Linear Impulsive Differential Equations Research Articles

On Oscillations of Nonautonomous Linear Impulsive Differential Equations with General Piecewise Constant Deviating Arguments

On Oscillations of Nonautonomous Linear Impulsive Differential Equations with General Piecewise Constant Deviating Arguments

Euler Method for a Class of Linear Impulsive Neutral Differential Equations

This paper presents a new numerical scheme for a class of linear impulsive neutral differential equations with constant coefficients based on the Euler method. We rigorously establish the first-order convergence of the proposed numerical approach. Additionally, the asymptotical stability of the exact solutions and numerical solutions of impulsive neutral differential equations are studied. To substantiate our findings, two illustrative examples are provided, demonstrating the theoretical conclusions of this paper.

Neural Network Methods Based on Efficient Optimization Algorithms for Solving Impulsive Differential Equations

In view of the fact that impulsive differential equations have the discreteness due to the impulse phenomenon, this paper proposes a single hidden layer neural network method based extreme learning machine and a physicsinformed neural network method combined with learning rate attenuation strategy to solve linear impulsive differential equations and nonlinear impulsive differential equations respectively. For the linear impulsive differential equations, firstly, the interval is segmented according to the impulse points, and a single hidden layer neural network model is constructed, the weight parameters of hidden layer are randomly set, the optimal output parameters and solution of the first segment are obtained by the extreme learning machine algorithm, then we calculate the initial value of the second segment according to the jumping equation, and the remaining segments are solved in turn in the same way. Although the single hidden layer neural network method proposed can solve linear equations with high accuracy, it is not suitable for solving nonlinear equations. Therefore, we propose the physics-informed neural network combined with learning rate attenuation strategy to solve the nonlinear impulsive differential equations, then the Adam algorithm and L-BFGS algorithm are combined to find the optimal approximate solution of each segment. Numerical examples show that the single hidden layer neural network method with Legendre polynomials as the activation function and the physics-informed neural network method combined with learning rate attenuation strategy can solve linear and nonlinear impulsive differential equations with higher accuracy.

A variation of parameters formula for nonautonomous linear impulsive differential equations with piecewise constant arguments of generalized type

A variation of parameters formula for nonautonomous linear impulsive differential equations with piecewise constant arguments of generalized type

Finite-Time Stability of Impulsive Fractional Differential Equations with Pure Delays

This paper introduces a novel concept of the impulsive delayed Mittag–Leffler-type vector function, an extension of the Mittag–Leffler matrix function. It is essential to seek explicit formulas for the solutions to linear impulsive fractional differential delay equations. Based on explicit formulas of the solutions, the finite-time stability results of impulsive fractional differential delay equations are presented. Finally, we present four examples to illustrate the validity of our theoretical results.

Numerical oscillation and non-oscillation analysis of the mixed type impulsive differential equation with piecewise constant arguments

The purpose of this paper is to study oscillation and non-oscillation of Runge–Kutta methods for linear mixed type impulsive differential equations with piecewise constant arguments. The conditions for oscillation and non-oscillation of numerical solutions are obtained. Also conditions under which Runge–Kutta methods can preserve the oscillation and non-oscillation of linear mixed type impulsive differential equations with piecewise constant arguments are obtained. Moreover, the interpolation function of numerical solutions is introduced and the properties of the interpolation function are discussed. It turns out that the zeros of the interpolation function converge to ones of the analytic solution with the same order of accuracy as that of the corresponding Runge–Kutta method. To confirm the theoretical results, the numerical examples are given.

Convergence of the Euler Method for Impulsive Neutral Delay Differential Equations

In this paper, we are concerned with a fixed stepsize Euler method for a class of linear impulsive neutral delay differential equations. By taking the partition nodes for the Euler scheme and employing the linear interpolation, we strictly prove the method is convergent of order one. Two examples illustrating the efficiency results are also presented.

Controllability and observability for linear quaternion-valued impulsive differential equations

In this paper, the controllability and observability for linear quaternion-valued impulsive differential equations (QIDEs) are investigated. We mainly establish some sufficient and necessary conditions for state controllability and state observability of linear QIDEs. By virtue of the isomorphism between quaternion vector space and complex variables space as well as the adjoint matrix of quaternion matrix, all the theoretical results in the sense of complex-valued and quaternion-valued are equivalent to each other. Finally, the validity of theoretical results acquired is indicated by two instances shown.

On stability for generalized linear differential equations and applications to impulsive systems

In this paper, we are interested in investigating notions of stability for generalized linear differential equations (GLDEs). Initially, we propose and revisit several definitions of stability and provide a complete characterization of them in terms of upper bounds and asymptotic behaviour of the transition matrix. In addition, we illustrate our stability results for GLDEs to linear periodic systems and linear impulsive differential equations. Finally, we prove that the well-known definitions of uniform asymptotic stability and variational asymptotic stability are equivalent to the global uniform exponential stability introduced in this article.

Existence and Ulam Type Stability for Impulsive Fractional Differential Systems with Pure Delay

Through literature retrieval and classification, it can be found that for the fractional delay impulse differential system, the existence and uniqueness of the solution and UHR stability of the fractional delay impulse differential system are rarely studied by using the polynomial function of the fractional delay impulse matrix. In this paper, we firstly introduce a new concept of impulsive delayed Mittag–Leffler type solution vector function, which helps us to construct a representation of an exact solution for the linear impulsive fractional differential delay equations (IFDDEs). Secondly, by using Banach’s and Schauder’s fixed point theorems, we derive some sufficient conditions to guarantee the existence and uniqueness of solutions of nonlinear IFDDEs. Finally, we obtain the Ulam–Hyers stability (UHs) and Ulam–Hyers–Rassias stability (UHRs) for a class of nonlinear IFDDEs.

Existence and uniqueness of periodic solution to second‐order impulsive differential equations

In this paper, existence and uniqueness of periodic solutions of second‐order semilinear impulsive differential equations are obtained. Some sufficient conditions that guarantee the existence of periodic solutions to second‐order linear nonhomogeneous impulsive differential equations are obtained. In addition, by using the appropriate fixed theorem and the theory of coincidence degree, some conditions ensuring the existence and uniqueness of periodic solution to second‐order semilinear impulsive differential systems are derived. Finally, some examples are provided to illustrate the availability of our results.

Controllability Results for First Order Impulsive Fuzzy Differential Systems

In this paper, we investigate the controllability of first-order impulsive fuzzy differential equations. Using the direct construction method, the controllability of first-order linear impulsive fuzzy differential equations is considered with a<0, the (c1) solution, and a<0, the (c2) solution, respectively.In addition, by employing the Banach fixed-point theorem, the controllability of first-order nonlinear impulsive fuzzy differential equations is studied. Finally, examples are presented to illustrate our theoretical results.

Smooth stable manifolds for the non-instantaneous impulsive equations with applications to Duffing oscillators.

In this paper, we present a theory of smooth stable manifold for the non-instantaneous impulsive differential equations on the Banach space or Hilbert space. Assume that the non-instantaneous linear impulsive evolution differential equation admits a uniform exponential dichotomy, we give the conditions of the existence of the global and local stable manifolds. Furthermore, -smoothness of the stable manifold is obtained, and the periodicity of the stable manifold is given. Finally, an application to nonlinear Duffing oscillators with non-instantaneous impulsive effects is given, to demonstrate the existence of stable manifold.

Results on the Behavior of the Solutions for Linear Impulsive Neutral Delay Differential Equations with Constant Coefficients

We have given some results regarding the behavior of solutions for first order linear impulsive neutral delay differential equations with constant coefficients. These results were obtained using two different real roots of the corresponding characteristic equation. Finally, two examples are given for solutions of impulsive neutral delay differential equations.

Ulam type stability for first-order linear and nonlinear impulsive fuzzy differential equations

This paper gives a number of Ulam type stability results of first-order linear and first-order nonlinear impulsive fuzzy differential equations in different cases by applying an appropriate inequality and the direct analysis method. First, the Ulam type stability results of linear impulsive fuzzy differentiable equations under a<0, (c1)-differentiable are given. Second, the Ulam type stability results of first-order nonlinear impulsive fuzzy differentiable equations under a<0, (c1)-differentiable and a<0, (c2)-differentiable are presented. Finally, examples are provided to illustrate our results. Our results transfer the corresponding results for impulsive ordinary differential equations to impulsive fuzzy ordinary differential equations.

Leighton and Wong type oscillation theorems for impulsive differential equations

We obtain the well-known Leighton and Wong oscillation theorems for a general class of second-order linear impulsive differential equations by making use of the recently established results on the existence of nonprincipal solutions. The results indicate that the oscillation character of solutions may be altered by the impulsive perturbations, which is not the case in most published works. Another difference is that the equations are quite general in the sense that the impulses are allowed to appear on both solutions and their derivatives. Examples are also given to illustrate the importance of the results.

Prescribed asymptotic behavior of second-order impulsive differential equations via principal and nonprincipal solutions

Finding solutions with prescribed asymptotic behavior is a classical problem for differential equations, which is also known as the asymptotic integration problem for differential equations. Very recent results have revealed that the problem is closely related to principal and nonprincipal solutions of a related homogeneous linear differential equation. Such solutions for second-order linear differential equations without impulse effects, first appeared in [W. Leighton, M. Morse, Trans. Amer. Math. Soc. 40 (1936), 252–286]. In the present work we first establish the concept of principal and nonprincipal solutions for second-order linear impulsive differential equations, and then use them to prove the existence of solutions for a class of second-order nonlinear impulsive differential equations, with prescribed asymptotic behavior at infinity in terms of a linear combination of these principal and nonprincipal solutions. Examples and numerical simulations are provided to illustrate the obtained results.

Oscillations in second order linear neutral impulsive differential equations with constant delays

Sufficient conditions are established for the oscillation of all solutions of a class of second order neutral differential equations with constant delay arguments and constant impulsive jumps for the cases where the coefficient p is a constant and when p is a variable. Examples are provided for clarity.

Quaternion-Valued Linear Impulsive Differential Equations

In this paper, we transfer the approach of finding the explicit solutions to linear impulsive differential equations (LIDEs) to quaternion-valued case. We develop the similar idea to find the solutions to LIDEs and give the details to derive the representations of the solutions to quaternion-valued LIDEs in the sense of complex numbers and quaternion. In addition, we present that two formulas are equivalent to each other. Finally, some examples are given to illustrate the theory approach.

Oscillations in second order linear neutral impulsive differential equations with constant delays

Oscillations in second order linear neutral impulsive differential equations with constant delays