Integro-differential System Research Articles

Existence and Stability of Neutral Stochastic Impulsive and Delayed Integro-Differential System via Resolvent Operator

In this paper, we present the existence of a mild solution for a class of a neutral stochastic integro-differential system over a Hilbert space. Such systems are influenced by both multiplicative and fractional noise, alongside non-instantaneous impulses, with a Hurst index H in the interval (12,1). Additionally, the systems under consideration feature state-dependent delays (SDDs). To address this, we develop an approach to reformulate the neutral stochastic integro-differential system, incorporating SDDs and non-instantaneous impulses, into an equivalent fixed-point (FP) problem via an appropriate integral operator. By integrating stochastic analysis with the theory of resolvent operators, we employ Banach’s FP theorem to establish both the existence and uniqueness of the solution. Furthermore, we explore the Ulam–Hyers–Rassias stability of the system. Lastly, we provide illustrative examples to demonstrate the practical applicability of our results.

Wavelet-based collocation methods for solving multi-dimensional integro-differential systems

This paper presents a comprehensive investigation of wavelet-based collocation methods for solving multi-dimensional integro-differential systems. The research addresses the challenging problem of finding numerical solutions to integro-differential equations (IDEs) that are essential in various scientific and engineering applications. We propose novel numerical approximations for infinitely dimensional problems, including fractional Cattaneo-Rayleigh waves and nonlinear diffusion problems defined on unbounded domains. The study introduces a unified computational strategy that directly applies to IDEs without discretization, particularly focusing on the integer and fractional-order Cattaneo-Rayleigh model of thermoelasticity in one-dimensional zonal regions. Through numerical experiments, we demonstrate that our wavelet collocation approach achieves high efficiency and superior accuracy compared to traditional methods. The results show particularly strong performance in handling systems with differential physical singularities, functional physical singularities, and vector-valued physical singularities. Our approach provides a valuable toolbox for addressing complex modeling challenges in biology, mechanical signal processing, and digital communications where partial integro-differential equations naturally arise.

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.

Controllability of impulsive fractional damped integrodifferential systems with distributed delays

Controllability of impulsive fractional damped integrodifferential systems with distributed delays

Two-step crystallization in a supersaturated solution with application to protein crystal growth: Theory and analytical solutions

A theory of two-step nucleation and evolution of crystals with allowance for their diffusion in the space of particle radii is developed. The theory is based on a two-step growth law of crystals appearing in dense liquid precursors, initially evolving within them with a diffusion-limited growth law, and then leaving them and evolving with another growth law. This two-step crystal growth law influences the integrodifferential system of kinetic and balance equations for the crystal-size distribution function and liquid supersaturation. The system of integrodifferential equations for the evolution of polydisperse ensemble of crystals is solved using the integral Laplace transform and saddle-point methods for the Weber–Volmer–Frenkel–Zel’dovich and Meirs kinetic mechanisms. A complete analytical solution to this non-linear system has been constructed in a parametric form. Namely, the particle-size distribution function, liquid supersaturation and time have been found as the functions of decision variable — the maximum size of crystals. The theory is in agreement with the experimental data on the growth of beta-lactoglobulin and insulin crystals.

A stochastic approach for the generation and combination of vertical and horizontal strong ground motion

Damage to structures during near-fault earthquakes often indicates that the vertical component was larger than what is usually considered in seismic design. Commonly, the vertical component of seismic ground motions is associated with the P-wave. In contrast, the horizontal seismic ground motions’ component is usually related to the arrival of the S-wave. However, findings of recent near-fault events indicate that both the horizontal and the vertical motions are a mixture of S and P-waves. Therefore, the need to generate three-component artificial accelerograms for structural design arises. This paper proposes a double aim: starting from the well-noted Snell’s law, a physical explanation of the mixture of S and P-waves is outlined. Then, a novel integrodifferential filter system for the stochastic simulation of the three components of strong motion is presented. The proposed study is applied to some theoretical applications as well as to two real Italian seismic events.

The averaging principle of Atangana–Baleanu fractional stochastic integro-differential systems with delay

The averaging principle of Atangana–Baleanu fractional stochastic integro-differential systems with delay

The Existence and Uniqueness Results of Solutions for a Fractional Hybrid Integro-differential System

This paper discuss two important results for a fractional hybrid boundary value problem of Riemann-Liouville integro-differential systems, the researches and the advance in this field and also the importance of this subject in the modeling of nonlinear real phenomena corresponding to a great variety of events gives the motivation to study this boundary value problem. The results are as follow, the first result consider the existence and uniqueness results of solutions for a fractional hybrid boundary value problem of Riemann-Liouville integro-differential system this result based on Krasnoslskii fixed point theorem for a sum of two operators, the second result is the uniqueness of solution for fractional hybrid boundary value problem of Riemann-Liouville integro-differential systems, the main result is based on Banach fixed point theorem, both results comes after transforming the system into Volterra integral system then transform again into operator system, then using fixed point theory to prove the results, this articule was ended buy an example to well illustrat the results and ideas of proof.

On the solvability of some systems of integro-differential equations with and without a drift

We prove the existence of solutions for some integro-differential systems containing equations with and without drift terms in the H 2 spaces by virtue of the fixed point technique when the elliptic equations contain second order differential operators with and without Fredholm property, on the whole real line or on a finite interval with periodic boundary conditions. Let us emphasise that the study of the system case is more complicated than of the scalar situation and requires to overcome more cumbersome technicalities.

On the fractional perturbed neutral integro-differential systems via deformable derivatives: an existence study

In this paper, we provide some appropriate conditions for the existence of solutions for a perturbed fractional neutral integro-differential system under the deformable derivative in a Banach space. Using the Banach contraction principle and Krasnoselskii’s fixed point theorem, we establish some new existence theorems. Moreover, we provide two numerical examples to demonstrate the applicability of the theoretical results

ON THE INITIAL VALUE PROBLEMS FOR NEUTRAL INTEGRO-DIFFERENTIAL SYSTEM WITHIN EXPONENTIAL KERNEL

This paper deals with the existence results for neutral integro-differential system (NIDS) through Caputo-Fabrizio (CF) operator. The solution to our addressing system, which can be considered innovative, is explained and proven in preliminary section. Certain novel existence findings are shown using fixed point approaches. Finally, two numerical examples are provided in the work to demonstrate the application of our theoretical findings.

Solving linear systems of fractional integro-differential equations by Haar and Legendre wavelets techniques

In this study, we present two highly effective approaches aimed at solving linear systems of equations, specifically focusing on the Fredholm and Volterra equations in fractional integro-differential formula. To tackle these challenging problems, we employ two distinct methodologies: the Haar wavelet technique (HWT) and the Legendre wavelet Technique (LWT). To accurately describe fractional derivatives with the equations, we adopt the Caputo sense, which provides a robust framework for handling fractional calculus. By applying HWT and LWT, we are able to transform the intricate equations of the fractional integro-differential system into algebraic equations. Fredholm and Volterra fractional integro-differential equations serve as benchmarks for evaluating the accuracy and efficiency of LWT and HWT. By comparing the approximate solutions obtained through these wavelet methods with the exact solutions, we are able to provide a comprehensive assessment of their performance. Through detailed numerical computations conducted using Mathematical 8,which demonstrate the effectiveness of both LWT and HWT in solving systems of Fredholm and Volterra equations in fractional integro-differential equations. The results obtained from these computations are presented through graphs, allowing for a clear and comparison between the approximate and exact solutions. The successful utilization of these wavelet methods provides researchers and practitioners with valuable tools for tackling complex mathematical problems in various fields of study. This study contributes to the progress of numerical techniques for solving of fractional integro-differential equations which aid in the development of more accurate and efficient computational approaches for analyzing and solving similar mathematical problems in the future.

Controllability of time-varying fractional dynamical systems with distributed delays in control

Abstract The objective of this study is to assess controllability results in the realm of Caputo fractional derivatives, specifically focusing on time-varying linear and nonlinear fractional dynamical systems with distributed control delays. In the setting of a linear system, using the Grammian matrix's positive definiteness makes it possible to ascertain both the necessary and sufficient conditions. It is possible to determine, for nonlinear systems, the sufficient conditions for the presence of a solution by making use of the Schauder fixed point theorem. In addition to that, the paper discusses the controllability of a nonlinear integrodifferential system for a particular case. A few examples and the graphs that correspond to them are shown here to make it easier to conceptualize the conclusions of the theoretical discussion.

A note on existence results for noninstantaneous impulsive integrodifferential systems

A note on existence results for noninstantaneous impulsive integrodifferential systems

Contributions to the Numerical Solutions of a Caputo Fractional Differential and Integro-Differential System

The primary goal of this research is to offer an efficient approach to solve a certain type of fractional integro-differential and differential systems. In the Caputo meaning, the fractional derivative is examined. This system is essential for many scientific disciplines, including physics, astrophysics, electrostatics, control theories, and the natural sciences. An effective approach solves the problem by reducing it to a pair of algebraically separated equations via a successful transformation. The proposed strategy uses first-order shifted Chebyshev polynomials and a projection method. Using the provided technique, the primary system is converted into a set of algebraic equations that can be solved effectively. Some theorems are proved and used to obtain the upper error bound for this method. Furthermore, various examples are provided to demonstrate the efficiency of the proposed algorithm when compared to existing approaches in the literature. Finally, the key conclusions are given.

Analysis and simulation of an integro-differential Lotka–Volterra model with variable reproduction rates and optimal control

In this work, we present an integro-differential system that generalizes the classical Lotka–Volterra model of competition. The model considers population heterogeneity with respect to reproduction rates, local diffusion in the aspect space and control interventions. We perform a rigorous mathematical analysis proving results on existence and uniqueness of solutions and on existence of optimal controls. We also perform numerical simulations illustrating different behaviors and the role of model parameters on survival and extinction of each population.

A Study of the Stability of Integro-Differential Volterra-Type Systems of Equations with Impulsive Effects and Point Delay Dynamics

This research relies on several kinds of Volterra-type integral differential systems and their associated stability concerns under the impulsive effects of the Volterra integral terms at certain time instants. The dynamics are defined as delay-free dynamics contriobution together with the contributions of a finite set of constant point delay dynamics, plus a Volterra integral term of either a finite length or an infinite one with intrinsic memory. The global asymptotic stability is characterized via Krasovskii–Lyapuvov functionals by incorporating the impulsive effects of the Volterra-type terms together with the effects of the point delay dynamics.

Weak convergence and stability of functional diffusion systems with singularly perturbed regime switching

This paper focuses on a class of functional diffusion systems with singularly perturbed regime switching, where the modulating Markov chain has a large state space and undergoes weak and strong interactions. By using the martingale method and weak convergence, this paper shows that the underlying system will weakly converge to a limit system, which is simpler than the original system. For a class of integro-differential diffusion system with singularly perturbed regime switching, as a class of special functional diffusion system, this paper demonstrates that if the limit system is moment exponentially stable, the original system with singular perturbation is also moment exponentially stable under suitable conditions. This result is interesting since the limit system is always simpler. Finally, an example is given to illustrate this result.

Mathematical definition of the fine-structure constant: A clue for fundamental couplings in astrophysics

Astrophysical tests of the stability—or not—of fundamental couplings (e.g., can the numerical value ∼1/137 of the fine-structure constant α = e2/ℏc vary with astronomical time?) are a very active area of observational research. Using a specific α-free non-relativistic and nonlinear isotropic quantum model compatible with its quantum electrodynamics (QED) counterpart yields the 99% accurate solution α = 7.364 × 10−3 vs its experimental value 7.297 × 10−3. The ∼1% error is due to the deliberate use of mean-field Hartree approximation involving lowest-order QED in the calculations. The present theory has been checked by changing the geometry of the model. Moreover, it fits the mathematical solution of the original nonlinear integro-differential Hartree system by use of a rapidly convergent series of nonlinear eigenstates [G. Reinisch, Phys. Lett. A 498, 129347 (2024)]. These results strongly suggest the mathematical transcendental nature—e.g., like for π or e—of α’s numerical value of ∼1/137 and, hence, its astrophysical as well as its cosmological stability.

Analysis of Abstract Partial Impulsive Integro-Differential System with Delay via Integrated Resolvent Operator

Analysis of Abstract Partial Impulsive Integro-Differential System with Delay via Integrated Resolvent Operator