Riemann-Liouville Operator Research Articles

Extremal Functions and Calderon’s Formulas for the Riemann-Liouville Two-wavelet Transform

The Riemann-Liouville operator has been extensively investigated and his witnessed a remarkable development in numerous fields of harmonic analysis. Knowing the fact of the study of the time-frequency analysis are both theoritically interesting and pratically useful, we investigated several problems for this subject on the setting of the Riemann-Liouville wavelet transform. Firstly, we introduce the notion of Riemann-Liouville two-wavelet and we present generalized version of Parseval’s, Plancherel’s, inversion and Calderon’s reproducing formulas. Next, using the theory of reproducing kernels, we give best estimates and an integral representation of the extremal functions related to the Riemann-Liouville wavelet transform on weighted Sobolev spaces.

On proportional hybrid operators in the discrete setting

In this article, we introduce a new nonlocal operator defined as a linear combination of the discrete fractional Caputo operator and the fractional sum operator. A new dual operator is also introduced by replacing the discrete fractional Caputo operator with the discrete fractional Riemann–Liouville operator. It is shown that it corresponds to a natural discretization of a proportional hybrid operator defined by the Riemann–Liouville operator instead of Caputo hybrid operator. We then analyze the most important properties of these operators, such as their inverse operator and the ‐transform, among others. As an application, we solve difference equations equipped with these operators and obtain explicit solutions for them in terms of trivariate Mittag‐Leffler sequences.

A study on the solvability of fractional integral equation in a Banach algebra via Petryshyn's fixed point theorem

This study focuses on the nonlinear fractional functional integral equation (FFIE) concerning the Riemann-Liouville operator. In certain weaker conditions, the authors demonstrate that the FFIE has a solution, which is defined within the Banach algebra 0 $ ]]> C [ 0 , a ] , a > 0 . Our analysis relies on the Petryshyn's fixed point theorem and the notion of measure of non-compactness (MNC). In addition, our results include numerous authors' work under less restrictive conditions. Furthermore, we provide an illustrative example of fractional functional integral equations to support our proven results.

Theoretical Results on Positive Solutions in Delta Riemann–Liouville Setting

This article primarily focuses on examining the existence and uniqueness analysis of boundary fractional difference equations in a class of Riemann–Liouville operators. To this end, we firstly recall the general solution of the homogeneous fractional operator problem. Then, the Green function to the corresponding fractional boundary value problems will be reconstructed, and homogeneous boundary conditions are used to find the unknown constants. Next, the existence of solutions will be studied depending on the fixed-point theorems on the constructed Green’s function. The uniqueness of the problem is also derived via Lipschitz constant conditions.

On the spaces introduced by N. K. Karapetyants and B. S. Rubin, and their connection with Lorentz–Zygmund spaces

We consider spaces introduced by N. K. Karapetyants and B. S. Rubin in 1982, to characterize, in particular, the image of the fractional integral Riemann–Liouville operator. These spaces lie near L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${L}^{\\infty }$$\\end{document}. We show that they coincide with well-known Lorentz–Zygmund spaces. This allows us to reformulate one result from N. K. Karapetyants and B. S. Rubin dealing with Riemann–Liouville fractional integral operator J0+α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${J}_{0+}^{\\alpha }$$\\end{document} defined on Lp0,1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${L}^{p}\\left(\ ext{0,1}\\right)$$\\end{document} (1<p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1<p<\\infty$$\\end{document}) in the borderline case α=1/p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha =1/p$$\\end{document}. Using of the well-developed theory of Lorentz–Zygmund spaces leads to new results on the fractional integral Riemann–Liouville operator.

Bicomplex Caputo Derivative: A Comparative Study with Bicomplex Riemann–Liouville Operators and Applications

Bicomplex Caputo Derivative: A Comparative Study with Bicomplex Riemann–Liouville Operators and Applications

Analytical insights into a fractional thin-film equation: exact solutions and dynamics

In this work, we examine a quadratic thin film equation with a constant negative absorption term. This equation extends a broad variety of the famous scalar reaction-diffusion equations appearing in nonlinear sciences and is derived from the estimations of lubrication theory to represent thin films of a Newtonian liquid dominated by surface tension effects. It is typically used to describe the behavior of light when it interacts with thin films, such as coatings on lenses or mirrors. The connection between thin film equations and optical quantum mechanics lies in the microscopic interactions between photons and the electrons in the thin film material. Employing the invariant subspace approach, we obtain explicit fractional exact solutions for the time-fractional case of the model containing the Riemann-Liouville derivative operator. Furthermore, we illustrate 3-D and 2-D plots of the obtained exact solutions for a better understanding of the physical phenomena.

A note on Kirchhoff type boundary value problem involving Riemann-Liouville fractional derivative

In this paper, we study some nonlinear Kirchhoff boundary value problems of fractional differential equations involving Riemann Liouville operator. Under appropriate assumptions on the functions in the given problem, we establish the existence of solutions using variational methods combined with the mountain pass theorem. Moreover, an illustrative example is presented to prove the validity of the main result.

Further Fractional Hadamard Integral Inequalities Utilizing Extended Convex Functions

This work investigates novel fractional Hadamard integral inequalities by utilizing extended convex functions and generalized Riemann-Liouville operators. By carefully using extended integral formulations, we not only find novel inequalities but also improve the accuracy of error bounds related to fractional Hadamard integrals. Our study broadens the applicability of these inequalities and shows that they are useful for a variety of convexity cases. Our results contribute to the advancement of mathematical analysis and provide useful information for theoretical comprehension as well as practical applications across several scientific directions.

Time-frequency analysis associated with the generalized Stockwell transform

The Riemann-Liouville operator has been extensively investigated and has witnessed a remarkable development in numerous fields of harmonic analysis. In this paper, we consider the Stockwell transform associated with the Riemann-Liouville operator. Knowing the fact that the study of the time-frequency analysis are both theoretically interesting and practically useful, we investigated several problems for this subject on the setting of this generalized Stockwell transform. Firstly, we study the boundedness and compactness of localization operators associated with the generalized Stockwell transform. Next, we explore the Shapiro uncertainty principle for the previous transform. Finally, the scalogram for the generalized Stockwell transform are introduced and studied at the end.

Fractional $ 3/8 $-Simpson type inequalities for differentiable convex functions

&lt;abstract&gt;&lt;p&gt;The main objective of this study is to establish error estimates of the new parameterized quadrature rule similar to and covering the second Simpson formula. To do this, we start by introducing a new parameterized identity involving the right and left Riemann-Liouville integral operators. On the basis of this identity, we establish some fractional Simpson-type inequalities for functions whose absolute value of the first derivatives are s-convex in the second sense. Also, we examine the special cases $ m = 1/2 $ and $ m = 3/8 $, as well as the two cases $ s = 1 $ and $ \alpha = 1 $, which respectively represent the classical convexity and the classical integration. By applying the definition of convexity, we derive larger estimates that only used the extreme points. Finally, we provide applications to quadrature formulas, special means, and random variables.&lt;/p&gt;&lt;/abstract&gt;

On Cauchy-type problems with weighted R-L fractional derivatives of a function with respect to another function and comparison theorems

&lt;abstract&gt;&lt;p&gt;The main aim of this paper is to study the Cauchy problem for nonlinear differential equations of fractional order containing the weighted Riemann-Liouville fractional derivative of a function with respect to another function. The equivalence of this problem and a nonlinear Volterra-type integral equation of the second kind have been presented. In addition, the existence and uniqueness of the solution to the considered Cauchy problem are proved using Banach's fixed point theorem and the method of successive approximations. Finally, we obtain a new estimate of the weighted Riemann-Liouville fractional derivative of a function with respect to functions at their extreme points. With the assistance of the estimate obtained, we develop the comparison theorems of fractional differential inequalities, strict as well as nonstrict, involving weighted Riemann-Liouville differential operators of a function with respect to functions of order $ \delta $, $ 0 &amp;lt; \delta &amp;lt; 1 $.&lt;/p&gt;&lt;/abstract&gt;

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;

An accelerated novel meshless coupled algorithm for Non-local nonlinear behavior in 2D/3D space-fractional GPEs

In this work, an accelerated novel meshless coupled algorithm is first presented for the numerical study of high-dimensional space-fractional nonlinear Schrödinger or Gross-Pitaevskii equations (SFNLS/GPEs). The proposed meshless method is motivated by the novel coupled concepts: (a) a time splitting-step is adopted to decompose the NLSE; (b) the spatial Riesz fractional derivative under the Riemann-Liouville operator is discretized by the combination of numerical integration and meshless weighted-least-square schemes; (c) an accurate absorbing boundary condition is employed to treat the unbounded domain. Meanwhile, a parallel algorithm based on the above approximated schemes is designed to reduce the computing cost on a CUDA-program-based GPU card. Subsequently, the validation and numerical convergence rate of the proposed method for SFNLSE are discussed and illustrated by solving two 2D/3D examples. Furthermore, the proposed parallel meshless algorithm is extended to predict the nonlinear dynamic behaviors dominated by the space-fractional Gross-Pitaevskii equations (SFGPEs), and the influence of nonlocal effect on the nonlinear phenomena is also discussed. The presented numerical results illustrate that the proposed algorithm for multi-dimensional SFNLSEs is reliability and high-efficient.

A Variation of uncertainty principles for the continuous wavelet transform connected with the Riemann–Liouville operator

A Variation of uncertainty principles for the continuous wavelet transform connected with the Riemann–Liouville operator

Boundary value problem for a degenerate equation with a Riemann-Liouville operator

Boundary value problem for a degenerate equation with a Riemann-Liouville operator

Approximation with continuous functions preserving fractal dimensions of the Riemann-Liouville operators of fractional calculus

Approximation with continuous functions preserving fractal dimensions of the Riemann-Liouville operators of fractional calculus

New integral inequalities involving generalized Riemann-Liouville fractional operators

"In this paper, using a generalized operator of the Riemann-Liouville type, defined and studied in a previous work, several integral inequalities for synchronous functions are established."

Operational calculus for Hilfer-Prabhakar operator Applications to inverse problems

Mikusiński's operational calculus has proven to be a powerful tool for tackling various fractional order differential equations. Notably, it has been extended to encompass the Hilfer-Prabhakar fractional order operator. By employing this operational calculus, we have been able to recover and connect previously derived results pertaining to the Hilfer, Riemann-Liouville, and Caputo operators. Furthermore, we address two inverse problems that involve determining space and time dependent source terms and diffusion concentration. Using our operational calculus results, we obtain series solutions for these inverse problems. It is essential to note that the inverse problems belong to the class of ill-posed problems according to Hadamard's definition. To establish the validity and uniqueness of the solutions to both inverse problems, we rigorously prove the existence and uniqueness results. Additionally, we provide specific examples of inverse problems, thereby illustrating the practical applications and versatility of the operational calculus approach.

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

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