scholarly journals Generalized Fractional-Order Bernoulli Functions via Riemann-Liouville Operator and Their Applications in the Evaluation of Dirichlet Series

2018 ◽  
Vol 2018 ◽  
pp. 1-5
Author(s):  
Jorge Sanchez-Ortiz
Keyword(s):  
Integral Operator ◽  
Dirichlet Series ◽  
Fractional Order ◽  
Generating Function ◽  
Liouville Operator ◽  
Fractional Integral ◽  
Generalized Functions ◽  
New Class ◽  
Bernoulli Functions ◽  
Class Of Functions

In this work, we define a new class of functions of the Bernoulli type using the Riemann-Liouville fractional integral operator and derive a generating function for these class generalized functions. Then, these functions are employed to derive formulas for certain Dirichlet series.

scholarly journals From fractional order equations to integer order equations

2017 ◽  
Vol 20 (6) ◽  
Author(s):  
Daniel Cao Labora ◽  
Rosana Rodríguez-López
Keyword(s):  
Integral Equation ◽  
Integral Operator ◽  
Vector Space ◽  
Integral Equations ◽  
Fractional Order ◽  
Fractional Integral ◽  
State Of The Art ◽  
Constant Coefficients ◽  
Fractional Order Integral ◽  
Ordinary Integral

AbstractThe main goal of this article is to show a new method to solve some Fractional Order Integral Equations (FOIE), more precisely the ones which are linear, have constant coefficients and all the integration orders involved are rational. The method essentially turns a FOIE into an Ordinary Integral Equation (OIE) by applying a suitable fractional integral operator.After discussing the state of the art, we present the idea of our construction in a particular case (Abel integral equation). After that, we propose our method in a general case, showing that it does work when dealing with a family of “additive” operators over a vector space. Later, we show that our construction is always possible when dealing with any FOIE under the above-mentioned hypotheses. Furthermore, it is shown that our construction is “optimal” in the sense that the OIE that we obtain has the least possible order.

scholarly journals New integral inequalities for strongly nonconvex functions involving Raina function

Filomat ◽  
2021 ◽  
Vol 35 (7) ◽  
pp. 2437-2456
Author(s):  
Artion Kashuri ◽  
Marcela Mihai ◽  
Muhammad Awan ◽  
Muhammad Noor ◽  
Khalida Noor
Keyword(s):  
Integral Operator ◽  
Fractional Integral ◽  
Integral Inequalities ◽  
Nonconvex Functions ◽  
Nonconvex Function ◽  
Special Cases ◽  
Type Integral ◽  
Generalized Fractional Integral ◽  
General Class Of Functions ◽  
Class Of Functions

In this paper, the authors defined a new general class of functions, the so-called strongly (h1,h2)-nonconvex function involving F??,?(?) (Raina function). Utilizing this, some Hermite-Hadamard type integral inequalities via generalized fractional integral operator are obtained. Some new results as a special cases are given as well.

scholarly journals New Hadamard-type inequalities via $(s,m_{1},m_{2})$-convex functions

2021 ◽  
Vol 31 (4) ◽  
pp. 597-612
Author(s):  
B. Bayraktar ◽  
S.I. Butt ◽  
Sh. Shaokat ◽  
J.E. Nápoles Valdés
Keyword(s):  
Integral Operator ◽  
Convex Functions ◽  
Fractional Integral ◽  
Fractional Integral Operator ◽  
Integral Inequalities ◽  
Class Of Functions

The article introduces a new concept of convexity of a function: $(s,m_{1},m_{2})$-convex functions. This class of functions combines a number of convexity types found in the literature. Some properties of $(s,m_{1},m_{2})$-convexities are established and simple examples of functions belonging to this class are given. On the basis of the proved identity, new integral inequalities of the Hadamard type are obtained in terms of the fractional integral operator. It is shown that these results give us, in particular, generalizations of a number of results available in the literature.

scholarly journals New Hadamard-type inequalities via $(s,m_{1},m_{2})$-convex functions

2021 ◽  
Vol 31 (4) ◽  
pp. 597-612
Author(s):  
B. Bayraktar ◽  
S.I. Butt ◽  
Sh. Shaokat ◽  
J.E. Napoles Valdes
Keyword(s):  
Integral Operator ◽  
Convex Functions ◽  
Fractional Integral ◽  
Fractional Integral Operator ◽  
Integral Inequalities ◽  
Class Of Functions

The article introduces a new concept of convexity of a function: $(s,m_{1},m_{2})$-convex functions. This class of functions combines a number of convexity types found in the literature. Some properties of $(s,m_{1},m_{2})$-convexities are established and simple examples of functions belonging to this class are given. On the basis of the proved identity, new integral inequalities of the Hadamard type are obtained in terms of the fractional integral operator. It is shown that these results give us, in particular, generalizations of a number of results available in the literature.

scholarly journals Study of new class of q-fractional integral operator

Filomat ◽  
2019 ◽  
Vol 33 (17) ◽  
pp. 5713-5721
Author(s):  
M. Momenzadeh ◽  
N.I. Mahmudov
Keyword(s):  
Differential Operator ◽  
Integral Operator ◽  
Fractional Integral ◽  
Fractional Integral Operator ◽  
Integral Approach ◽  
Cauchy Integral ◽  
New Class ◽  
Fractional Differential Operator ◽  
Fractional Differential

In this paper, we study on the new class of q-fractional integral operator. In the aid of iterated Cauchy integral approach to fractional integral operator, we applied tp f(t) in these integrals and a new class of q-fractional integral operator with parameter p, is introduced. Recently, the q-analogue of fractional differential integral operator is studied and all of the operators defined in these studies are q-analogue of Riemann fractional differential operator. We show that our new class of operator generalize all the operators in use, and additionally, it can cover the q-analogue of Hadamard fractional differential operator, as well. Some properties of this operator are investigated.

scholarly journals Some new generalizations of Ostrowski type inequalities for s-convex functions via fractional integral operators

Filomat ◽  
2018 ◽  
Vol 32 (16) ◽  
pp. 5595-5609
Author(s):  
Erhan Set
Keyword(s):  
Integral Operator ◽  
Convex Functions ◽  
Fractional Integral ◽  
Present Note ◽  
Integral Operators ◽  
Fractional Integral Operator ◽  
Integral Inequalities ◽  
Fractional Integral Operators ◽  
Special Cases ◽  
Class Of Functions

Remarkably a lot of Ostrowski type inequalities involving various fractional integral operators have been investigated by many authors. Recently, Raina [34] introduced a new generalization of the Riemann-Liouville fractional integral operator involving a class of functions defined formally by F? ?,?(x)=??,k=0 ?(k)/?(?k + ?)xk. Using this fractional integral operator, in the present note, we establish some new fractional integral inequalities of Ostrowski type whose special cases are shown to yield corresponding inequalities associated with Riemann-Liouville fractional integral operators.

scholarly journals Modified Laguerre wavelet based Galerkin method for fractional and fractional-order delay differential equations

2019 ◽  
Vol 23 (Suppl. 1) ◽  
pp. 13-21 ◽  
Author(s):  
Aydin Secer ◽  
Neslihan Ozdemir
Keyword(s):  
Approximate Solution ◽  
Integral Operator ◽  
Differential Equations ◽  
Galerkin Method ◽  
Fractional Order ◽  
Delay Differential Equations ◽  
Fractional Integral ◽  
Approximate Solutions ◽  
Delay Differential ◽  
The Given

The application of modified Laguerre wavelet with respect to the given conditions by Galerkin method to an approximate solution of fractional and fractional-order delay differential equations is studied in this paper. For the concept of fractional derivative is used Caputo sense by using Riemann-Liouville fractional integral operator. The presented method here is tested on several problems. The approximate solutions obtained by presented method are compared with the exact solutions and is shown to be a very efficient and powerful tool for obtaining approximate solutions of fractional and fractional-order delay differential equations. Some tables and figures are presented to reveal the performance of the presented method.

scholarly journals Riemann–Liouville Operator in Weighted Lp Spaces via the Jacobi Series Expansion

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 75 ◽  
Author(s):  
Maksim V. Kukushkin
Keyword(s):  
Integral Operator ◽  
Liouville Operator ◽  
Fractional Integral ◽  
Jacobi Polynomials ◽  
Fractional Integral Operator ◽  
Proved Theorem ◽  
Jacobi Series ◽  
Integrable Functions ◽  
Operator Action ◽  
Square Integrable Functions

In this paper, we use the orthogonal system of the Jacobi polynomials as a tool to study the Riemann–Liouville fractional integral and derivative operators on a compact of the real axis. This approach has some advantages and allows us to complete the previously known results of the fractional calculus theory by means of reformulating them in a new quality. The proved theorem on the fractional integral operator action is formulated in terms of the Jacobi series coefficients and is of particular interest. We obtain a sufficient condition for a representation of a function by the fractional integral in terms of the Jacobi series coefficients. We consider several modifications of the Jacobi polynomials, which gives us the opportunity to study the invariant property of the Riemann–Liouville operator. In this direction, we have shown that the fractional integral operator acting in the weighted spaces of Lebesgue square integrable functions has a sequence of the included invariant subspaces.

scholarly journals Generalized operator for Alexander integral operator

2020 ◽  
Vol 12 (2) ◽  
pp. 294-306
Author(s):  
H. Özlem Güney ◽  
Shigeyoshi Owa
Keyword(s):  
Power Series ◽  
Integral Operator ◽  
Unit Disc ◽  
Fractional Integral ◽  
Open Unit ◽  
Open Unit Disc ◽  
Boundary Points ◽  
Closed Unit Disc ◽  
Generalized Operator ◽  
Class Of Functions

AbstractLet Tn be the class of functions f which are defined by a power series f\left( z \right) = z + {a_{n + 1}}{z^{n + 1}} + {a_n}2{z^{n + 2}} + \ldots for every z in the closed unit disc \bar {\mathbb{U}}. With m different boundary points zs, (s = 1,2,...,m), we consider αm ∈ eiβ𝒜−j−λf(𝕌), here 𝒜−j−λ is the generalized Alexander integral operator and 𝕌 is the open unit disc. Applying 𝒜−j−λ, a subclass Bn(αm,β,ρ; j, λ) of Tn is defined with fractional integral for functions f. The object of present paper is to consider some interesting properties of f to be in Bn(αm,β,ρ; j, λ).

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