Riemann-Liouville Fractional Integrals Research Articles

Fractional integral inequalities for s-convex functions in the first sense

Certain fractional integral inequalities via general fractional integral operator are presented for s-convex functions in the first sense. Also, they are restated in terms of the Riemann-Liouville and Hadamard fractional integrals and illustrated.

Some Results of R-Matrix Functions and Their Fractional Calculus

In this study, we explore various fractional integral properties of R-matrix functions using the Hilfer fractional derivative operator within the framework of fractional calculus. We introduce the θ integral operator and extend its definition to include the R matrix functions. The composition of Riemann–Liouville fractional integral and differential operators is determined using the θ-integral operator. Additionally, we investigate the compositional properties of θ-integral operators, and we establish their inversion, offering new insights into their structural and functional characteristics.

Fixed Point Theorems on Controlled Orthogonal δ‐Metric‐Type Spaces and Applications to Fractional Integrals

In this article, we introduce a notion of controlled orthogonal δ‐metric‐type spaces with an example. Further, we prove a contraction theorem and a generalized fixed point theorem in controlled orthogonal δ‐metric‐type spaces. Finally, we illustrate two applications of the obtained fixed point results on the Atangana–Baleanu fractional integrals and the Riemann–Liouville fractional integrals. These applications showcase the versatility and efficacy of the developed theoretical framework in addressing real‐world mathematical problems.

Some novel inequalities of Weddle’s formula type for Riemann–Liouville fractional integrals with their applications to numerical integration

Some novel inequalities of Weddle’s formula type for Riemann–Liouville fractional integrals with their applications to numerical integration

Fractional Inequalities for Exponentially s-Convex Functions on Time Scales

In this paper, we present new integral inequalities involving exponentially s-convex functions in the second sense on time scales. By utilizing the delta Riemann-Liouville fractional integral and the fractional Taylor formula, we establish upper bounds for functions that are n-times rd-continuously Δ-differentiable with exponentially s-convex properties. Our results provide novel insights into the theory of time scales, bridging the gap between discrete and continuous calculus. The application of fractional calculus on time scales is explored, and several well-known inequalities are employed to derive the main findings. These results have potential implications for further studies in fractional dynamic calculus and other related fields. AMS Subject Classification: 39A10, 39A11, 39A20.

A new Approach of Generalized Fractional Integrals in Multiplicative Calculus and Related Hermite–Hadamard-Type Inequalities with Applications

Abstract The primary goal of this paper is to define Katugampola fractional integrals in multiplicative calculus. A novel method for generalizing the multiplicative fractional integrals is the Katugampola fractional integrals in multiplicative calculus. The multiplicative Hadamard fractional integrals are also novel findings of this research and may be derived from the special situations of Katugampola fractional integrals. These integrals generalize to multiplicative Riemann–Liouville fractional integrals and multiplicative Hadamard fractional integrals. Moreover, we use the Katugampola fractional integrals to prove certain new Hermite–Hadamard and trapezoidal-type inequalities for multiplicative convex functions. Additionally, it is demonstrated that several of the previously established inequalities are generalized from the newly derived inequalities. Finally, we give some computational analysis of the inequalities proved in this paper.

The Essential Gronwall Inequality Demands the $\left(\rho,\varphi \right) -$Fractional Operator with Applications in Economic Studies

Gronwall's inequalities are important in the study of differential equations and integral inequalities. Gronwall inequalities are a valuable mathematical technique with several applications. They are especially useful in differential equation analysis, stability research, and dynamic systems modeling in domains spanning from science and math to biology and economics. In this paper, we present new generalizations of Gronwall inequalities of integral versions. The proposed results involve $( \rho ,\varphi)-$Riemann-Liouville fractional integral with respect to another function. Some applications on differential equations involving $( \rho ,\varphi)-$Riemann-Liouville fractional integrals and derivatives are established.

Error Analysis for Semilinear Stochastic Subdiffusion with Integrated Fractional Gaussian Noise

We analyze the error estimates of a fully discrete scheme for solving a semilinear stochastic subdiffusion problem driven by integrated fractional Gaussian noise with a Hurst parameter H∈(0,1). The covariance operator Q of the stochastic fractional Wiener process satisfies ∥A−ρQ1/2∥HS < ∞ for some ρ∈[0,1), where ∥·∥HS denotes the Hilbert–Schmidt norm. The Caputo fractional derivative and Riemann–Liouville fractional integral are approximated using Lubich’s convolution quadrature formulas, while the noise is discretized via the Euler method. For the spatial derivative, we use the spectral Galerkin method. The approximate solution of the fully discrete scheme is represented as a convolution between a piecewise constant function and the inverse Laplace transform of a resolvent-related function. By using this convolution-based representation and applying the Burkholder–Davis–Gundy inequality for fractional Gaussian noise, we derive the optimal convergence rates for the proposed fully discrete scheme. Numerical experiments confirm that the computed results are consistent with the theoretical findings.

Novel inequalities involving exponentially (α, m)-convex function and fractional integrals

In this paper, an identity involving Riemann–Liouville fractional integrals is newly produced. Some new extensions of integral inequalities are proved with the help of Hölder, power-mean integral inequalities and newly established identity. Whereas, the absolute values of first derivatives of real valued functions, which are appeared in these inequalities, are exponentially ( α , m ) -convex. Some special cases are discussed and graphical evidences are presented for validity of proved results.

The Riemann-Liouville fractional integral in Bochner-Lebesgue spaces III

In this manuscript, we examine the continuity properties of the Riemann-Liouville fractional integral for order α=1/p, where p>1, mapping from Lp(t0,t1;X) to the Banach space BMO(t0,t1;X)∩K(p−1)/p(t0,t1;X). This improvement, refines a result by Hardy-Littlewood. To achieve this, we study properties between spaces BMO(t0,t1;X) and K(p−1)/p(t0,t1;X). Additionally, we obtained the boundedness of the fractional integral of order α≥1 from L1(t0,t1;X) into the Riemann-Liouville fractional Sobolev space WRLs,p(t0,t1;X).

NEW FRACTIONAL INTEGRAL INEQUALITIES FOR EXPONENTIALLY (h,m)-PREINVEX INTERVAL-VALUED FUNCTIONS

We introduce the concept of exponentially [Formula: see text]-preinvex interval-valued functions (EP-([Formula: see text])-IVFs) by using the pseudo-order relation, and derive some interesting properties. Based on such a definition, we present some new Hermite–Hadamard and Hermite–Hadamard–Fejér-type inequalities for [Formula: see text]-Riemann–Liouville fractional integrals. These inequalities not only generalize some existing research findings in the literature, but also provide a new direction for the further study of interval integral estimation. Additionally, graphical representations and numerical estimates of some examples are given to illustrate the validity of the main results.

ANALYSIS ON MULTIPLICATIVELY (P,m)-SUPERQUADRATIC FUNCTIONS AND RELATED FRACTIONAL INEQUALITIES WITH APPLICATIONS

In this work, we, for the first time, establish a class of multiplicatively [Formula: see text]-superquadratic function and look into its various features. In the light of these features, we come up with the several integer order integral inequalities in the frame of multiplicative calculus. Moreover, we develop the fractional version of Hermite–Hadamard’s type inequalities involving midpoints and end points for multiplicatively [Formula: see text]-superquadratic function with respect to multiplicatively [Formula: see text]-Riemann–Liouville fractional integrals. By choosing different values for the parameters of such integral operators, we acquire a simple version of integral inequalities of Hermite–Hadamard’s type as well as its fractional form via multiplicatively Riemann–Liouville fractional integrals for multiplicatively [Formula: see text]-superquadratic function. The findings are confirmed by graphical illustration by taking appropriate examples into account. The study is further enhanced by the addition of applications of special means and first-type modified Bessel functions. The new results clearly provide extensions and improvements of the work available in the literature.

Fractional Milne-type inequalities for twice differentiable functions for Riemann–Liouville fractional integrals

Fractional Milne-type inequalities for twice differentiable functions for Riemann–Liouville fractional integrals

A novel explicit fast numerical scheme for the Cauchy problem for integro-differential equations with a difference kernel and its application

The present study focuses on designing a second-order novel explicit fast numerical scheme for the Cauchy problem incorporating memory associated with an evolutionary equation, where the integral term's kernel is a discrete difference operator. The Cauchy problem under consideration is related to a real finite-dimensional Hilbert space and includes a self-adjoint operator that is both positive and definite. We introduce a transformative technique for converting the Cauchy problem incorporating memory, into a local evolutionary system of equations by approximating the difference kernel using the sum of exponentials (SoE) approach. A second-order explicit scheme is then constructed to solve the local system. We thoroughly investigate the stability of this explicit scheme, and present the necessary conditions for the stability of the scheme. Moreover, we extended our investigation to encompass time-fractional diffusion-wave equations (TFDWEs) involving a fractional Caputo derivative with an order ranging between (1,2). Initially, we transform the main TFDWE model into a new model that incorporates the fractional Riemann-Liouville integral. Subsequently, we expand the applicability of our idea to develop an explicit fast numerical algorithm for approximating the model. The stability properties of this fast scheme for solving TFDWEs are assessed. Numerical simulations including a two-dimensional Cauchy problem as well as one-dimensional and two-dimensional TFDWE models are provided to validate the accuracy and experimental order of convergence of the schemes.

Hermite–Hadamard inequalities for Riemann–Liouville fractional integrals

Abstract In this paper, we prove some new inequalities of Hermite–Hadamard type for differentiable functions with h-convex derivatives. It is also shown that the newly established inequalities are extension of the existing inequalities in the literature. Finally, we give applications of the new results and outline some future problems.

Fractional Sobolev type spaces of functions of two variables via Riemann-Liouville derivatives

We introduce and study the spaces of fractionally absolutely continuous functions of two variables of any order and the fractional Sobolev type spaces of functions of two variables. Our approach is based on the Riemann-Liouville fractional integrals and derivatives. We investigate relations between these spaces as well as between the Riemann-Liouville and weak derivatives.

Hermite–Hadamard-type inequalities arising from tempered fractional integrals including twice-differentiable functions

UDC 517.5 We propose a new method for the investigation of integral identities according to tempered fractional operators. In addition, we prove the midpoint-type and trapezoid-type inequalities by using twice-differentiable convex functions associated with tempered fractional integral operators. We use the well-known Hölder inequality and the power-mean inequality in order to obtain inequalities of these types. The resulting Hermite–Hadamard-type inequalities are generalizations of some investigations in this field, involving Riemann–Liouville fractional integrals.

Improved Hermite–Hadamard Inequality Bounds for Riemann–Liouville Fractional Integrals via Jensen’s Inequality

Improved Hermite–Hadamard Inequality Bounds for Riemann–Liouville Fractional Integrals via Jensen’s Inequality

On Newton–Cotes Formula-Type Inequalities for Multiplicative Generalized Convex Functions via Riemann–Liouville Fractional Integrals with Applications to Quadrature Formulas and Computational Analysis

In this article, we develop multiplicative fractional versions of Simpson’s and Newton’s formula-type inequalities for differentiable generalized convex functions with the help of established identities. The main motivation for using generalized convex functions lies in their ability to extend results beyond traditional convex functions, encompassing a broader class of functions, and providing optimal approximations for both lower and upper bounds. These inequalities are very useful in finding the error bounds for the numerical integration formulas in multiplicative calculus. Applying these results to the Quadrature formulas demonstrates their practical utility in numerical integration. Furthermore, numerical analysis provides empirical evidence of the effectiveness of the derived findings. It is also demonstrated that the newly proven inequalities extend certain existing results in the literature.

An effective numerical method for solving fractional delay differential equations using fractional-order Chelyshkov functions

In this paper, the fractional-order Chelyshkov functions (FCHFs) and Riemann-Liouville fractional integrals are utilized to find numerical solutions to fractional delay differential equations, by transforming the problem into a system of algebraic equations with unknown FCHFs coefficients. An error bound of FCHFs approximation is estimated and its convergence is also demonstrated. The effectiveness and accuracy of the presented method are established through several examples. The resulting solution is accurate and agrees with the exact solution, even if the exact solution is not a polynomial. Moreover, comparisons between the obtained numerical results and those recently reported in the literature are shown.