Riemann-Liouville Fractional Integrals Research Articles

Some new parameterized inequalities based on Riemann-Liouville fractional integrals

In this article, we first obtain an identity that we will use throughout the article. With the help of this equality, new inequalities involving a real parameter are established for Riemann-Liouville fractional integrals. For this purpose, properties of the differentiable convex function, H?lder inequality, and power-mean inequality are used. In addition, new results are established with special choices of parameters in all proven inequalities. Our results are supported by examples and graphs. It is shown that some of these results generalize the trapezoid type and Newton-type inequalities.

HÖLDER CONTINUITY AND BOX DIMENSION FOR THE MIXED RIEMANN–LIOUVILLE FRACTIONAL INTEGRAL

In this paper, we investigate the Hölder continuity and the estimate of the Box dimension of [Formula: see text], which is called the mixed Riemann–Liouville fractional integral of the continuous function [Formula: see text]. We focus on the case that [Formula: see text] is [Formula: see text]th-order Hölder continuous, where [Formula: see text]. By using the approximated integral, we obtain that, [Formula: see text] is [Formula: see text]th-order Hölder continuous, and the Box dimension of the graph of [Formula: see text] is less than or equal to [Formula: see text], provided that [Formula: see text]. Here [Formula: see text]. By using Stein’s Lemma, we prove that [Formula: see text] is [Formula: see text]th-order Hölder continuous, provided that [Formula: see text], and the Box dimension of the graph of [Formula: see text] is less than or equal to [Formula: see text]. Moreover, we also illustrate that the latter conclusion is sharp.

New Chebyshev-type inequalities for the generalized Riemann-Liouville fractional integral with respect to an increasing function

New Chebyshev-type inequalities for the generalized Riemann-Liouville fractional integral with respect to an increasing function

HERMITE–HADAMARD TYPE INEQUALITIES FOR KATUGAMPOLA FRACTIONAL INTEGRALS

In the paper, basing on the Katugampola fractional integrals ${}^\rho\mathcal{K}^\alpha_{a+}f$ and ${}^\rho\mathcal{K}^\alpha_{b-}f$ with $f\in\mathfrak{X}_c^p(a,b)$, the authors establish the Hermite-Hadamard type inequalities for convex functions, give their left estimates, and apply these newly-established inequalities to special means of real numbers. When $\rho\to1$, these results become the corresponding ones for the Riemann-Liouville fractional integrals.

On the Optimality Condition for Optimal Control of Caputo Fractional Differential Equations with State Constraints

We consider the fractional optimal control problem with state constraints. The fractional calculus of derivatives and integrals can be viewed as generalizations of their classical ones to any arbitrary real order. In our problem setup, the dynamic constraint is captured by the Caputo fractional differential equation with order α ∈ (0, 1), and the objective functional is formulated by the left Riemann-Liouville fractional integral with order β ≥ 1. In addition, there are terminal and running state constraints; while the former is described by initial and final states within a convex set, the latter is given by an explicit instantaneous inequality state constraint. We obtain the maximum principle, the first-order necessary optimality condition, for the problem of this paper. Due to the inherent complex nature of the fractional control problem, the presence of the terminal and running state constraints, and the generalized standing assumptions, the maximum principle of this paper is new in the optimal control problem context, and its proof requires to develop new variational and duality analysis using fractional calculus and functional analysis, together with the Ekeland variational principle and the spike variation.

Fractional Maclaurin-type inequalities for multiplicatively convex functions and multiplicatively P-functions

In this paper, we present a fractional integral identity, and then based upon it we establish the Maclaurin?s inequalities for multiplicatively convex functions and multiplicatively P-functions via multiplicative Riemann-Liouville fractional integrals.

Differential sandwich theorems involving Riemann-Liouville fractional integral of $ q $-hypergeometric function

<abstract><p>The development of certain aspects of geometric function theory after incorporating fractional calculus and $ q $-calculus aspects is obvious and indisputable. The study presented in this paper follows this line of research. New results are obtained by applying means of differential subordination and superordination theories involving an operator previously defined as the Riemann-Liouville fractional integral of the $ q $-hypergeometric function. Numerous theorems are stated and proved involving the fractional $ q $-operator and differential subordinations for which the best dominants are found. Associated corollaries are given as applications of those results using particular functions as best dominants. Dual results regarding the fractional $ q $-operator and differential superordinations are also considered and theorems are proved where the best subordinants are given. Using certain functions known for their remarkable geometric properties applied in the results as best subordinant, interesting corollaries emerge. As a conclusion of the investigations done by applying the means of the two dual theories considering the fractional $ q $-operator, several sandwich-type theorems combine the subordination and superordiantion established results.</p></abstract>

Fractional midpoint-type inequalities for twice-differentiable functions

In this research article, we obtain an identity for twice differentiable functions whose second derivatives in absolute value are convex. By using this identity, we prove several left Hermite-Hadamardtype inequalities for the case of Riemann-Liouville fractional integrals. Furthermore, we provide our results by using special cases of obtained theorems.

A numerical study for solving multi-term fractional-order differential equations

In this article, we extended operational matrices using orthonormal Boubaker polynomials of Riemann-Liouville fractional integration and Caputo derivative to find numerical solution of multi-term fractional-order differential equations (FDE). The proposed method is utilized to convert FDE into a system of algebraic equations. The convergence of the method is proved. Examples are given to explain the simplicity, computational time and accuracy of the method.

SOME HERMITE-HADAMARD TYPE INEQUALITIES INVOLVING FRACTIONAL INTEGRAL OPERATORS

The aim of this article is to give new generalized Hermite-Hadamard type inequalities involving the Riemann-Liouville fractional integral for functions whose absolute values of third derivatives are s-convex in the second sense. In order to do that an integral identity for three times differentiable mapping involving fractional integral operators is established. Several consequences are then presented in some special cases.

"On the generalized trapezoid and midpoint type inequalities involving Euler's beta function}"

"The main object of this paper is to present some generalizations of fractional integral inequalities involving Euler's beta function of Hermite-Hadamard type which cover the previously published result such as Riemann integral, Riemann-Liouville fractional integral, k-Riemann-Liouville fractional integral."

Modified Inequalities on Center-Radius Order Interval-Valued Functions Pertaining to Riemann–Liouville Fractional Integrals

In this paper, we shall discuss a newly introduced concept of center-radius total-ordered relations between two intervals. Here, we address the Hermite–Hadamard-, Fejér- and Pachpatte-type inequalities by considering interval-valued Riemann–Liouville fractional integrals. Interval-valued fractional inequalities for a new class of preinvexity, i.e., cr-h-preinvexity, are estimated. The fractional operator is used for the first time to prove such inequalities involving center–radius-ordered functions. Some numerical examples are also provided to validate the presented inequalities.

Non-polynomial fractional spline method for solving Fredholm integral equations

A new type of non-polynomial fractional spline function for approximating solutions of Fredholm-integral equations has been presented. For this purpose, we used a new idea of fractional continuity conditions by using the Caputo fractional derivative and the Riemann Liouville fractional integration to generate fractional spline derivatives. Moreover, the convergence analysis is studied with proven theorems. The approach is also well-explained and supported by four computational numerical findings, which show that it is both accurate and simple to apply.

High-order orthogonal spline collocation ADI scheme for a new complex two-dimensional distributed-order fractional integro-differential equation with two weakly singular kernels

We present an alternating direction implicit scheme for the numerical solution of the distributed-order fractional integro-differential equation with two weakly singular kernels. Orthogonal spline collocation with piecewise Hermite bicubics is used for spatial discretization. The weighted and shifted Grünwald difference formula is employed to discretize the distributed-order time-fractional derivative combined with the second-order quadrature convolution rule introduced by Lubich for the Riemann–Liouville fractional integral. We prove the stability of the proposed scheme and derive error estimates. We also describe an efficient implementation of the scheme and show the numerical results demonstrating the accuracy and convergence rates in norm.

New Fractional Integral Inequalities Pertaining to Caputo–Fabrizio and Generalized Riemann–Liouville Fractional Integral Operators

Integral inequalities have accumulated a comprehensive and prolific field of research within mathematical interpretations. In recent times, strategies of fractional calculus have become the subject of intensive research in historical and contemporary generations because of their applications in various branches of science. In this paper, we concentrate on establishing Hermite–Hadamard and Pachpatte-type integral inequalities with the aid of two different fractional operators. In particular, we acknowledge the critical Hermite–Hadamard and related inequalities for n-polynomial s-type convex functions and n-polynomial s-type harmonically convex functions. We practice these inequalities to consider the Caputo–Fabrizio and the k-Riemann–Liouville fractional integrals. Several special cases of our main results are also presented in the form of corollaries and remarks. Our study offers a better perception of integral inequalities involving fractional operators.

On the Generalized Inequalities for Co-Ordinated Convex Functions

Abstract The aim of this paper is to establish some generalized integral inequalities for convex functions of 2-variables on the co-ordinat. Then, we will give a generalized identity and with the help of this integral identity, we will investigate some integral inequalities connected with the right hand side of the Hermite-Hadamard type inequalities involving Riemann integrals and Riemann-Liouville fractional integrals.

Numerical simulation for generalized space-time fractional Klein–Gordon equations via Gegenbauer wavelet

Abstract This paper investigates numerical solution of generalized space-time fractional Klein–Gordon equations (GSTFKGE) by using Gegenbauer wavelet method (GWM). The developed method makes use of fractional order integral operator (FOIO) for Gegenbauer wavelet, which is constructed by employing the definition of Riemann–Liouville fractional integral (RLFI) operator and Laplace transformation. The present algorithm is based on Gegenbauer wavelet jointly with FOIO to convert a GSTFKGE into a system of equations which is solved by using Newton’s technique. Additionally, the upper bound of error norm of the proposed method is calculated to validate the theoretical authenticity of the developed method. The comparison of numerical outcomes with the existing results in the literature and graphical illustrations show the accuracy and reliability of our method.

A study of a nonlinear Riemann–Liouville coupled integro-differential system with coupled nonlocal fractional integro-multipoint boundary conditions

Abstract We discuss the existence of solutions for a boundary value problem of nonlinear coupled Riemann–Liouville fractional integro-differential equations equipped with coupled nonlocal fractional integro-multipoint boundary conditions. The standard tools of the modern functional analysis are employed to derive the desired results for the problem at hand. The case of nonlinearities depending on the Riemann–Liouville fractional integrals is also discussed. Examples illustrating the obtained results are presented.

A new analytical method to simulate the mutual impact of space-time memory indices embedded in (1 + 2)-physical models

Abstract In the present article, we geometrically and analytically examine the mutual impact of space-time Caputo derivatives embedded in (1 + 2)-physical models. This has been accomplished by integrating the residual power series method (RPSM) with a new trivariate fractional power series representation that encompasses spatial and temporal Caputo derivative parameters. Theoretically, some results regarding the convergence and the error for the proposed adaptation have been established by virtue of the Riemann–Liouville fractional integral. Practically, the embedding of Schrödinger, telegraph, and Burgers’ equations into higher fractional space has been considered, and their solutions furnished by means of a rapidly convergent series that has ultimately a closed-form fractional function. The graphical analysis of the obtained solutions has shown that the solutions possess a homotopy mapping characteristic, in a topological sense, to reach the integer case solution where the Caputo derivative parameters behave similarly to the homotopy parameters. Altogether, the proposed technique exhibits a high accuracy and high rate of convergence.

Applications of Riemann–Liouville Fractional Integral of q-Hypergeometric Function for Obtaining Fuzzy Differential Sandwich Results

Studies regarding the two dual notions are conducted in this paper using Riemann–Liouville fractional integral of q-hypergeometric function for obtaining certain fuzzy differential subordinations and superordinations. Fuzzy best dominants and fuzzy best subordinants are given in the theorems investigating fuzzy differential subordinations and superordinations, respectively. Moreover, corollaries are stated by considering particular functions with known geometric properties as fuzzy best dominant and fuzzy best subordinant in the proved results. The study is completed by stating fuzzy differential sandwich theorems followed by related corollaries combining the results previously established concerning fuzzy differential subordinations and superordinations.