Benchmarking of ODE methods has a long tradition. Several sets of test problems have been developed and new problems are still collected. So, a whole variety of problems can be used to check the eeciency of a method under investigation. In general, eeciency is measured by the amount of work which is necessary to get a reliable solution for a prescribed accuracy. In order to quantify the term \a...

Ordinary differential equation (ODE) models, e.g. the susceptible-infected-recovered model, are widely used in engineering, ecology, and epidemiology. Many ODE models

In this paper we introduce some stability criteria for impulsive fuzzy system of differential equations with finite delay in states. Firstly, a new comparison principle for fuzzy differential system compared to crisp ordinary differential equation, based on a notion of upper quasi-monotone nondecreasing, in N dimentional state space is presented. Furthermore, in order to analyze the stability o...

We study symmetry and holomorphy of the third-order ordinary differential equation satisfied by the third Painlevé Hamiltonian.

Polynomial and rational solutions for linear ordinary differential equations can be obtained by algorithmic methods. For instance, the maple package DEtools provides efficient functions polysols and ratsols to find polynomial and rational solutions for a given linear ordinary differential equation with rational function coefficients. A natural analogue of the notion of linear ordinary different...

Using geometric methods for linearizing systems of second order cubically semi-linear ordinary differential equations and third order quintically semi-linear ordinary differential equations, we extend to the fourth order by differentiating the third order equation. This yields criteria for linearizability of a class of fourth order semi-linear ordinary differential equations, which have not bee...

Using an existing ordinary differential equation model which describes the interaction of the immune system with the human immunodeficiency virus (HIV), we introduce chemotherapy in an early treatment setting through a dynamic treatment and then solve for an optimal chemotherapy strategy. The control represents the percentage of effect the chemotherapy has on the viral production. Using an obje...

We consider the problem of estimating a nonlinear state-space model whose state process is driven by an ordinary differential equation (ODE) or a stochastic differential equation (SDE), with discrete-time data. We propose a new estimation method by minimizing the conditional least squares (CLS) with the conditional mean function computed approximately via unscented Kalman filter (UKF). We deriv...

Given a first order autonomous algebraic ordinary differential equation, we present a method to compute all formal series solutions. Furthermore, when the ground field is the field of the complex numbers, the computed formal power series solutions are indeed convergent in suitable neighborhoods. keywordAlgebraic autonomous differential equation, algebraic curve, local parametrization, formal po...

In this paper a higher-order numerical solution of a non-linear Volterra integro-differential equation is discussed. Example of this question has been solved numerically using the Runge-Kutta-Verner method for Ordinary Differential Equation (ODE) part and Newton-Cotes formulas for integral parts.