arXiv subjects

Ismail T. Huseynov

Publications and source records attributed to Ismail T. Huseynov.

8 recordsLinked to original sources

arXiv2026

Explicit representation of solutions to a linear wave equation with time delay

This paper develops an explicit spectral representation for solutions of a one-dimensional linear wave equation with a constant time delay. The model is considered on a bounded interval with non-homogeneous Dirichlet boundary data and a prescribed history function. To accommodate the loss of global smoothness in time caused by delay terms, solutions are understood in a \textit{stepwise classical sense}, allowing jump discontinuities in the second time derivative at multiples of the delay while maintaining continuity of the solution and its first time derivative. By combining separation of variables with Sturm-Liouville expansions, the delayed PDE is reduced to a family of scalar second-order delay differential equations. Using delay-dependent fundamental solutions, we derive closed-form representation formulas for the modal dynamics and reconstruct the PDE solution as a Fourier series. Convergence conditions guaranteeing uniform convergence and admissibility of termwise differentiation in space are established. A numerical example demonstrates the practical computation of truncated series solutions and their visualization.

math.AP↗

arXiv2022

Perturbation theory and linear partial differential equations with delay

Functional evolution equations are used in the modeling of numerous physical processes. In this work, our main tool is perturbation theory of strongly continuous semigroups. The advantage of this technique is that one can provide functional evolution equations with the explicit representation formulas of the solution. First, we introduce a closed form of the fundamental solution of the evolution equation with a discrete delay using the delayed Dyson-Phillips series. Then we set up the analytical representation formulas of the classical solutions of linear homogeneous/non-homogeneous evolution equations with a constant delay in a Banach space. In the special case, when a strongly continuous group $\left\lbrace \mathcal{T}(t)\right\rbrace_{t\in \mathbb{R}}$ commutes with a bounded linear operator $A_{1}$, we obtain an elegant formula for the fundamental solution using the powers of the resolvent operator of $A_{0}$. Furthermore, we consider delay evolution equations with permutable/non-permutable linear bounded operators and derive crucial results in terms of non-commutative analysis. Finally, we present an example, in the context of a one-dimensional heat equation with a discrete delay to demonstrate the applicability of our theoretical results and give some comparisons with existing results.

math.FA↗

arXiv2021

Perturbation of fractional strongly continuous cosine family operators

Perturbation theory has long been a very useful tool in the hands of mathematicians and physicists. The purpose of this paper is to prove some perturbation results for infinitesimal generators of fractional strongly continuous cosine families. That is, we impose sufficient conditions such that $\mathscr{A}$ is the infinitesimal generator of a fractional strongly continuous cosine family in a Banach space $\mathcal{X}$, and $\mathscr{B}$ is a bounded linear operator in $\mathcal{X}$, then $\mathscr{A}+\mathscr{B}$ is also the infinitesimal generator of a fractional strongly continuous cosine family in $\mathcal{X}$. Our results coincide with the classical ones when $α=2$.

math.AP↗

arXiv2021

Perturbation theory for fractional evolution equations in a Banach space

A strong inspiration for studying perturbation theory for fractional evolution equations comes from the fact that they have proven to be useful tools in modeling many physical processes. In this paper, we study fractional evolution equations of order $α\in (1,2]$ associated with the infinitesimal generator of an operator fractional cosine function generated by bounded time-dependent perturbations in a Banach space. We show that the abstract fractional Cauchy problem associated with the infinitesimal generator $A$ of a strongly continuous fractional cosine function remains uniformly well-posed under bounded time-dependent perturbation of $A$. We also provide some necessary special cases.

math.AP↗

arXiv2021

A new technique for solving Sobolev type fractional multi-order evolution equations

A strong inspiration for studying Sobolev type fractional evolution equations comes from the fact that have been verified to be useful tools in the modeling of many physical processes. We introduce a novel technique for solving Sobolev type fractional evolution equations with multi-orders in a Banach space. We propose a new Mittag-Leffler type function which is generated by linear bounded operators and investigate their properties which are productive for checking the candidate solutions for multi-term fractional differential equations. Furthermore, we propose an exact analytical representation of solutions for multi-dimensional fractional-order dynamical systems with nonpermutable and permutable matrices.

math.AP↗

arXiv2020

Fractional Leibniz integral rules for Riemann-Liouville and Caputo fractional derivatives and their applications

In recent years, the theory for Leibniz integral rule in the fractional sense has not been able to get substantial development. As an urgent problem to be solved, we study a Leibniz integral rule for Riemann-Liouville and Caputo type differentiation operators with general fractional-order of $n-1 <α\leq n$, $n \in \mathbb{N}$ . A rule of fractional differentiation under integral sign with general order is necessary and applicable tool for verification by substitution for candidate solutions of inhomogeneous multi-term fractional differential equations. We derive explicit analytical solutions of generalized Bagley-Torvik equations in terms of recently defined bivariate Mittag-Leffler type functions that based on fractional Green's function method and verified solutions by substitution in accordance by applying the fractional Leibniz integral rule. Furthermore, we study an oscillator equation as a special case of differential equations with multi-orders via the Leibniz integral rule.

math.CA↗

arXiv2020

A natural extension of Mittag-Leffler function associated with a triple infinite series

We establish a new natural extension of Mittag-Leffler function with three variables which is so called "trivariate Mittag-Leffler function". The trivariate Mittag-Leffler function can be expressed via complex integral representation by putting to use of the eminent Hankel's integral. We also investigate Laplace integral relation and convolution result for a univariate version of this function. Moreover, we present fractional derivative of trivariate Mittag-Leffler function in Caputo type and we also discuss Riemann- Liouville type fractional integral and derivative of this function. The link of trivariate Mittag-Leffler function with fractional differential equation systems involving different fractional orders is necessary on certain applications in physics. Thus, we provide an exact analytic solutions of homogeneous and inhomogeneous multi-term fractional differential equations by means of a newly defined trivariate Mittag-Leffler functions.

math.CA↗

arXiv2020

Controllability of fractional stochastic delay dynamical systems

In this paper, we consider Caputo type fractional stochastic time-delay system with permutable matrices. We derive stochastic analogue of variation of constants formula via a newly defined delayed Mittag-Leffer type matrix function. Thus, we investigate new results on existence and uniqueness of mild solutions with the help of weighted maximum norm to fractional stochastic time-delay differential equations whose coefficients satisfy standard Lipschitz conditions. The main points in the proof are to apply Ito's isometry and martingale representation theorem, and to show the notion of a coincidence between the integral equation and the mild solution. Finally, we study complete controllability results for linear and nonlinear fractional stochastic delay dynamical systems with Wiener noise.

math.DS↗