representations. The proof of the results are…
We consider the semiconjugate factorization and reduction of order for non-autonomous, nonlinear, higher order difference equations containing linear arguments. These equations have appeared in several mathematical models in biology and…
We continue to investigate which polynomials can possibly occur as factors in the denominators of rational solutions of a given partial linear difference equation. In an earlier article we had introduced the distinction between periodic and…
This article deals with the second order linear differential equations with entire coefficients. We prove some results involving conditions on coefficients so that the order of growth of every non-trivial solution is infinite.