Integro-differential System Research Articles

On the admissibility and input–output representation for a class of Volterra integro-differential systems

On the admissibility and input–output representation for a class of Volterra integro-differential systems

Trajectory controllability of neutral stochastic integrodifferential equations with mixed fractional Brownian motion

This work focuses on the existence and trajectory (T-)controllability of mixed fractional Brownian motion (fBm) with the Hurst index and neutral stochastic integrodifferential equations (NSIDEs) with deviating argument and fBm. Stochastic integrodifferential equations (SIDEs) are solved in Hilbert space using stochastic analysis, the resolvent operator, and Krasnoselskii's fixed point theorem (KFPT). Furthermore, providing adequate assumptions, the T-controllability of the considered system is organised by using extended Gronwall's inequality. We demonstrate the theoretical insights and numerical simulations are included which is unique and makes this work more interesting. The obtained results generalise existing results from [Chalishajar, D. N., George, R. K., & Nandakumaran, A. K. (2010). Trajectory controllability of nonlinear integro-differential system. Journal of Franklin Institute, 347(7), 1065–1075.; Durga, N., Muthukumar, P., & Malik, M. (2022). Trajectory controllability of Hilfer fractional neutral stochastic differential equation with deviated argument and mixed fractional Brownian motion. Optimisation, 1–27.; Muslim, M., & George, R. K. (2019). Trajectory controllability of the nonlinear systems governed by fractional differential equations. Differential Equations and Dynamical Systems, 27, 529–537.; Dhayal, R., Malik, M., & Abbas, S. (2021). Approximate and trajectory controllability of fractional stochastic differential equation with non-instantaneous impulses and Poisson jumps. Asian Journal of Control, 23(6), 2669–2680.].

Existence and exact controllability results of nonlocal integro-differential neutral stochastic system with finite delay

Existence and exact controllability results of nonlocal integro-differential neutral stochastic system with finite delay

Bounded on the semi-axis multiperiodic solution of a linear finite-hereditarity integro-differential equation of parabolic type

The question of the existence of a solution of linear integro-differential systems of parabolic type limited on the semiaxis in a spatial variable and multiperiodic in time variables was considered. Sufficient conditions of multiperiodic oscillations in time variables in a linear homogeneous equation with a boundary condition and in a linear inhomogeneous equation were established. A linear homogeneous and inhomogeneous finitehereditarity integro-differential equation of convective-diffusion type were investigated.

New results on approximate controllability of fractional delay integrodifferential systems of order 1 < r < 2 with Sobolev-type

In this manuscript, we look at fractional evolution integrodifferential systems with infinite delay of order r∈(1,2) with control problems. To begin, we establish the mild solution for the given system. Further, we look at the approximate controllability outcomes utilizing multivalued functions, Sobolev-type, nonlocal circumstances, fractional analysis, fixed point technique, and cosine families. Finally, an application for generating the primary outcome theory has been created.

Optimal control of conformable fractional neutral stochastic integrodifferential systems with infinite delay

The main concern of the manuscript deals with the optimal control problem of conformable fractional neutral stochastic integrodifferential systems with infinite delay. This study is motivated by SAR and RAR systems advantage of providing control over many factors such as power, frequency, phase, polarization, incidence angle, spatial resolution and swath width, all of which are important when designing and operating a radar system. Initially, we investigate the existence of mild solutions for the conformable fractional stochastic integrodifferential equations with infinite delay using stochastic analysis techniques and Banach fixed point theorem. In the later part we establish the existence of mild solutions of the conformable fractional neutral stochastic integrodifferential system with infinite time delay. Furthermore, the existence of optimal control of the corresponding Lagrange optimal control problem is investigated. An example is provided to illustrate the applications of the obtained results. We explain the limitation we have with the existence software, and developed a numerical scheme to justify the theory.

A non-trivial solution for a p-Schrödinger–Kirchhoff-type integro-differential system by non-smooth techniques

A non-trivial solution for a p-Schrödinger–Kirchhoff-type integro-differential system by non-smooth techniques

Analysis of Fractional Integro-differential System with Impulses

Analysis of Fractional Integro-differential System with Impulses

Non-Instantaneous Impulsive Fractional Neutral Functional Stochastic Integro-Differential System with Measure of Non-compactness

Non-Instantaneous Impulsive Fractional Neutral Functional Stochastic Integro-Differential System with Measure of Non-compactness

Existence and Trajectory Controllability of Conformable Fractional Neutral Stochastic Integrodifferential Systems with Infinite Delay

Existence and Trajectory Controllability of Conformable Fractional Neutral Stochastic Integrodifferential Systems with Infinite Delay

A study on the approximate controllability results of fractional stochastic integro-differential inclusion systems via sectorial operators

The study deals with the findings of the outcome of the approximate controllability results of inclusion type fractional stochastic system in Banach space with the order of the fractional system varrho in (1,2). At first, we implement Bohnenblust-Karlin's fixed point technique to deduce the required conditions on which the fractional system with inital conditions is approximately controllable, and there by, we postulate the sufficient conditions for extending the obtained results to the system with nonlocal conditions.

Influence of rapidly oscillating inhomogeneities in the formation of additional boundary layers for singularly perturbed integro-differential systems

In this paper, the Lomov's regularization method is generalized to a singularly perturbed integro-differential equation with a fractional derivative and with a rapidly oscillating inhomogeneity. The main goal of the work is to reveal the influence of a rapidly changing kernel on the structure of the asymptotics of the problem solution and to study additional boundary functions that are generated by rapidly oscillating inhomogeneities.

On the Existence and Multiplicity of Classical and Weak Solutions of a Hamiltonian Integro-Differential System and Their Equivalence Relation

This paper is devoted to the study of existence and multiplicity of weak solutions to a Hamiltonian integro-differential system. The main tool used is the theory of min–max based on Mountain-Pass theorem. Hamiltonian integro-differential considered system is of Fredholm type and the imposed Dirichlet boundary conditions are occurred at the integral bounds. Furthermore, we demonstrate some cases in which the weak solutions are equivalent with classical solutions

Approximate Controllability of Nonlinear Fractional Stochastic Systems Involving Impulsive Effects and State Dependent Delay

This paper studies the analysis of approximate controllability for the fractional order neutral stochastic impulsive integro-differential systems involving nonlocal condition and State Dependent Delay (SDD). Sufficient conditions are designed to illustrate the evaluation of approximate controllability. It is exhibited that the proposed protocol can explicitly drive the results by Krasnoselskii's fixed point technique and semigroup theory. As a final point, the derived scheme is validated through an example.

Impulsive integro-differential systems involving conformable fractional derivative in Banach space

Impulsive integro-differential systems involving conformable fractional derivative in Banach space

Unconditional superconvergence analysis of an energy-stable C–N fully discrete scheme for the nonlinear magnetic diffusion model with memory

The unconditional superconvergence behavior of an energy-stable Crank–Nicolson (C–N) fully discrete scheme with nonconforming EQ1rot finite element method (FEM) is investigated for the nonlinear integro-differential system associated with the diffusion of the magnetic field. The stability of the numerical solution in the broken H1-norm is proved by the Gagliardo–Nirenberg (G–N) inequality, and the unique solvability is demonstrated by the Brouwer fixed point theorem. In which the boundness of the L∞-norm and the inverse inequality are eliminated. The combination of the Green formula and two special properties of EQ1rot element leads to the unconditional superconvergence estimate without any restriction between the time step and space partition size. Finally, the validity of the theoretical findings is supported by some numerical results.

An alternating direction implicit compact finite difference scheme for the multi-term time-fractional mixed diffusion and diffusion–wave equation

An alternating direction implicit(ADI) compact finite difference scheme for the two-dimensional multi-term time-fractional mixed diffusion and diffusion–wave equation is given in this paper. By using the Riemann–Liouville fractional integral operator on both sides of the original equation, we obtain a time-fractional integro–differential equation with the order of the highest derivative being one. Using the weighted and shifted Grünwald formulas of Riemann–Liouville fractional derivative and fractional integral, and the Crank–Nicolson approximation, a temporal second-order approximation is obtained for the equivalent integrodifferential system. In the spatial direction, a fourth-order compact approximation is employed to give a fully discrete scheme. Using the splitting techniques for higher dimensional problems, we derive the fully discrete ADI Crank–Nicolson scheme. With the positive definiteness of the related time discrete coefficients and spatial operators, the stability and convergence (second-order in time and fourth-order in space) in the discrete L2-norm are proved by using energy method. Two numerical examples are given to demonstrate the efficiency and accuracy of the proposed method.

An exploration of controllability on Hilfer fractional system via integral contractor

The primary focus of this article is to provide sufficient conditions for the controllability of Hilfer fractional neutral integrodifferential systems. By using the theories on fractional calculus, semigroup theory, and fixed point theorem method, the results are proved through integral contractors for the nonlinear neutral evolution equation in a non‐dense domain. In the end, the derived outcomes are validated through suitable illustrations.

Singularly Perturbed Integro-Differential Systems with Kernels Depending on Solutions of Differential Equations

We consider integro-differential equations (IDEs) with a rapidly oscillating inhomogeneity and with a Volterra-type integral operator whose kernels can contain both a classical rapidly decreasing exponential (the simplest case) and fundamental solutions of differential systems (the general case). Difficulty in constructing a regularized (according to S.A. Lomov) asymptotics in the general case is due to the complex asymptotic structure of the fundamental solution matrix (Cauchy matrix) of the homogeneous differential system. In the present paper, we first construct a regularized asymptotics of the Cauchy matrix, which is then used to construct a regularized asymptotics of the solution of the IDE.

Approximate controllability results for the Sobolev type fractional delay impulsive integrodifferential inclusions of order $${r} \in (1,2)$$ via sectorial operator

In this study, we investigate nonlocal problems for a class of the Sobolev type fractional delay impulsive integrodifferential inclusions of order $${r} \in (1,2)$$ via sectorial operator in Hilbert space. The results are obtained by engaging of mixed Volterra-Fredholm integrodifferential systems, sectorial operator of type $$(P,\kappa ,{r},\gamma )$$ and Martelli’s fixed point theorem under the assumption that the corresponding linear system is approximately controllable. Finally, an example is given to illustrate the effectiveness of the results obtained.