Fractional Power Of Operators Research Articles

A Study on Stochastic Differential Equation Using Fractional Power of Operator in the Semigroup Theory

A Study on Stochastic Differential Equation Using Fractional Power of Operator in the Semigroup Theory

Complete controllability of nonlinear fractional neutral functional differential equations

This paper is concerned with the complete controllability of a nonlinear fractional neutral functional differential equation. Some sufficient conditions are established for the complete controllability of the nonlinear fractional system. The conditions are established based on the fractional power of operators and the fixed-point theorem under the assumption that the associated linear system is completely controllable. Finally, an example is presented to illustrate our main result.

Some methods for solving equations with an operator function and applications for problems with a fractional power of an operator

Several applied problems are characterized by the need to numerically solve equations with an operator function (matrix function). In particular, in the last decade, mathematical models with a fractional power of an elliptic operator and numerical methods for their study have been actively discussed. Computational algorithms for such non-standard problems are based on approximations by the operator function. In this case, an approximate solution is determined by solving auxiliary standard problems. This paper discusses the main directions for constructing acceptable approximations of operator functions when solving equations.The most widespread are the approaches using various options for rational approximation. Also, we note the methods that relate to approximation by exponential sums. We propose to use a new approach, which is based on the application of approximation by exponential products. The solution of an equation with an operator function is based on the transition to standard stationary or evolutionary problems. Estimates of the accuracy of the approximate solution of the operator equation at known absolute or relative errors of approximation functions are obtained. The influence of the error in the solution of auxiliary problems is investigated separately.General approaches are illustrated by a problem with a fractional power of the operator. The first class of methods is based on the integral representation of the operator function under rational approximation, approximation by exponential sums, and approximation by exponential products. The second class of methods is associated with solving an auxiliary Cauchy problem for some evolutionary equation. In this case, we can distinguish a method based on approximation by exponential products.

Existence and Stability Results of Stochastic Differential Equations with Non-instantaneous Impulse and Poisson jumps

Abstract This paper focuses on a new class of non-instantaneous impulsive stochastic differential equations generated by mixed fractional Brownian motion with poisson jump in real separable Hilbert space. A set of sufficient conditions are generated based on the stochastic analysis technique, analytic semigroup theory of linear operators, fractional power of operators, and fixed point theory to obtain existence and uniqueness results of mild solutions for the considered system. Furthermore, the asymptotic behaviour of the system is investigated. Finally, an example is proposed to validate the obtained results.

Factorized Schemes for First and Second Order Evolution Equations with Fractional Powers of Operators

Abstract Many non-local processes are modeled using mathematical models that include fractional powers of elliptic operators. The approximate solution of stationary problems with fractional powers of operators is most often based on rational approximations introduced in various versions for a fractional power of the self-adjoint positive operator. The purpose of this work is to use such approximations for the approximate solution of nonstationary problems. We consider Cauchy problems for the first and second order differential-operator equations in finite-dimensional Hilbert spaces. Estimates for the proximity of an approximate solution to an exact one are obtained when specifying the absolute and relative errors of the approximation of the fractional power of the operator. We construct splitting schemes based on the additive representation with a rational approximation of the operator’s fractional power. The stability and accuracy of factorized two-level additive operator-difference schemes for the first order evolution equation and three-level schemes for a second order equation are established.

An approximate representation of a solution to fractional elliptical BVP via solution of parabolic IVP

A boundary value problem for fractional power of an elliptic operator is addressed. To solve it, integral representation through solution of a standard problem for parabolic equation is used. Quadrature generalized Gauss–Laguerre formulas are constructed. The effect of key parameters on the accuracy of approximate solution of both the number of nodes in the quadrature formula and the fractional power of operator is investigated. Computational experiments for a model two-dimensional problem with fractional power of elliptic operator are carried out.

Approximate controllability of noninstantaneous impulsive Hilfer fractional integrodifferential equations with fractional Brownian motion

We introduce the investigation of approximate controllability for a new class of nonlocal and noninstantaneous impulsive Hilfer fractional neutral stochastic integrodifferential equations with fractional Brownian motion. An appropriate set of sufficient conditions is derived for the considered system to be approximately controllable. For the main results, we use fractional calculus, stochastic analysis, fractional power of operators and Sadovskii’s fixed point theorem. At the end, an example is also given to show the applicability of our obtained theory.

Existence and exponential stability in the pth moment for impulsive neutral stochastic integro-differential equations driven by mixed fractional Brownian motion

This paper consider the existence, uniqueness and exponential stability in the pth moment of mild solution for impulsive neutral stochastic integro-differential equations driven simultaneously by fractional Brownian motion and by standard Brownian motion. Based on semigroup theory, the sufficient conditions to ensure the existence and uniqueness of mild solutions are obtained in terms of fractional power of operators and Banach fixed point theorem. Moreover, the pth moment exponential stability conditions of the equation are obtained by means of an impulsive integral inequality. Finally, an example is presented to illustrate the effectiveness of the obtained results.

Exponential stability behavior of neutral stochastic integrodifferential equations with fractional Brownian motion and impulsive effects

In this paper, by employing the fractional power of operators, semigroup theory, and fixed point strategy we obtain some new criteria ensuring the existence and exponential stability of a class of impulsive neutral stochastic integrodifferential equations driven by a fractional Brownian motion. We establish some new sufficient conditions that ensure the exponential stability of mild solution in the mean square moment by utilizing an impulsive integral inequality. Also, we provide an example to show the efficiency of the obtained theoretical result.

EXISTENCE OF SOLUTIONS FOR IMPULSIVE NEUTRAL FUNCTIONAL INTEGRO DIFFERENTIAL EQUATIONS WITH NONLOCAL CONDITIONS

In this paper, by using fractional power of operators and Sadovskii’s fixed point theorem, we study the existence of mild solution for a certain class of impulsive neutral functional integrodifferential equations with nonlocal conditions. The results we obtained are a generalization and continuation of the recent resultson this issue.

Existence of Solutions of Abstract Nonlinear Mixed Functional Integrodifferential equation with nonlocal conditions

Abstract In this paper we discuss the existence of mild and strong solutions of abstract nonlinear mixed functional integrodifferential equation with nonlocal condition by using Sadovskii’s fixed point theorem and theory of fractional power of operators.

A generalized Gronwall inequality and its application to fractional neutral evolution inclusions

This paper deals with the fractional neutral evolution differential inclusions. The existence results are established by using the fractional power of operators and a fixed point theorem for multivalued map. Moreover, we present a new generalized Gronwall inequality with singularity, which is an important tool in the proof of solvability.

Cyber-physical systems as general distributed parameter systems: three types of fractional order models and emerging research opportunities

Cyber-physical systems (CPSs) are man-made complex systems coupled with natural processes that, as a whole, should be described by distributed parameter systems (DPSs) in general forms. This paper presents three such general models for generalized DPSs that can be used to characterize complex CPSs. These three different types of fractional operators based DPS models are: fractional Laplacian operator, fractional power of operator or fractional derivative. This research investigation is motivated by many fractional order models describing natural, physical, and anomalous phenomena, such as sub-diffusion process or super-diffusion process. The relationships among these three different operators are explored and explained. Several potential future research opportunities are then articulated followed by some conclusions and remarks.

On Existence of Solutions of Impulsive Nonlinear Functional Neutral Integro-Differential Equations With Nonlocal Condition

AbstractIn the present paper, we investigate the existence, uniqueness and continuous dependence of mild solutions of an impulsive neutral integro-differential equations with nonlocal condition in Banach spaces. We use Banach contraction principle and the theory of fractional power of operators to obtain our results.

EXISTENCE OF SOLUTIONS FOR NEUTRAL FUNCTIONAL VOLTERRA-FREDHOLM INTEGRODIFFERENTIAL EQUATIONS

In this paper, we study the existence of mild solutions of nonlinear neutral functional VolterraFredholm integrodifferential equations with nonlocal conditions. The results are obtained by using fractional power of operators and Sadovskii’s fixed point theorem.

Existence of Solutions for Fractional Partial Neutral Stochastic Functional Integro-Differential Inclusions with State-Dependent Delay and Analytic Resolvent Operators

We investigate the existence of mild solutions for a class of fractional partial neutral stochastic functional integro-differential inclusions with state-dependent delay and analytic α-resolvent operators in Hilbert spaces. Sufficient conditions for the existence are established by using the nonlinear alternative of Leray–Schauder type for multivalued maps due to O’Regan and the fractional power of operators. An example is provided to illustrate the obtained theory.

Sobolev Type Fractional Dynamic Equations and Optimal Multi-Integral Controls with Fractional Nonlocal Conditions

Abstract In We prove existence and uniqueness of mild solutions to Sobolev type fractional nonlocal dynamic equations in Banach spaces. The Sobolev nonlocal condition is considered in terms of a Riemann-Liouville fractional derivative. A Lagrange optimal control problem is considered, and existence of a multi-integral solution obtained. Main tools include fractional calculus, semigroup theory, fractional power of operators, a singular version of Gronwall's inequality, and Leray-Schauder fixed point theorem. An example illustrating the theory is given.

Approximate controllability of fractional delay dynamic inclusions with nonlocal control conditions

We introduce a nonlocal control condition and the notion of approximate controllability for fractional order quasilinear control inclusions. Approximate controllability of a fractional control nonlocal delay quasilinear functional differential inclusion in a Hilbert space is studied. The results are obtained by using the fractional power of operators, multi-valued analysis, and Sadovskii’s fixed point theorem. Main result gives an appropriate set of sufficient conditions for the considered system to be approximately controllable. As an example, a fractional partial nonlocal control functional differential inclusion is considered.

Existence results for Riemann-Liouville fractional neutral evolution equations

In this paper, by using the fractional power of operators and the theory of measure of noncompactness, we discuss a class of fractional neutral evolution equations with Riemann-Liouville fractional derivative. We establish sufficient conditions for the existence of mild solutions for fractional neutral evolution equations in the cases C 0 semigroup is compact or noncompact. We give an example to illustrate the applications of the abstract results.

Control problems for semilinear neutral differential equations in Hilbert spaces.

We construct some results on the regularity of solutions and the approximate controllability for neutral functional differential equations with unbounded principal operators in Hilbert spaces. In order to establish the controllability of the neutral equations, we first consider the existence and regularity of solutions of the neutral control system by using fractional power of operators and the local Lipschitz continuity of nonlinear term. Our purpose is to obtain the existence of solutions and the approximate controllability for neutral functional differential control systems without using many of the strong restrictions considered in the previous literature. Finally we give a simple example to which our main result can be applied.