The left Riemann-Liouville fractional Hermite-Hadamard type inequalities for convex functions

2019 ◽  
Vol 69 (4) ◽  
pp. 773-784
Author(s):  
Mehmet Kunt ◽  
Dünya Karapinar ◽  
Sercan Turhan ◽  
İmdat İşcan
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Fractional Integration ◽  
New Approach

Abstract In this paper, with a new approach, a new fractional Hermite-Hadamard type inequalities for convex functions is obtained by using only the left Riemann-Liouville fractional integral. Also, to have new fractional trapezoid and midpoint type inequalities for the differentiable convex functions, two new equalities are proved. Our results generalize earlier studies. We expect that this study will be lead to the new fractional integration studies for Hermite-Hadamard type inequalities.

scholarly journals On new inequalities for h-convex functions via Riemann-Liouville fractional integration

Filomat ◽  
2013 ◽  
Vol 27 (4) ◽  
pp. 559-565 ◽  
Author(s):  
Mevlüt Tunç
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Fractional Integration ◽  
New Inequalities

In this paper, some new inequalities of the Hermite-Hadamard type for h-convex functions via Riemann-Liouville fractional integral are given.

scholarly journals On New Inequalities via Riemann-Liouville Fractional Integration

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Mehmet Zeki Sarikaya ◽  
Hasan Ogunmez
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Fractional Integration ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
New Inequalities

We extend the Montgomery identities for the Riemann-Liouville fractional integrals. We also use these Montgomery identities to establish some new integral inequalities. Finally, we develop some integral inequalities for the fractional integral using differentiable convex functions.

A note on fractional integral operator associated with multiindex Mittag-Leffler functions

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1931-1939 ◽  
Author(s):  
Junesang Choi ◽  
Praveen Agarwal
Keyword(s):  
Integral Operator ◽  
Fractional Calculus ◽  
Fractional Integral ◽  
Fractional Integration ◽  
Fractional Integral Operator ◽  
Integration Formula ◽  
Special Cases

Recently Kiryakova and several other ones have investigated so-called multiindex Mittag-Leffler functions associated with fractional calculus. Here, in this paper, we aim at establishing a new fractional integration formula (of pathway type) involving the generalized multiindex Mittag-Leffler function E?,k[(?j,?j)m;z]. Some interesting special cases of our main result are also considered and shown to be connected with certain known ones.

scholarly journals New Modifications of Integral Inequalities via ℘-Convexity Pertaining to Fractional Calculus and Their Applications

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1753
Author(s):  
Saima Rashid ◽  
Aasma Khalid ◽  
Omar Bazighifan ◽  
Georgia Irina Oros
Keyword(s):  
Integral Operator ◽  
Convex Functions ◽  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Type Inequality ◽  
Fractional Integral Operator ◽  
Integral Inequalities ◽  
Convex Mappings ◽  
Special Cases ◽  
Harmonically Convex Functions

Integral inequalities for ℘-convex functions are established by using a generalised fractional integral operator based on Raina’s function. Hermite–Hadamard type inequality is presented for ℘-convex functions via generalised fractional integral operator. A novel parameterized auxiliary identity involving generalised fractional integral is proposed for differentiable mappings. By using auxiliary identity, we derive several Ostrowski type inequalities for functions whose absolute values are ℘-convex mappings. It is presented that the obtained outcomes exhibit classical convex and harmonically convex functions which have been verified using Riemann–Liouville fractional integral. Several generalisations and special cases are carried out to verify the robustness and efficiency of the suggested scheme in matrices and Fox–Wright generalised hypergeometric functions.

Orthonormal piecewise Bernoulli functions: Application for optimal control problems generated using fractional integro-differential equations

2022 ◽  
pp. 107754632110593
Author(s):  
Mohammad Hossein Heydari ◽  
Mohsen Razzaghi ◽  
Zakieh Avazzadeh
Keyword(s):  
Optimal Control ◽  
Fractional Integral ◽  
Original Problem ◽  
Optimal Control Problems ◽  
Integration Method ◽  
Direct Method ◽  
Fractional Integration ◽  
Basis Functions ◽  
Control Problems ◽  
Bernoulli Functions

In this study, the orthonormal piecewise Bernoulli functions are generated as a new kind of basis functions. An explicit matrix related to fractional integration of these functions is obtained. An efficient direct method is developed to solve a novel set of optimal control problems defined using a fractional integro-differential equation. The presented technique is based on the expressed basis functions and their fractional integral matrix together with the Gauss–Legendre integration method and the Lagrange multipliers algorithm. This approach converts the original problem into a mathematical programming one. Three examples are investigated numerically to verify the capability and reliability of the approach.

scholarly journals MIXED FRACTIONAL INTEGRATION AND DIFFERENTIATION AS RECIPROCAL OPERATIONS

2021 ◽  
Vol 6 (2(35)) ◽  
pp. 22-29
Author(s):  
Dilshod Barakaev
Keyword(s):  
Fractional Derivative ◽  
Fractional Integral ◽  
Broad Class ◽  
Fractional Integration ◽  
Class Of Functions

We study the question of the composition of the mixed fractional integral and the mixed fractional derivative in sufficiently broad class of functions. The treatment formula for mixed fractional derivative is obtained.

scholarly journals $k$-fractional integral inequalities of Hadamard type for exponentially $(s,m)$-convex functions

2021 ◽  
Vol 6 (1) ◽  
pp. 882-892
Author(s):  
Atiq Ur Rehman ◽  
◽  
Ghulam Farid ◽  
Sidra Bibi ◽  
Chahn Yong Jung ◽  
...  
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Integral Inequalities

scholarly journals Numerical Investigation of Fractional-Order Differential Equations via φ -Haar-Wavelet Method

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
F. M. Alharbi ◽  
A. M. Zidan ◽  
Muhammad Naeem ◽  
Rasool Shah ◽  
Kamsing Nonlaopon
Keyword(s):  
Differential Equations ◽  
Fractional Integral ◽  
Numerical Technique ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Nonlinear Problems ◽  
Haar Wavelet ◽  
Mathematical Tool ◽  
Suggested Technique ◽  
Operational Matrix Of Integration

In this paper, we propose a novel and efficient numerical technique for solving linear and nonlinear fractional differential equations (FDEs) with the φ -Caputo fractional derivative. Our approach is based on a new operational matrix of integration, namely, the φ -Haar-wavelet operational matrix of fractional integration. In this paper, we derived an explicit formula for the φ -fractional integral of the Haar-wavelet by utilizing the φ -fractional integral operator. We also extended our method to nonlinear φ -FDEs. The nonlinear problems are first linearized by applying the technique of quasilinearization, and then, the proposed method is applied to get a numerical solution of the linearized problems. The current technique is an effective and simple mathematical tool for solving nonlinear φ -FDEs. In the context of error analysis, an exact upper bound of the error for the suggested technique is given, which shows convergence of the proposed method. Finally, some numerical examples that demonstrate the efficiency of our technique are discussed.

Initialization of Identification of Fractional Model by Output-Error Technique

2015 ◽  
Vol 11 (2) ◽  
Author(s):  
Abir Khadhraoui ◽  
Khaled Jelassi ◽  
Jean-Claude Trigeassou ◽  
Pierre Melchior
Keyword(s):  
Numerical Simulation ◽  
Least Squares ◽  
Fractional Order ◽  
Fractional Integration ◽  
New Approach ◽  
Fractional Model ◽  
Output Error

A bad initialization of output-error (OE) technique can lead to an inappropriate identification results. In this paper, we introduce a solution to this problem; the basic idea is to estimate the parameters and the fractional order of the noninteger system by a new approach of least-squares (LS) method based on repeated fractional integration to initialize OE technique. It will be shown that LS method offers a good initialization to OE algorithm and leads to acceptable identification results. The performance of the proposed method is shown through numerical simulation examples.

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