Riemann-Liouville Fractional Operator Research Articles

On existence of certain delta fractional difference models

The discretization of initial and boundary value problems and their existence behaviors are of great significance in various fields. This paper explores the existence of a class of self-adjoint delta fractional difference equations. The study begins by demonstrating the uniqueness of an initial value problem of delta Riemann–Liouville fractional operator type. Based on this result, the uniqueness of the self-adjoint equation will be examined and determined. Next, we define the Cauchy function based on the delta Riemann–Liouville fractional differences. Accordingly, the solution of the self-adjoint equation will be investigated according to the delta Cauchy function. Furthermore, the research investigates the uniqueness of the self-adjoint equation including the component of Green’s functions of and examines how this equation has only a trivial solution. To validate the theoretical analysis, specific examples are conducted to support and verify our results

The cubic nonlinear fractional Schrödinger equation on the half-line

We study the cubic nonlinear fractional Schrödinger equation with Lévy indices 43<α<2 posed on the half-line. More precisely, we define the notion of a solution for this model and we obtain a result of local-well-posedness almost sharp in the sense of index of regularity required for the solutions with respect for known results on the full real line R. Also, we prove for the same model that the solution of the nonlinear part is smoother than the initial data and the corresponding linear solution. To get our results we use the Colliander and Kenig approach based on the Riemann–Liouville fractional operator combined with Fourier restriction method of Bourgain (1993) and some ideas of the recent work of Erdoğan et al. (2019). The method applies to both focusing and defocusing nonlinearities. As a consequence of our analysis we prove a smoothing effect for the cubic nonlinear fractional Schrödinger equation posed in full line R for the case of the low regularity assumption, which was point out at the recent work (Erdoğan et al., 2019).

The first boundary value problem for the fractional diffusion equation in a degenerate angular domain

This article addresses the problems observed in branching fractal structures, where super-slow transport processes can occur, a phenomenon described by diffusion equations with a fractional time derivative. The characteristic feature of these processes is their extremely slow relaxation rate, where a physical quantity changes more gradually than its first derivative. Such phenomena are sometimes categorized as processes with “residual memory”. The study presents a solution to the first boundary problem in an angular domain degenerating into a point at the initial moment of time for a fractional diffusion equation with the RiemannLiouville fractional differentiation operator with respect to time. It establishes the existence theorem of the problem under investigation and constructs a solution representation. The need for understanding these super-slow processes and their impact on fractal structures is identified and justified. The paper demonstrates how these processes contribute to the broader understanding of fractional diffusion equations, proving the theorem’s existence and formulating a solution representation.

New Results on (r,k,μ)-Riemann–Liouville Fractional Operators in Complex Domain with Applications

This paper introduces fractional operators in the complex domain as generalizations for the Srivastava–Owa operators. Some properties for the above operators are also provided. We discuss the convexity and starlikeness of the generalized Libera integral operator. A condition for the convexity and starlikeness of the solutions of fractional differential equations is provided. Finally, a fractional differential equation is converted into an ordinary differential equation by wave transformation; illustrative examples are provided to clarify the solution within the complex domain.

Solving hybrid functional-fractional equations originating in biological population dynamics with an effect on infectious diseases

&lt;abstract&gt;&lt;p&gt;This paper study was designed to establish solutions for mixed functional fractional integral equations that involve the Riemann-Liouville fractional operator and the Erdélyi-Kober fractional operator to describe biological population dynamics in Banach space. The results rely on the measure of non-compactness and theoretical concepts from fractional calculus. Darbo's fixed-point theorem for Banach spaces has been utilized. Moreover, the solvability of a specific non-linear integral equation that models the spread of infectious diseases with a seasonally varying periodic contraction rate has been explored by using the Banach contraction principle. Finally, two numerical examples demonstrate the practical application of these findings in the realm of fractional integral equation theory.&lt;/p&gt;&lt;/abstract&gt;

Weighted Fractional Hermite–Hadamard Integral Inequalities for up and down Ԓ-Convex Fuzzy Mappings over Coordinates

Due to its significant influence on numerous areas of mathematics and practical sciences, the theory of integral inequality has attracted a lot of interest. Convexity has undergone several improvements, generalizations, and extensions over time in an effort to produce more accurate variations of known findings. This article’s main goal is to introduce a new class of convexity as well as to prove several Hermite–Hadamard type interval-valued integral inequalities in the fractional domain. First, we put forth the new notion of generalized convexity mappings, which is defined as UD-Ԓ-convexity on coordinates with regard to fuzzy-number-valued mappings and the up and down (UD) fuzzy relation. The generic qualities of this class make it novel. By taking into account different values for Ԓ, we produce several known classes of convexity. Additionally, we create some new fractional variations of the Hermite–Hadamard (HH) and Pachpatte types of inequalities using the concepts of coordinated UD-Ԓ-convexity and double Riemann–Liouville fractional operators. The results attained here are the most cohesive versions of previous findings. To demonstrate the importance of the key findings, we offer a number of concrete examples.

New integral inequalities involving generalized Riemann-Liouville fractional operators

New integral inequalities involving generalized Riemann-Liouville fractional operators

Generalization of Riemann-Liouville Fractional Operators in Bicomplex Space and Applications

Generalization of Riemann-Liouville Fractional Operators in Bicomplex Space and Applications

Non-Gaussian Measures in Infinite Dimensional Spaces: the Gamma-Grey Noise

In the context of non-Gaussian analysis, Schneider [29] introduced grey noise measures, built upon Mittag-Leffler functions; analogously, grey Brownian motion and its generalizations were constructed (see, for example, [6, 7, 9, 27]). In this paper, we construct and study a new non-Gaussian measure, by means of the incomplete-gamma function (exploiting its complete monotonicity). We label this measure Gamma-grey noise and we prove, for it, the existence of Appell system. The related generalized processes, in the infinite dimensional setting, are also defined and, through the use of the Riemann-Liouville fractional operators, the (possibly tempered) Gamma-grey Brownian motion is consequently introduced. A number of different characterizations of these processes are also provided, together with the integro-differential equation satisfied by their transition densities. They allow to model anomalous diffusions, mimicking the procedures of classical stochastic calculus.

On a General Formulation of the Riemann–Liouville Fractional Operator and Related Inequalities

In this paper, we present a general formulation of the Riemann–Liouville fractional operator with generalized kernels. Many of the known operators are shown to be particular cases of the one we present. In this new framework, we prove several known integral inequalities in the literature.

I.V-CR-γ-Convex Functions and Their Application in Fractional Hermite–Hadamard Inequalities

In recent years, the theory of convexity has influenced every field of mathematics due to its unique characteristics. Numerous generalizations, extensions, and refinements of convexity have been introduced, and one of them is set-valued convexity. Interval-valued convex mappings are a special type of set-valued maps. These have a close relationship with symmetry analysis. One of the important aspects of the relationship between convex and symmetric analysis is the ability to work on one field and apply its principles to another. In this paper, we introduce a novel class of interval-valued (I.V.) functions called CR-γ-convex functions based on a non-negative mapping γ and center-radius ordering relation. Due to its generic property, a set of new and known forms of convexity can be obtained. First, we derive new generalized discrete and integral forms of Jensen’s inequalities using CR-γ-convex I.V. functions. We employ this definition and Riemann-Liouville fractional operators to develop new fractional versions of Hermite-Hadamard’s, Hermite-Hadamard-Fejer, and Pachpatte’s type integral inequalities. We examine various key properties of this class of functions by considering them as special cases. Finally, we support our findings with interesting examples and graphical representations.

Fast numerical scheme for the time-fractional option pricing model with asset-price-dependent variable order

We provide a fast numerical technique for a time-fractional option pricing model with asset-price-dependent variable order. Due to the complicated variable-order fractional derivative and its related fast approximations, the temporal coefficients are coupled with the inner product of the finite element method and lose monotonicity, which introduces uncommon difficulties in numerical analysis. In addition, the Riemann-Liouville fractional operators are often used in option pricing models, but its variable-order case gets far less attention than the corresponding Caputo-type problems. We prove error estimates for the proposed fast method and show that the computational cost is almost linear with respect to the temporal steps, which is much faster than the quadratic growth of the time-stepping solver. Numerical experiments are performed to illustrate the theoretical findings.

On Novel Fractional Integral and Differential Operators and Their Properties

The main goal of this paper is to describe the new version of extended Bessel–Maitland function and discuss its special cases. Then, using the aforementioned function as their kernels, we develop the generalized fractional integral and differential operators. The convergence and boundedness of the newly operators and compare them with the existing operators such as the Saigo and Riemann–Liouville fractional operators are explored. The integral transforms of newly defined and generalized fractional operators in terms of the generalized Fox–Wright function are presented. Additionally, we discuss a few exceptional cases of the main result.

Exponentially Convex Functions on the Coordinates and Novel Estimations via Riemann-Liouville Fractional Operator

In this study, the modification of the concept of exponentially convex function, which is a general version of convex functions, given on the coordinates, is recalled. With the help of an integral identity which includes the Riemann-Liouville (RL) fractional integral operator, new Hadamard-type inequalities are proved for exponentially convex functions on the coordinates. Many special cases of the results are discussed.

Operational matrix-based technique treating mixed type fractional differential equations via shifted fifth-kind Chebyshev polynomials

The theory of mixed fractional operators is still an uncovered area in fractional modelling. These multi-sided operators result by combining two fractional derivatives with different kernels, that is, the right-sided Caputo's and the left-sided Riemann–Liouville's fractional operators in a sequential manner. Although it may capture different memory phenomenons, there is no general approach to approximate the solution of the mixed type fractional differential equations (MFDEs). In this article, the authors introduce a novel numerical technique that relies on obtaining the matrix representation of the mixed operators. This was done by considering a new extension of the shifted Chebyshev polynomials of the fifth-kind (SFCPs) with a variable domain as a basis of space. Then we employ the spectral collocation method together with the operational matrix to transform the MFDE into the corresponding system of algebraic equations. The convergence analysis of the proposed technique was studied in terms of the generalized Taylor's formula and the Gram determinant. Our findings facilitate further development of this theory by providing such a method of approximation.

Hermite Hadamard Type Inequalities Involving (k-p) Fractional Operator with (α, h − m) − p convexity

We establish various fractional convex inequalities of the Hermite-Hadamard type which generalize the previously obtained results in the literature. Various types of such inequalities are obtained and given as corollaries. The main motivation of the paper is to generalize the recently published results in terms of the (α, h − m) − p convexity with k-p Riemann Liouville fractional operator. The application of H¨olders inequality is given in tandem with the k-p fractional operator of the convex type.

Solvability of a mixed problem with the integral gluing condition for a loaded equation with the Riemann–Liouville fractional operator

In this paper, we study boundary value problems with an integral gluing condition for a loaded equation of parabolic–hyperbolic type. The existence and uniqueness of the problem under study are proved based on the unique solvability obtained from integral and loaded integral equations.

An effective method for solving nonlinear integral equations involving the Riemann-Liouville fractional operator

&lt;abstract&gt;&lt;p&gt;In this paper, under some conditions in the Banach space $ C ([0, \beta], \mathbb{R}) $, we establish the existence and uniqueness of the solution for the nonlinear integral equations involving the Riemann-Liouville fractional operator (RLFO). To establish the requirements for the existence and uniqueness of solutions, we apply the Leray-Schauder alternative and Banach's fixed point theorem. We analyze Hyers-Ulam-Rassias (H-U-R) and Hyers-Ulam (H-U) stability for the considered integral equations involving the RLFO in the space $ C([0, \beta], \mathbb{R}) $. Also, we propose an effective and efficient computational method based on Laguerre polynomials to get the approximate numerical solutions of integral equations involving the RLFO. Five examples are given to interpret the method.&lt;/p&gt;&lt;/abstract&gt;

New developments in fractional integral inequalities via convexity with applications

&lt;abstract&gt;&lt;p&gt;The main objective of this article is to build up a new integral equality related to Riemann Liouville fractional (RLF) operator. Based on this integral equality, we show numerous new inequalities for differentiable convex as well as concave functions which are similar to celebrated Hermite-Hadamard and Simpson's integral inequalities. The present outcomes of this paper are a unification and generalization of the comparable results in the literature on Hermite-Hadamard and Simpson's integral inequalities. Furthermore as applications in numerical analysis, we find some means, q-digamma function and modified Bessel function type inequalities.&lt;/p&gt;&lt;/abstract&gt;

Semigroup property of fractional differential operators and its applications

&lt;p style='text-indent:20px;'&gt;We establish partial semigroup property of families of Riemann-Liouville and Caputo fractional differential operators. Using this result we prove theorems on reduction of multi-term fractional differential systems to single-term and multi-order systems. As an application we obtain existence and uniqueness of solution to multi-term Caputo fractional differential systems.&lt;/p&gt;