Liouville Fractional Operator Research Articles

Novel Estimations of Hadamard-Type Integral Inequalities for Raina’s Fractional Operators

In the present paper, utilizing a wide class of fractional integral operators (namely the Raina fractional operator), we develop novel fractional integral inequalities of the Hermite–Hadamard type. With the help of the well-known Riemann–Liouville fractional operators, s-type convex functions are derived using the important results. We also note that some of the conclusions of this study are more reasonable than those found under certain specific conditions, e.g., s=1, λ=α, σ(0)=1, and w=0. In conclusion, the methodology described in this article is expected to stimulate further research in this area.

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).

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.

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.

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.

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.

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.

Existence study of semilinear fractional differential equations and inclusions for multi–term problem under Riemann–Liouville operators

In the present paper, we study the existence and uniqueness of the solution for multi–term fractional boundary value problem under Riemann–Liouville fractional operators by using Banach's fixed point theorem. Afterward, we investigate some existence results for a semilinear fractional differential inclusions multi–term problem by using some notions and properties on set‐valued maps together with classical fixed point theorems due to Leray–Schauder. The cases when the set‐valued function has convex as well as nonconvex values are considered. Finally, some examples are given to illustrate our main results.

Analysis of Tempered Fractional Calculus in Hölder and Orlicz Spaces

Here, we propose a general framework covering a wide variety of fractional operators. We consider integral and differential operators and their role in tempered fractional calculus and study their analytic properties. We investigate tempered fractional integral operators acting on subspaces of L1[a,b], such as Orlicz or Hölder spaces. We prove that in this case, they map Orlicz spaces into (generalized) Hölder spaces. In particular, they map Hölder spaces into the same class of spaces. The obtained results are a generalization of classical results for the Riemann–Liouville fractional operator and constitute the basis for the use of generalized operators in the study of differential and integral equations. However, we will show the non-equivalence differential and integral problems in the spaces under consideration.

An accurate method for fractional optimal control problems governed by nonlinear multi-delay systems

This study introduces a novel framework based on a combination of block-pulse and fractional-order Chebyshev functions. The new framework is a generalization of the fractional-order Chebyshev functions called the fractional hybrid functions. An accurate method is designed for solving nonlinear fractional optimal control problems with fractional multi-delay systems. Two essential linear operators, specifically, the fractional derivative operator and the fractional integral operator are introduced by implementing the Caputo and the Riemann–Liouville fractional operators. The two mentioned operators have a fundamental impact on reducing the computational complexity of the problem under study. Furthermore, these operators enable us to simply transform the principal problem into a new optimization one. Due to the structure of the fractional framework, we can construct an accurate solution for an extensive family of fractional multi-delay systems. By using a generalization of Taylor’s theorem, we prove that the proposed framework is convergent. The reliability, feasibility and accuracy of the new fractional framework are validated through examining a wide range of nontrivial examples.

Riemann–Liouville fractional operators of bicomplex order and its properties

In this paper, we construct the Riemann–Liouville fractional integral and differential operator of bicomplex order and illustrate some examples to calculate the fractional integration and differentiation of bicomplex order of some elementary bicomplex‐valued functions. Also, we discuss some properties of these operators by proving analogues of the semigroup property, Leibniz rule, chain rule, etc.

New “Conticrete” Hermite–Hadamard–Jensen–Mercer Fractional Inequalities

The theory of symmetry has a significant influence in many research areas of mathematics. The class of symmetric functions has wide connections with other classes of functions. Among these, one is the class of convex functions, which has deep relations with the concept of symmetry. In recent years, the Schur convexity, convex geometry, probability theory on convex sets, and Schur geometric and harmonic convexities of various symmetric functions have been extensively studied topics of research in inequalities. The present attempt provides novel portmanteauHermite–Hadamard–Jensen–Mercer-type inequalities for convex functions that unify continuous and discrete versions into single forms. They come as a result of using Riemann–Liouville fractional operators with the joint implementations of the notions of majorization theory and convex functions. The obtained inequalities are in compact forms, containing both weighted and unweighted results, where by fixing the parameters, new and old versions of the discrete and continuous inequalities are obtained. Moreover, some new identities are discovered, upon employing which, the bounds for the absolute difference of the two left-most and right-most sides of the main results are established.

A new approach for the generalized fractional Casson fluid model with Newtonian heating described by the modified Riemann–Liouville fractional operator

This research is about the transfer of heat of a generalized fractional Casson fluid on an unsteady boundary layer that is passing through an infinite oscillating plate, in a vertical direction combined with the Newtonian heating. The results are obtained by using a Modified Riemann–Liouville fractional derivative. The present fluid model starts with the governing equations which are then converted to a system of partial differential equations (linear) by using some suitable nondimensional variables. Using the method of integral balance and the Laplace transform technique, an analytical solution is obtained. The velocity and temperature expressions are derived, and the effects of modeling parameters are shown in tables and graphs to validate the obtained theoretical results.

Implicit Hybrid Fractional Boundary Value Problem via Generalized Hilfer Derivative

This research paper is dedicated to the study of a class of boundary value problems for a nonlinear, implicit, hybrid, fractional, differential equation, supplemented with boundary conditions involving general fractional derivatives, known as the ϑ-Hilfer and ϑ-Riemann–Liouville fractional operators. The existence of solutions to the mentioned problem is obtained by some auxiliary conditions and applied Dhage’s fixed point theorem on Banach algebras. The considered problem covers some symmetry cases, with respect to a ϑ function. Moreover, we present a pertinent example to corroborate the reported results.

Numerical solution using radial basis functions for multidimensional fractional partial differential equations of type Black–Scholes

In this paper, as far as the authors know, for the first time, a one-dimensional partial differential model is generalized using fractional differential operators and the same principle that provides the dimensional invariance of the radial basis functions methodology, resulting in a multidimensional fractional partial differential model that can be solved using a numerical scheme of radial basis functions. A radial basis functions scheme is proposed to solve numerically, on different node configurations, multidimensional fractional partial differential equations, both in space and in time. Using the QR factorization, a way to reduce the condition number of the interpolation matrices of the proposed scheme is presented, the resulting scheme is used to numerically solve the diffusion equation that may be obtained from the Black–Scholes model, as well as some generalizations of this diffusion model with fractional differential operators and multiple dimensions. The Caputo fractional derivative is discretized with an order error $${\mathcal {O}}(\mathrm{{d}}t^{n-\alpha +1})$$ , with $$(n-1)<\alpha \le n$$ . The examples of fractional partial differential equations that are presented involve the Caputo fractional operator in the temporal part due to the memory phenomenon, and the Riemann–Liouville fractional operator in the spatial part due to the property of nonlocality.

Hermite–Hadamard Type Inequalities Involving k-Fractional Operator for (h¯,m)-Convex Functions

The principal motivation of this paper is to establish a new integral equality related to k-Riemann Liouville fractional operator. Employing this equality, we present several new inequalities for twice differentiable convex functions that are associated with Hermite–Hadamard integral inequality. Additionally, some novel cases of the established results for different kinds of convex functions are derived. This fractional integral sums up Riemann–Liouville and Hermite–Hadamard’s inequality, which have a symmetric property. Scientific inequalities of this nature and, particularly, the methods included have applications in different fields in which symmetry plays a notable role. Finally, applications of q-digamma and q-polygamma special functions are presented.

The fractional comparative study of the non-linear directional couplers in non-linear optics

A new extended direct algebraic method is applied to extract new traveling wave solutions for non-linear directional couplers with the optical meta-materials. This model studies light distribution from the main fiber into other branch fibers, which can be one or more than one. So, for this reason, twin couplers and multiple core couplers are studied in this paper. Further two cases in multiple core couplers namely multiple core couplers near the neighbor and multiple core couplers with a complete neighbor are exhaustively discussed. The non-linearity type that is considered here is Kerr Law. The obtained solutions set is more generalized than the existing solution as it is transformed into fractional-order derivative because it gives the solutions of problems with heavy tails or infinity fluctuations. Furthermore, the obtained solutions contain almost all types of solitons such as dark, bright, dark-bright, rational, periodic, breather, and singular soliton solutions, etc. Moreover, the obtained solutions are fractional solitons with Atangana-Baleanu and modified Riemann–Liouville fractional operator. These non-linear directional couplers also act as limiters in intensity-dependent switches. The propagation constants of the modes of a system are changed by non-linearity. In order to discuss the physical nature of couplers, 2D and 3D figures are also shown.