On adjoint operators for fractional differentiation operators

2020 ◽  
pp. 284-289
Author(s):  
Garik G. Petrosyan
Keyword(s):  
Fractional Derivatives ◽  
Linear Manifold ◽  
Fractional Differentiation ◽  
Finite Segment ◽  
The Right ◽  
Adjoint Operators

On a linear manifold of the space of square summable functions on a finite segment vanishing at its ends, we consider the operator of left-sided Caputo fractional differentiation. We prove that the adjoint for it is the operator of right-sided Caputo fractional differentiation. Similar results are established for the Riemann–Liouville fractional differentiation operators. We also demonstrate that the operator, which is represented as the sum of the left-sided and the right-sided fractional differentiation operators is self adjoint. The known properties of the Caputo and Riemann–Liouville fractional derivatives are used to substantiate the results.

On Finite Part Integrals and Hadamard-Type Fractional Derivatives

2018 ◽  
Vol 13 (9) ◽  
Author(s):  
Li Ma ◽  
Changpin Li
Keyword(s):  
Fractional Derivative ◽  
Singular Integral ◽  
Fractional Derivatives ◽  
Continuous Functions ◽  
Finite Part ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Fundamental Properties ◽  
Logarithmic Series ◽  
The Right

This paper is devoted to investigating the relation between Hadamard-type fractional derivatives and finite part integrals in Hadamard sense; that is to say, the Hadamard-type fractional derivative of a given function can be expressed by the finite part integral of a strongly singular integral, which actually does not exist. Besides, our results also cover some fundamental properties on absolutely continuous functions, and the logarithmic series expansion formulas at the right end point of interval for functions in certain absolutely continuous spaces.

scholarly journals On the Nonlocal Problems in Time for Time-Fractional Subdiffusion Equations

2022 ◽  
Vol 6 (1) ◽  
pp. 41
Author(s):  
Ravshan Ashurov ◽  
Yusuf Fayziev
Keyword(s):  
Fractional Derivatives ◽  
Separable Hilbert Space ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Value Problem ◽  
Nonlocal Problems ◽  
Existence Of A Solution ◽  
Left Hand ◽  
Liouville Derivative ◽  
The Right

The nonlocal boundary value problem, dtρu(t)+Au(t)=f(t) (0<ρ<1, 0<t≤T), u(ξ)=αu(0)+φ (α is a constant and 0<ξ≤T), in an arbitrary separable Hilbert space H with the strongly positive selfadjoint operator A, is considered. The operator dt on the left hand side of the equation expresses either the Caputo derivative or the Riemann–Liouville derivative; naturally, in the case of the Riemann–Liouville derivatives, the nonlocal boundary condition should be slightly changed. Existence and uniqueness theorems for solutions of the problems under consideration are proved. The influence of the constant α on the existence of a solution to problems is investigated. Inequalities of coercivity type are obtained and it is shown that these inequalities differ depending on the considered type of fractional derivatives. The inverse problems of determining the right-hand side of the equation and the function φ in the boundary conditions are investigated.

On the Right Hamiltonian for Singular Perturbations: General Theory

1997 ◽  
Vol 09 (05) ◽  
pp. 609-633 ◽  
Author(s):  
Hagen Neidhardt ◽  
Valentin Zagrebnov
Keyword(s):  
General Theory ◽  
Symmetric Operator ◽  
Stability Domain ◽  
Bounded Operators ◽  
Friedrichs Extension ◽  
Abstract Problem ◽  
Dense Domain ◽  
Adjoint Extension ◽  
The Right ◽  
Adjoint Operators

Let the pair of self-adjoint operators {A≥0,W≤0} be such that: (a) there is a dense domain [Formula: see text] such that [Formula: see text] is semibounded from below (stability domain), (b) the symmetric operator [Formula: see text] is not essentially self-adjoint (singularity of the perturbation), (c) the Friedrichs extension [Formula: see text] of [Formula: see text] is maximal with respect to W, i.e., [Formula: see text]. [Formula: see text]. Let [Formula: see text] be a regularizing sequence of bounded operators which tends in the strong resolvent sense to W. The abstract problem of the right Hamiltonian is: (i) to give conditions such that the limit H of self-adjoint regularized Hamiltonians [Formula: see text] exists and is unique for any self-adjoint extension [Formula: see text] of [Formula: see text], (ii) to describe the limit H. We show that under the conditions (a)–(c) there is a regularizing sequence [Formula: see text] such that [Formula: see text] tends in the strong resolvent sense to unique (right Hamiltonian) [Formula: see text], otherwise the limit is not unique.

Systems of Nonlinear Fractional Differential Equations

2015 ◽  
Vol 18 (1) ◽  
Author(s):  
Tadeusz Jankowski
Keyword(s):  
Differential Equations ◽  
Iterative Method ◽  
Unique Solution ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Nonlinear Fractional Differential Equations ◽  
Fractional Differential ◽  
The Right ◽  
Linear Fractional Differential Equations

AbstractUsing the iterative method, this paper investigates the existence of a unique solution to systems of nonlinear fractional differential equations, which involve the right-handed Riemann-Liouville fractional derivatives $D^{q}_{T}x$ and $D^{q}_{T}y$. Systems of linear fractional differential equations are also discussed. Two examples are added to illustrate the results.

scholarly journals On the Kolmogorov forward equations within Caputo and Riemann-Liouville fractions derivatives

2016 ◽  
Vol 24 (3) ◽  
pp. 5-19
Author(s):  
Mohsen Alipour ◽  
Dumitru Baleanu
Keyword(s):  
Exact Solutions ◽  
Fractional Derivatives ◽  
Laplace Transforms ◽  
Caputo Fractional Derivatives ◽  
The Right ◽  
Forward Equations ◽  
Kolmogorov Forward Equations

AbstractIn this work, we focus on the fractional versions of the well-known Kolmogorov forward equations. We consider the problem in two cases. In case 1, we apply the left Caputo fractional derivatives for α ∈ (0, 1] and in case 2, we use the right Riemann-Liouville fractional derivatives on R+, for α ∈ (1, +∞). The exact solutions are obtained for the both cases by Laplace transforms and stable subordinators.

scholarly journals Response of Fractionally Damped Beams with General Boundary Conditions Subjected to Moving Loads

2012 ◽  
Vol 19 (3) ◽  
pp. 333-347 ◽  
Author(s):  
R. Abu-Mallouh ◽  
I. Abu-Alshaikh ◽  
H.S. Zibdeh ◽  
Khaled Ramadan
Keyword(s):  
Dynamic Response ◽  
Boundary Conditions ◽  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Initial Conditions ◽  
General Boundary ◽  
General Boundary Conditions ◽  
Damping Characteristics ◽  
The Laplace Transform ◽  
The Right

This paper presents the transverse vibration of Bernoulli-Euler homogeneous isotropic damped beams with general boundary conditions. The beams are assumed to be subjected to a load moving at a uniform velocity. The damping characteristics of the beams are described in terms of fractional derivatives of arbitrary orders. In the analysis where initial conditions are assumed to be homogeneous, the Laplace transform cooperates with the decomposition method to obtain the analytical solution of the investigated problems. Subsequently, curves are plotted to show the dynamic response of different beams under different sets of parameters including different orders of fractional derivatives. The curves reveal that the dynamic response increases as the order of fractional derivative increases. Furthermore, as the order of the fractional derivative increases the peak of the dynamic deflection shifts to the right, this yields that the smaller the order of the fractional derivative, the more oscillations the beam suffers. The results obtained in this paper closely match the results of papers in the literature review.

scholarly journals SOLVABILITY OF THE CAUCHY PROBLEM FOR EQUATIONS WITH RIEMANN–LIOUVILLE’S FRACTIONAL DERIVATIVES

2018 ◽  
Vol 62 (4) ◽  
pp. 391-397 ◽  
Author(s):  
Petr P. Zabreiko ◽  
Svetlana V. Ponomareva
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Fixed Point ◽  
Unique Solution ◽  
Metric Space ◽  
Fractional Derivatives ◽  
Volterra Equation ◽  
Complete Metric Space ◽  
The Cauchy Problem ◽  
The Right

In this article we study the solvability of the analogue of the Cauchy problem for ordinary differential equations with Riemann–Liouville’s fractional derivatives with a nonlinear restriction on the right-hand side of functions in certain spaces. The conditions for solvability of the problem under consideration in given function spaces, as well as the conditions for existence of a unique solution are given. The study uses the method of reducing the problem to the second-kind Volterra equation, the Schauder principle of a fixed point in a Banach space, and the Banach-Cachoppoli principle of a fixed point in a complete metric space.

scholarly journals On Some Inequalities Involving Liouville–Caputo Fractional Derivatives and Applications to Special Means of Real Numbers

Mathematics ◽  
2018 ◽  
Vol 6 (10) ◽  
pp. 193 ◽  
Author(s):  
Bessem Samet ◽  
Hassen Aydi
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Fractional Derivatives ◽  
Real Numbers ◽  
Caputo Fractional Derivatives ◽  
Parallel Development ◽  
Special Means ◽  
The Right ◽  
Class Of Functions

We are concerned with the class of functions f ∈ C 1 ( [ a , b ] ; R ) , a , b ∈ R , a < b , such that c D a α f is convex or c D b α f is convex, where 0 < α < 1 , c D a α f is the left-side Liouville–Caputo fractional derivative of order α of f and c D b α f is the right-side Liouville–Caputo fractional derivative of order α of f. Some extensions of Dragomir–Agarwal inequality to this class of functions are obtained. A parallel development is made for the class of functions f ∈ C 1 ( [ a , b ] ; R ) such that c D a α f is concave or c D b α f is concave. Next, an application to special means of real numbers is provided.

scholarly journals Numerical Study of Fractional Mathieu Differential Equation Using Radial Basis Functions

2020 ◽  
Vol 7 (4) ◽  
pp. 568-576
Author(s):  
Hojjat Ghorbani ◽  
Yaghoub Mahmoudi ◽  
Farhad Dastmalchi Saei
Keyword(s):  
Differential Equation ◽  
Radial Basis Functions ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Numerical Study ◽  
Mathieu Equation ◽  
Basis Functions ◽  
Damping Factor ◽  
Radial Basis ◽  
The Right

In this paper, we introduce a method based on Radial Basis Functions (RBFs) for the numerical approximation of Mathieu differential equation with two fractional derivatives in the Caputo sense. For this, we suggest a numerical integration method for approximating the improper integrals with a singularity point at the right end of the integration domain, which appear in the fractional computations. We study numerically the affects of characteristic parameters and damping factor on the behavior of solution for fractional Mathieu differential equation. Some examples are presented to illustrate applicability and accuracy of the proposed method. The fractional derivatives order and the parameters of the Mathieu equation are changed to study the convergency of the numerical solutions.

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