Impulsive Conditions Research Articles

Solving fractional integro-differential equations with delay and relaxation impulsive terms by fixed point techniques

This paper presents a systematic approach to investigating the existence of solutions for fractional integro-differential equation systems incorporating delay and relaxation impulsive terms. By employing suitable definitions of fractional derivatives, we establish physically interpretable boundary conditions. To account for abrupt state changes, impulsive conditions are integrated into the model. The system is transformed into an equivalent integral equation, facilitating the application of Banach and Schaefer fixed-point theorems to prove the existence and uniqueness of solutions. The practical applicability of our findings is demonstrated through an illustrative example.

Existence results for some fractional order coupled systems with impulses and nonlocal conditions on the half line

Abstract. In this paper, we deal with initial value problems for coupled systems of nonlinear fractional differential equations, subject to coupled nonlocal initial and impulsive conditions on the half line. Global existence-uniqueness results are obtained under weak conditions allowing the reaction part of the problem to increase indefinitely with time. Our approach relies mainly to some fixed point theorem of Perov’s type in generalized gauge spaces. The obtained results improve, generalize and complement many existing results in the literature. An example illustrating our main finding is also given. Mathematics Subject Classification (2010): 26A33, 93C23, 35E15, 47H10. Keywords: Fractional differential equation, coupled systems, generalized spaces in Perov’s sens, nonlocal initial conditions, impulses.

Existenceof Heteroclinic Solutions in Nonlinear Differential Equations of the Second-Order Incorporating Generalized Impulse Effects with the Possibility of Application to Bird Population Growth

This work considers the existence of solutions of the heteroclinic type in nonlinear second-order differential equations with ϕ-Laplacians, incorporating generalized impulsive conditions on the real line. For the construction of the results, it was only imposed that ϕ be a homeomorphism, using Schauder’s fixed-point theorem, coupled with concepts of L1-Carathéodory sequences and functions along with impulsive points equiconvergence and equiconvergence at infinity. Finally, a practical part illustrates the main theorem and a possible application to bird population growth.

Mild and classical solutions to fractional Cauchy problem on time scales

In this manuscript, aims to examine the uniqueness and existence of impulsive non-local conditions, as well as classical and mild solutions to the fractional Cauchy problem in time scales. The main results are derived by the Gronwall inequality and the Banach fixed point theorem in time scales. Eventually, we offered an example to demonstrate the suitability of the conclusions on the fractional Cauchy issue on a time scale.

Matched Stochastic Resonance Enhanced Underwater Passive Sonar Detection under Non-Gaussian Impulsive Background Noise.

Remote passive sonar detection with low-frequency band spectral lines has attracted much attention, while complex low-frequency non-Gaussian impulsive noisy environments would strongly affect the detection performance. This is a challenging problem in weak signal detection, especially for the high false alarm rate caused by heavy-tailed impulsive noise. In this paper, a novel matched stochastic resonance (MSR)-based weak signal detection model is established, and two MSR-based detectors named MSR-PED and MSR-PSNR are proposed based on a theoretical analysis of the MSR output response. Comprehensive detection performance analyses in both Gasussian and non-Gaussian impulsive noise conditions are presented, which revealed the superior performance of our proposed detector under non-Gasussian impulsive noise. Numerical analysis and application verification have revealed the superior detection performance with the proposed MSR-PSNR detector compared with energy-based detection methods, which can break through the high false alarm rate problem caused by heavy-tailed impulsive noise. For a typical non-Gasussian impulsive noise assumption with α=1.5, the proposed MSR-PED and MSR-PSNR can achieve approximately 16 dB and 22 dB improvements, respectively, in the detection performance compared to the classical PED method. For stronger, non-Gaussian impulsive noise conditions corresponding to α=1, the improvement in detection performance can be more significant. Our proposed MSR-PSNR methods can overcome the challenging problem of a high false alarm rate caused by heavy-tailed impulsive noise. This work can lay a solid foundation for breaking through the challenges of underwater passive sonar detection under non-Gaussian impulsive background noise, and can provide important guidance for future research work.

Mixed problem for a linear differential equation of parabolic type with nonlinear impulsive conditions

Mixed problem for a linear differential equation of parabolic type with nonlinear impulsive conditions

Improved prediction of wave overtopping rates at vertical seawalls with recurve retrofitting

This study investigates the reduction in overtopping discharge along a vertical seawall through the implementation of a recurve retrofitting. A comprehensive set of physical modelling experiments were undertaken in a laboratory-scale wave flume at the University of Warwick, to investigate the wave overtopping processes under both swell and storm wave conditions. The tests measured overtopping discharges for impulsive and non-impulsive wave conditions. The effects of geometrical design of recurve retrofitting on overtopping reduction are examined by four configurations with varying overhang length and recurve hight. The study revealed that the reduction in overtopping is primarily determined by the length of the overhang in the recurve wall, while the influence of the recurve height is limited. A longer overhang length results in a more substantial decrease in overtopping discharges on the seawall crest. The results also highlight the role of incident wave steepness and the crest freeboard on the overtopping mitigation performance of the recurve walls. A new enhanced methodology is proposed to predict the wave overtopping from vertical seawalls with recurve retrofitting., considering the effects of freeboard and wave steepness. The findings of this study provide new important insight in the role of retrofitting as a robust intervention to improve the wave overtopping mitigation performance of seawalls. The predictive empirical formulae proposed by this study facilitate readily and accurate estimation of overtopping rates as a function of retrofitting geometrical design, allowing for wider application of retrofitting solutions.

MIXED PROBLEM FOR A NONLINEAR IMPULSIVE DIFFERENTIAL EQUATION OF PARABOLIC TYPE

In this paper, we consider a nonlinear impulsive parabolic type partial differential equation with nonlinear impulsive conditions. Dirichlet type boundary value conditions with respect to spatial variable is used, and eigenvalues and eigenfunctions of the spectral problem are founded. The Fourier method of the separation of variables is applied. A countable system of nonlinear functional equations is obtained with respect to the Fourier coefficients of the unknown function. A theorem on a unique solvability of the countable system of nonlinear functional equations is proved by the method of successive approximations. A criteria of uniqueness and existence of a solution for the nonlinear impulsive mixed problem is obtained. A solution of the mixed problem is derived in the form of the Fourier series. The absolute and uniform convergence of the Fourier series is proved.

Numerical study on dynamic cracking characteristic and mechanism of quasi-brittle materials under impulsive loading

This study focuses on investigating the dynamic cracking behaviors of quasi-brittle materials under impulsive loading using the extended non-ordinary state-based peridynamic (NOSB-PD) model. This extended model incorporates two stress-based failure criteria: Mohr-Coulomb and maximum tensile stress criteria, to represent shear and tensile fractures of quasi-brittle materials, respectively. By analyzing the evolution characteristics of the maximum principal stress obtained from the NOSB-PD model, the fracture mechanisms of quasi-brittle materials subjected to impulsive loading are explored. The research reveals that tensile failure predominantly governs the initiation and propagation processes of quasi-brittle materials under impulsive loading. The stress concentration effect, transfer effect, and dissipation effect observed in the maximum principal stress fields provide insights into the cracking mechanism during the failure processes of such materials. To validate the findings, the crack initiation modes and crack propagation paths obtained from the NOSB-PD model are compared with those obtained from the existing experimental and numerical approaches. The results demonstrate that the extended NOSB-PD model is effective in addressing fracture problems associated with quasi-brittle materials under impulsive loading conditions.

An investigation of Fractional Mixed Functional Integro-Differential Equations with Impulsive Conditions

An investigation of Fractional Mixed Functional Integro-Differential Equations with Impulsive Conditions

Protection exhibited by hearing protection devices using test fixtures under extended shock tube exposure ranges

Hearing protection devices (HPDs) are used in a variety of environments to mitigate the risk of hearing injury, particularly in environments with pervasive impulsive noise. Evaluation of HPD performance for these conditions can be performed with acoustic test fixtures to ensure human subjects are not exposed to injurious levels of noise. Impulse Peak Insertion Loss (IPIL) and Impulse Spectral Insertion Loss (ISIL), as defined in ASA/ANSI S12.42, are used to quantify and compare performance of HPDs in exposure environments created by different impulsive sources. To simulate a range of exposures relevant across industries, multiple shock tubes were used to generate waveforms with A-durations ranging from 0.1 to 6.0 ms and peak pressures ranging from 130 to 183 dB. Protection provided by each device varied depending on the source conditions with passive and active devices producing fundamentally different responses depending on source characteristics. Ultimately, evaluation of devices under a variety of impulsive noise conditions allows for a more representative evaluation of HPD performance, bridging the gap between standard testing protocols and real-world acoustic environments, thus enabling better informed HPD selection.

A Study on Approximate Controllability of Ψ-Caputo Fractional Differential Equations with Impulsive Effects

In this article, we studied the approximate controllability of Ψ-Caputo fractional differential systems. We prove the sufficient conditions for an abstract Cauchy problem invloving infinite delay, impulsive and nonlocal conditions. The result is shown by means of the infinitesimal operator, semigroup theory, fractional calculus, and Schauder’s fixed point theorem. First, we prove the existence of the mild solution and demonstrate that the Ψ-Caputo fractional system is approximately controllable. Finally, an example is given to analyse the obtained results.

A study on the existence results of boundary value problems of fractional relaxation integro-differential equations with impulsive and delay conditions in Banach spaces

<abstract><p>The aim of this paper was to provide systematic approaches to study the existence of results for the system fractional relaxation integro-differential equations. Applied problems require definitions of fractional derivatives, allowing the utilization of physically interpretable boundary conditions. Impulsive conditions serve as basic conditions to study the dynamic processes that are subject to sudden changes in their state. In the process, we converted the given fractional differential equations into an equivalent integral equation. We constructed appropriate mappings and employed the Schaefer's fixed-point theorem and the Banach fixed-point theorem to show the existence of a unique solution. We presented an example to show the applicability of our results.</p></abstract>

A NEW ALGORITHM FOR SOLVING TWO-LINKED BOUNDARY VALUE PROBLEMS WITH IMPULSIVE CONDITIONS

A NEW ALGORITHM FOR SOLVING TWO-LINKED BOUNDARY VALUE PROBLEMS WITH IMPULSIVE CONDITIONS

Lipschitz Stability for Impulsive Riemann–Liouville Fractional Differential Equations

Initial and impulsive conditions for initial value problems of systems of nonlinear impulsive Riemann–Liouville fractional differential equations are introduced. The case when the lower limit of the fractional derivative is changed at each time point of the impulses is studied. In the case studied, the solution has a singularity at the initial time and at any point of the impulses. This leads to the need to appropriately generalize the classical concept of Lipschitz stability. Two derivative types of Lyapunov functions are utilized in order to deduce sufficient conditions for the new stability concept. Three examples are provided for illustration purpose of the theoretical results.

Study on the controllability of Hilfer fractional differential system with and without impulsive conditions via infinite delay

In this manuscript, we investigate the controllability of two different kinds of Hilfer fractional differential equations with an almost sectorial operator and infinite delay. First, we demonstrate the exact controllability of the Hilfer fractional system using the measure of noncompactness. Next, we develop the results for the controllability of the system under impulsive conditions. Finally, to show how the key findings may be utilised, applications are presented.

A Novel Approach to Partial Discharge Detection Under Repetitive Unipolar Impulsive Voltage

Partial discharge (PD) detection is an important method to verify the reliability of the insulation structure of high voltage power electronics devices. Although there are many researches on PD detection inside power modules, they mainly focus on traditional electrical detection methods and 50 Hz working conditions, which do not meet the repetitive unipolar impulsive voltage conditions of power modules, and traditional methods cannot solve the electromagnetic interference (EMI) problem under working conditions. This paper proposes a novel approach to PD detection in power modules. This approach designs an optic fiber sensing (OFS) system based on optic theory, and conducts the detailed mathematical theoretical deduction. In addition, A partial discharge inception voltage (PDIV) signal processing method based on PD acoustic signal is proposed. Finally, the proposed approach is applied to a power module. The PD signal and PDIV is successfully captured without EMI interference and new insulation problems.

Impulsive regular q-Dirac systems

This article concerns a regular $q$-Dirac system under impulsive conditions. We study the existence of solutions, symmetry of the corresponding operator, eigenvalues and eigenfunctions of the system. Also we obtain Green's function and its basic properties. For more informatin see https://ejde.math.txstate.edu/Volumes/2023/74/abstr.html

Fractional Dirac system with impulsive conditions

In this article, fractional Dirac systems are considered under impulsive conditions and their basic properties are investigated. In this context, the symmetry, eigenvalues, and eigenfunctions of the system were examined and then the existence and uniqueness of the solutions of the system were proved. Finally, the characteristic function of the system is shown.

VIOLENT AND IMPULSIVE WAVE OVERTOPPING AT VERTICAL WALLS WITH LARGE FREEBOARDS

Vertical walls are designed and built to protect coastal areas or harbours. Typically, these structures are installed in relatively deep water where incoming waves are reflected without breaking. In such cases the available prediction methods for wave overtopping discharge are well established. When the relative water depth h/Hm0 becomes smaller, waves start shoaling on the foreshore and breaking can occur leading to impulsive conditions. The present overtopping prediction methods (EurOtop, 2018) under impulsive conditions have been proposed by Van der Meer and Bruce (2014) including data from the VOWS project (Bruce et al.; 2001). Wave overtopping is strongly related to the relative freeboard Rc/Hm0, and for impulsive overtopping EurOtop suggests that for Rc/Hm0 greater than 3 there still might be significant overtopping but there is no data to validate such an extrapolation. Also the well-known overtopping graphs of Goda (2000) show that overtopping could be present for very large freeboards and impulsive waves. The objective of the present research is to investigate the mean discharge and volumes per wave from impulsive and violent overtopping at vertical walls with very large freeboards (up to Rc/Hm0 = 10).