Riemann-Liouville Fractional Integrals Research Articles

Hermite–Hadamard-type Inequalities for hbar-preinvex Interval-Valued Functions via Fractional Integral

We present a comprehensive study on Hermite–Hadamard-type inequalities for interval-valued functions that are hbar-preinvex, using the Riemann–Liouville fractional integral. Our research extends and generalizes some existing results found in the literature. In addition, we provide accurate proofs for the main theorems originally derived by Srivastava et al. in their publication titled “Hermite–Hadamard Type Inequalities for Interval-Valued Preinvex Functions via Fractional Integral Operators" (Int. J. Comput. Int. Sys. 15(1):8, 2022). Finally, we illustrate our findings through a practical example to demonstrate the validity of our results.

A Comprehensive Review on the Fejér-Type Inequality Pertaining to Fractional Integral Operators

A review of the results on the fractional Fejér-type inequalities, associated with different families of convexities and different kinds of fractional integrals, is presented. In the numerous families of convexities, it includes classical convex functions, s-convex functions, quasi-convex functions, strongly convex functions, harmonically convex functions, harmonically quasi-convex functions, quasi-geometrically convex functions, p-convex functions, convexity with respect to strictly monotone function, co-ordinated-convex functions, (θ,h−m)−p-convex functions, and h-preinvex functions. Included in the fractional integral operators are Riemann–Liouville fractional integral, (k−p)-Riemann–Liouville, k-Riemann–Liouville fractional integral, Riemann–Liouville fractional integrals with respect to another function, the weighted fractional integrals of a function with respect to another function, fractional integral operators with the exponential kernel, Hadamard fractional integral, Raina fractional integral operator, conformable integrals, non-conformable fractional integral, and Katugampola fractional integral. Finally, Fejér-type fractional integral inequalities for invex functions and (p,q)-calculus are also included.

A Second-Order Difference Scheme for Solving a Class of Fractional Differential Equations

Introduction. Increasing accuracy in the approximation of fractional integrals, as is known, is one of the urgent tasks of computational mathematics. The purpose of this study is to create and apply a second-order difference analog to approximate the fractional Riemann-Liouville integral. Its application is investigated in solving some classes of fractional differential equations. The difference analog is designed to approximate the fractional integral with high accuracy.Materials and Methods. The paper considers a second-order difference analogue for approximating the fractional Riemann-Liouville integral, as well as a class of fractional differential equations, which contains a fractional Caputo derivative in time of the order belonging to the interval (1, 2).Results. To solve the above equations, the original fractional differential equations have been transformed into a new model that includes the Riemann-Liouville fractional integral. This transformation makes it possible to solve problems efficiently using appropriate numerical methods. Then the proposed difference analogue of the second order approximation is applied to solve the transformed model problem.Discussion and Conclusions. The stability of the proposed difference scheme is proved. An a priori estimate is obtained for the problem under consideration, which establishes the uniqueness and continuous dependence of the solution on the input data. To evaluate the accuracy of the scheme and verify the experimental order of convergence, calculations for the test problem were carried out.

Properties of a subclass of analytic functions defined by Riemann-Liouville fractional integral applied to convolution product of multiplier transformation and Ruscheweyh derivative

Abstract The contribution of fractional calculus in the development of different areas of research is well known. This article presents investigations involving fractional calculus in the study of analytic functions. Riemann-Liouville fractional integral is known for its extensive applications in geometric function theory. New contributions were previously obtained by applying the Riemann-Liouville fractional integral to the convolution product of multiplier transformation and Ruscheweyh derivative. For the study presented in this article, the resulting operator is used following the line of research that concerns the study of certain new subclasses of analytic functions using fractional operators. Riemann-Liouville fractional integral of the convolution product of multiplier transformation and Ruscheweyh derivative is applied here for introducing a new class of analytic functions. Investigations regarding this newly introduced class concern the usual aspects considered by researchers in geometric function theory targeting the conditions that a function must meet to be part of this class and the properties that characterize the functions that fulfil these conditions. Theorems and corollaries regarding neighborhoods and their inclusion relation involving the newly defined class are stated, closure and distortion theorems are proved, and coefficient estimates are obtained involving the functions belonging to this class. Geometrical properties such as radii of convexity, starlikeness, and close-to-convexity are also obtained for this new class of functions.

Bernoulli wavelet least squares support vector regression: Robust numerical method for systems of fractional differential equations

In this study, a new hybrid method is developed to solve linear or nonlinear systems of fractional differential equations using Bernoulli wavelets (Bws) and the least squares support vector regression (LS‐SVR). The numerical methods based on operational matrices for solving various kinds of fractional equations have been widely studied in the last decade. In contrast to the existing methods, here we derive the operator of fractional integration, aiming to remove the approximation error. For this purpose, we present Bw operator of Riemann–Liouville fractional integration and use it in our scheme. In the proposed technique, we approximate the unknown functions via Bws, and then with the help of Bw operator of fractional integration and LS‐SVR, we reduce the problem to an algebraic system. In this way, we simplify the computation of the considered system. The error analysis of our method is proposed. Finally, we demonstrate the applicability of the present scheme by solving several numerical examples.

A Note on Fractional Simpson Type Inequalities for Twice Differentiable Functions

ABSTRACT In this paper, an equality is proved for twice differentiable convex functions involving Riemann–Liouville fractional integral. With the help of this equality, there are established several fractional Simpson type inequalities for functions whose second derivatives in absolute value are convex. By using special cases of the main results, previously obtained Simpson type inequalities are found for the Riemann–Liouville fractional integral.

A Cotangent Fractional Derivative with the Application

In this work, we present a new type of fractional derivatives (FD) involving exponential cotangent function in their kernels called Riemann–Liouville Dσ,γ and Caputo cotangent fractional derivatives CDσ,γ, respectively, and their corresponding integral Iσ,γ. The advantage of the new fractional derivatives is that they achieve a semi-group property, and we have special cases; if γ=1 we obtain the Riemann–Liouville FD (RL-FD), Caputo FD (C-FD), and Riemann–Liouville fractional integral (RL-FI). We give some theorems and lemmas, and we give solutions to linear cotangent fractional differential equations using the Laplace transform of the Dσ,γ, CDσ,γ and Iσ,γ. Finally, we give the application of this new type on the SIR model. This new type of fractional calculus can help other researchers who still work on the actual subject.

New Versions of Midpoint Inequalities Based on Extended Riemann–Liouville Fractional Integrals

This study aims to prove some midpoint-type inequalities for fractional extended Riemann–Liouville integrals. Crucial equality is proven to build new results. Using this equality, several midpoint-type inequalities are established via differentiable convex functions and the proposed extended fractional operators. To be more specific, the well-known Hölder, Jensen, and power mean integral inequalities are employed in the demonstrated inequalities. Additionally, many remarks based on specific selections of the main results are presented. Moreover, to illustrate the key conclusions, a few instances are provided.

Hybrid fractional-order optimal control problem for immuno-chemotherapy with gene therapy and time-delay: numerical treatments

ABSTRACT This paper investigates the optimal control problem for the tumor-immune system with time-delay in the presence of gene therapy and immuno chemotherapy. Riemann-Liouville fractional order integrals and Caputo fractional order derivatives are combined to form a hybrid fractional order operator. For consistency with the physical model problem, a new parameter is presented. A stability and bifurcation analysis of the proposed model is performed. The positivity, boundedness, and existence of optimal control for the proposed model are discussed. Grünwald-Letnikov nonstandard finite difference method with the discretization of the hybrid fractional order operator is constructed to solve the proposed model. Examples and comparative studies are presented to demonstrate the simplicity of the approximation approaches and the applicability of the utilized methods.

Controlled S-Metric-Type Spaces and Applications to Fractional Integrals

In this paper, we introduce controlled S-metric-type spaces and give some of their properties and examples. Moreover, we prove the Banach fixed point theorem and a more general fixed point theorem in this new space. Finally, using the new results, we give two applications on Riemann–Liouville fractional integrals and Atangana–Baleanu fractional integrals.

On the numerical solution of fractional differential equations under white noise processes

The aim of this paper is to elucidate some relevant aspects concerning the numerical solution of stochastic differential equations in structural and mechanical applications. Specifically, the attention is focused on those differential problems involving fractional operators to model the viscoelastic behavior of the structural/mechanical components and involving a white noise process as stochastic input. Starting from the consideration that the Grünwald–Letnikov based integration scheme, that is a step-by-step procedure often invoked in literature to discretize and integrate the aforementioned differential equations, is not properly employed due to the discontinuous nature of the input, an alternative numerical integration scheme is proposed. The latter is based on the Riemann–Liouville fractional integral and relies on the parabolic piecewise approximation of the response function to be integrated, leading to a more effective and more advantageous solution than that provided by the Grünwald–Letnikov based integration scheme. This is demonstrated analyzing the case study of a fractional Euler–Bernoulli beam and comparing the numerical results with those obtained by an analytical solution available in literature.

Optimal error analysis of space–time second-order difference scheme for semi-linear non-local Sobolev-type equations with weakly singular kernel

In this paper, we construct and analyze a Crank–Nicolson difference scheme for solving the semilinear Sobolev-type equation with the Riemann–Liouville fractional integral of order α∈(0,1). The proposed scheme consists of two main stages. First, to compensate for the singular behavior of the exact solution at the initial time t=0, a Crank–Nicolson scheme and the product trapezoidal integration rule are proposed on graded meshes for time discretization. Second, a classical finite difference operator on uniform meshes is used for space discretization. Under suitable assumptions of the regularity condition, the stability and convergence analysis of this scheme in a new norm are given by the energy argument. It is shown that how the mesh grading exponent γ affects the temporal convergence rate of the presented scheme in the final convergence result. This scheme can attain a second-order accuracy in both time and space by choosing an optimal grading parameter γ. Numerical results confirm that the error analysis is sharp, and comparisons with results on uniform temporal meshes are included to indicate the effectiveness of our method.

The n-Point Composite Fractional Formula for Approximating Riemann–Liouville Integrator

In this paper, we aim to present a novel n-point composite fractional formula for approximating a Riemann–Liouville fractional integral operator. With the use of the definite fractional integral’s definition coupled with the generalized Taylor’s formula, a novel three-point central fractional formula is established for approximating a Riemann–Liouville fractional integrator. Such a new formula, which emerges clearly from the symmetrical aspects of the proposed numerical approach, is then further extended to formulate an n-point composite fractional formula for approximating the same operator. Several numerical examples are introduced to validate our findings.

Weighted Hardy–Littlewood–Sobolev-type inequality for ψ-Riemann–Liouville fractional integrals

In this paper, we present a new extension for Hardy–Littlewood-type inequalities using the Riemann–Liouville fractional integral with respect to another function in the weighted Lebesgue space Lψp[a,b]. These estimates are very important in the study of fractional differential equations with variational form and in the construction of fractional spaces of Sobolev type.

FRACTIONAL HERMITE–HADAMARD INEQUALITY, SIMPSON’S AND OSTROWSKI’S TYPE INEQUALITIES FOR CONVEX FUNCTIONS WITH RESPECT TO A PAIR OF FUNCTIONS

We consider the convexity with respect to a pair of functions and establish a Hermite–Hadamard type inequality for Riemann–Liouville fractional integrals. Moreover, we derive some new Simpson’s and Ostrowski’s type inequalities for differentiable convex mapping with respect to a pair of functions. We also show that the newly established inequalities are the extension of some existing inequalities. Finally, we consider some mathematical examples and graphs to show the validity of the newly established inequalities.

Vector-valued fractal functions: Fractal dimension and fractional calculus

There are many research available on the study of a real-valued fractal interpolation function and fractal dimension of its graph. In this paper, our main focus is to study the dimensional results for a vector-valued fractal interpolation function and its Riemann–Liouville fractional integral. Here, we give some results which ensure that dimensional results for vector-valued functions are quite different from real-valued functions. We determine interesting bounds for the Hausdorff dimension of the graph of a vector-valued fractal interpolation function. We also obtain bounds for the Hausdorff dimension of the associated invariant measure supported on the graph of a vector-valued fractal interpolation function. Next, we discuss more efficient upper bound for the Hausdorff dimension of measure in terms of probability vector and contraction ratios. Furthermore, we determine some dimensional results for the graph of the Riemann–Liouville fractional integral of a vector-valued fractal interpolation function.

Hermite-Hadamard Like Inequalities for Exponentially Subadditive Functions via Fractional Integrals

In this work, through the Riemann-Liouville fractional integrals, we give Hermite-Hadamard type inequalities for exponentially sub-additive functions. For the product of exponentially sub-additive functions, we present fractional integral inequalities. It is also shown that the results proved here are the refinements and extensions of several existing results in the field of Hermite-Hadamard like inequalities.

Analytical and dimensional properties of fractal interpolation functions on the Sierpiński gasket

In this article, we construct the fractal interpolation functions (FIFs) on the Sierpiński gasket (SG) by taking data set at the nth level. We discuss the oscillation space, $$\mathcal {L}_q$$ space and some other function spaces on SG, and determine some conditions under which, this FIF belongs to these spaces. We obtain the fractal dimension of the graph of the FIF and the Hausdorff dimension of the invariant measure supported on the graph of the FIF. We also prove that this FIF has finite energy. After that, we discuss the restrictions of the FIF on the bottom line segment of SG and prove that the restriction is again an FIF on the bottom line. We estimate the fractal dimension of the graph of the Riemann-Liouville fractional integral of the restrictions of the FIF on the bottom line segment of SG. Lastly, we prove that the Hausdorff and box-counting dimension of the graph of the restriction of any harmonic function on the bottom line segment of SG are 1 and for the Riemann- Liouville fractional integral of this restriction function, these dimensions are also 1.

Some new integral inequalities for higher-order strongly exponentially convex functions

Integral inequalities with generalized convexity play an important role in both applied and theoretical mathematics. The theory of integral inequalities is currently one of the most rapidly developing areas of mathematics due to its wide range of applications. In this paper, we study the concept of higher-order strongly exponentially convex functions and establish a new Hermite–Hadamard inequality for the class of strongly exponentially convex functions of higher order. Further, we derive some new integral inequalities for Riemann–Liouville fractional integrals via higher-order strongly exponentially convex functions. These findings include several well-known results and newly obtained results as special cases. We believe that the results presented in this paper are novel and will be beneficial in encouraging future research in this field.

A Meyer-Itô formula for stable processes via fractional calculus

The infinitesimal generator of a one-dimensional strictly alpha -stable process can be represented as a weighted sum of (right and left) Riemann-Liouville fractional derivatives of order alpha and one obtains the fractional Laplacian in the case of symmetric stable processes. Using this relationship, we compute the inverse of the infinitesimal generator on Lizorkin space, from which we can recover the potential if alpha in (0,1) and the recurrent potential if alpha in (1,2). The inverse of the infinitesimal generator is expressed in terms of a linear combination of (right and left) Riemann-Liouville fractional integrals of order alpha . One can then state a class of functions that give semimartingales when applied to strictly stable processes and state a Meyer-Itô theorem with a non-zero (occupational) local time term, providing a generalization of the Tanaka formula given by Tsukada [1]. This result is used to find a Doob-Meyer (or semimartingale) decomposition for |X_t - x|^{gamma } with X a recurrent strictly stable process of index alpha and gamma in (alpha -1,alpha ), generalizing the work of Engelbert and Kurenok [2] to the asymmetric case.