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A review of the basic Hansen's ideal frame algorithms for accurate numerical integration of perturbed elliptic motion is carried out. The fundamental approaches rely on the use of nonsingular variables and differ in the ways in which the ellipse in the orbital plane is determined. It is well known that the accuracy of the propagation of the orbit geometry is notably increased when using time-regularization techniques to transform the independent variable. However, this is at the expense of adding a differential equation to compute the time, which gathers the Lyapunov-type instabilities that are removed from the coordinates. The asynchronism resulting from errors in the numerical integration of the time may be palliated with the use of time elements, to which end a constant and a linear nonsingular time element are presented, which are new to our knowledge. [ABSTRACT FROM AUTHOR]

This paper deals with the Ulam stability of linear differential equations by using the method of variation of parameters, which provides a unified method to study the Ulam stability problem of linear differential equations of n-th order with constant and nonconstant coefficients. As an application of the main results, we also obtain the Hyers–Ulam stability of the Cauchy–Euler differential equations of second order, third order and n-th order. Our results make up to some deficiencies in the relevant literature. [ABSTRACT FROM AUTHOR]

How should our students think about external forcing in differential equations setting, and how can we help them gain intuition? To address this question, we share a variety of problems and projects that explore the dynamics of the undamped forced spring–mass system. We provide a sequence of discovery-based exercises that foster physical and mathematical intuition about polynomial forcing, as we build tools and techniques (including Green's function) to explore the amazing behavior of We encourage the insightful use of a Computer Algebra System, and provide a paired supplemental Maple worksheet. [ABSTRACT FROM AUTHOR]

The method of variation of parameters can be found in most undergraduate textbooks on differential equations. The method leads to solutions of the non-homogeneous equation of the form, a sum of function products using solutions to the homogeneous equationand. Why though, might varying parameters of solutions to the homogeneous equation be a feasible approach? An insight is needed, even for a good guess. This paper is for instructors teaching a first course in undergraduate differential questions, where the students already have covered first-order linear equations and second-order homogeneous linear equations. We provide a variation to the standard textbook approach to variation of parameters, intended to help the student find their way toward a key insight: “Aha! Let's try …” We point to several reasons why it can be important for students to learn variation of parameters. [ABSTRACT FROM PUBLISHER]

Abstract: The problem that motivates the considerations here is the construction of mathematical models of natural phenomena that depend upon past states. The paper divides naturally into two parts: in the first, we expound the inter-connection between ordinary differential equations, delay-differential equations, neutral delay-differential equations and integral equations (with emphasis on certain linear cases). As we show, this leads to a natural hierarchy of model complexity when such equations are used in mathematical and computational modelling, and to the possibility of reformulating problems either to facilitate their numerical solution or to provide mathematical insight, or both. Volterra integral equations include as special cases the others we consider. In the second part, we develop some practical and theoretical consequences of results given in the first part. In particular, we consider various approaches to the definition of an adjoint, we establish (notably, in the context of sensitivity analysis for neutral delay-differential equations) rôles for well-defined adjoints and ‘quasi-adjoints’, and we explore relationships between sensitivity analysis, the variation of parameters formulae, the fundamental solution and adjoints. [Copyright &y& Elsevier]

This paper investigates the problem of estimating synchronization errors and its application to uniform synchronization with an error bound for the general master-slave chaos synchronization scheme via feedback control, which is subjected to disturbances by unknown but bounded channel noise. Based on the Lyapunov function and nonlinear parametric variation techniques, estimation formulae for synchronization errors are derived. It is possible to synchronize two master-slave chaotic systems with a relatively small error bound, even in the case with unknown but bounded noisy disturbances. After the theoretical analysis, some representative examples and their numerical simulations are given for illustration. [ABSTRACT FROM AUTHOR]

We define the principal matrix solution Z(t, s) of the linear Volterra vector integro-differential equation x′(t) = A(t)x(t) + ∫st B(t, u)x(u)du in the same way that it is defined for x′ = A(t)x and prove that it is the unique matrix solution of ∂/∂t Z(t, s) = A(t)Z(t, s) + ∫st B(t, u)Z(u, s)du, Z(s, s)=I. Furthermore, we prove that the solution of x′(t) = A(t)x(t) + ∫τt B(t, u)x(u)du + f(t), x(τ) = x0 is unique and given by the variation of parameters formula x(t) = Z(t, τ)x0 + ∫τt Z(t, s)f(s)ds. We also define the principal matrix solution R(t, s) of the adjoint equation r′(s) = -r(s)A(s) - ∫st r(u)B(u, s)du and prove that it is identical to Grossman and Miller's resolvent, which is the unique matrix solution of ∂/∂t R(t, s) = -R(t, s)A(s) - ∫st R(t, u)B(u, s)du, R(t, t)=I. Finally, we prove that despite the difference in their definitions R(t, s) and Z(t, s) are in fact identical. [ABSTRACT FROM AUTHOR]

Abstract: This paper establishes several criteria on stability in terms of two measures for impulsive integro-differential equations with fixed moments of impulse effects by using the generalized variation of parameters technique unified with the Lyapunov second method. [Copyright &y& Elsevier]

This article investigates the relationship between an unperturbed integrodifferential system and a perturbed integrodifferential system that have different initial positions and an initial time difference (ITD). Variation of parameter techniques are employed to obtain integral formulae and to define Lyapunov-like functions. [ABSTRACT FROM AUTHOR]

Abstract: In spite of being a classical method for solving differential equations, the method of variation of parameters continues having a great interest in theoretical and practical applications, as in astrodynamics. In this paper we analyse this method providing some modifications and generalised theoretical results. Finally, we present an application to the determination of the ephemeris of an artificial satellite, showing the benefits of the method of variation of parameters for this kind of problems. [Copyright &y& Elsevier]

Abstract: Nonlinear variation of parameters formula for set differential equations in a metric space is developed investigating necessary tools such as differentiability of solutions relative to initial values. Other needed results are also proven. [Copyright &y& Elsevier]

In this article we show that the conditions imposed by Lagrange and other authors for finding a particular solution for a non-homogeneous linear differential equation of order n ≥ 2 are convenient at times, but not always the best, and that a particular solution may be obtained by imposing much more general conditions. [ABSTRACT FROM AUTHOR]

This paper studies absolute stability of Lurie systems with impulsive effects. Using the method of Lyapunov functions and the variation of parameters technique, we establish sufficient conditions for absolute stability. [Copyright &y& Elsevier]