Polynomial solutions to second order linear homogeneous ordinary differential equations. Properties and approximation

Second order linear homogeneous ODEs which generalize those considered in the orthogonal polynomial theory are studied. A method and an algorithm to approach their polynomial solutions are also given.     

Asymptotic behaviour of the solutions of second order functional differential equations

This paper presents integral criteria to determine the asymptotic behaviour of the solutions of second order nonlinear differential equations of the type y^″(x)+q(x)f(y(x))=0, with q(x)>0 and f(y) odd and positive for y>0, as x tends to +∞. It also compares them with the results obtained by Chanturia (1975) in [11] for the same problem.     

Computer-assisted reduction of second order linear differential equations

So as to explore the possibilities of representing its solutions in terms of special functions, and using factorization techniques, a process is defined to decide whether a second order linear differential equation with polynomial coefficients can be brought to the hypergeometric or the confluent hypergeometric equation by a rational change of variable. In the first case, an upper bound has to be provided for the degree of the numerator of the rational function which defines the change of variable.     

A note on oscillation criteria of second order nonlinear neutral delay differential equations

In recent years, many results on oscillation criteria of second order nonlinear neutral delay differential equations have been obtained, but some of these criteria are incorrect due to a mistake in a crucial auxiliary result. This note examines several of these results, provides some counterexamples and points out what causes the problems.     

Linear systems of first order ordinary differential equations: fuzzy initial conditions

 We present two types of fuzzy solutions to linear systems of first order differential equations having fuzzy initial conditions. The first solution, called the extension principle solution, fuzzifies the crisp solution and then checks to see if its α-cuts satisfy the differential equations. The second solution, called the classical solution, solves the fuzzified differential equations and then checks to see if the solution always defines a fuzzy number. Three applications are presented: (1) predator–prey models; (2) the spread of infectious diseases; and (3) modeling an arms race.     

New solutions for ordinary differential equations

This paper introduces a new method for solving ordinary differential equations (ODEs) that enhances existing methods that are primarily based on finding integrating factors and/or point symmetries. The starting point of the new method is to find a non-invertible mapping that maps a given ODE to a related higher-order ODE that has an easily obtained integrating factor. As a consequence, the related higher-order ODE is integrated. Fixing the constant of integration, one then uses existing methods to solve the integrated ODE. By construction, each solution of the integrated ODE yields a solution of the given ODE. Moreover, it is shown when the general solution of an integrated ODE yields either the general solution or a family of particular solutions of the given ODE. As an example, new solutions are obtained for an important class of nonlinear oscillator equations. All solutions presented in this paper cannot be obtained using the current Maple ODE solver.     

Stability of linear systems governed by second order vector differential equations

We consider linear non-autonomous systems governed by second order vector ordinary differential equations. Explicit stability conditions are derived.     

Solving and factoring boundary problems for linear ordinary differential equations in differential algebras

We present a new approach for expressing and solving boundary problems for linear ordinary differential equations in the language of differential algebras. Starting from an algebra with a derivation and integration operator, we construct an algebra of linear integro-differential operators that is expressive enough for specifying regular boundary problems with arbitrary Stieltjes boundary conditions as well as their solution operators.     

General linear methods for ordinary differential equations

General linear methods were introduced as the natural generalizations of the classical Runge–Kutta and linear multistep methods. They have potential applications, especially for stiff problems. This paper discusses stiffness and emphasises the need for efficient implicit methods for the solution of stiff problems. In this context, a survey of general linear methods is presented, including recent results on methods with the inherent RK stability property.     

A method for solving nth order fuzzy linear differential equations

In this paper, an analytic method (eigenvalue–eigenvector method) for solving nth order fuzzy differential equations with fuzzy initial conditions is considered. In this method, three cases are introduced, in each case, it is shown that the solution of differential equation is a fuzzy number. In addition, the method is illustrated by solving several numerical examples.     

Solving large systems of linear ordinary differential equations on a vector computer

This paper is concerned with the development, analysis and implementation on a computer consisting of two vector processors of the arithmetic mean method for solving numerically large sparse sets of linear ordinary differential equations. This method has second-order accuracy in time and is stable.

The special class of differential equations that arise in solving the diffusion problem by the method of lines is considered. In this case, the proposed method has been tested on the CRAY X-MP/48 utilizing two CPUs. The numerical results are largely in keeping with the theory; a speedup factor of nearly two is obtained.     

On a class of positive linear differential equations with infinite delay

We give an explicit criterion for positivity of the solution semigroup of linear differential equations with infinite delay and a Perron-Frobenius type theorem for positive equations. Furthermore, a novel criterion for the exponential asymptotic stability of positive equations is presented. Finally, we provide a sufficient condition for the exponential asymptotic stability of positive equations subjected to structured perturbations. A simple example is given to illustrate the obtained results.     

Differential periodic Riccati equations: Existence and uniqueness of nonnegative definite solutions

In this paper we consider the differential periodic Riccati equation. All the periodic nonnegative definite solutions are characterized in the more general case, providing a method for constructing them. The method is obtained from the study of the invariant subspaces of the monodromy matrix of the associated Hamiltonian system, and from the relations between these invariant subspaces and the controllability and unobservability subspaces. Finally, the method is applied to obtain necessary and sufficient conditions for the existence of any periodic nonnegative definite solution and to study the existence and uniqueness of minimal, maximal, stabilizing, and strong solutions.This work has been partially supported by Spanish DGICYT Grant No. PB91-O535.     

On d-solvability for linear differential equations

The aim of this paper is to investigate the possibility of solving a linear differential equation of degree n$n$ by means of differential equations of degree less than or equal to a fixed d$d$, 1≤d<n$1\le d<n$. This paper recovers and extends work of G. Fano, M. F. Singer and E. Compoint. Representations of algebraic Lie algebras are the main tool.