Liouville Fractional Integral Operator Research Articles

A fractional hybrid function composed of block-pulses and fractional Fibonacci polynomials for solving multiterm fractional variable-order differential equations

This paper presents the development of a fractional hybrid function composed of block-pulse functions and Fibonacci polynomials (FHBPF) for the numerical solution of multiterm variable-order fractional differential equations. By replacing x→xα in FHBPF and utilizing incomplete beta functions, we construct the method with a focus on fractional derivatives in the Caputo sense and fractional integrals in the Riemann–Liouville sense. A key advantage of the proposed method is its exact computation of the Riemann–Liouville fractional integral operator. Using the Newton–Cotes collocation method, we transform the differential equations into systems of algebraic equations, which are then solved with traditional techniques such as Newton’s iterative method. An error analysis method is also introduced. Several numerical examples are provided to demonstrate the effectiveness and simplicity of the approach, highlighting its efficiency even for relatively small base sizes.

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.

Analysis of (P,m)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(P,\\mathrm{m})$\\end{document}-superquadratic function and related fractional integral inequalities with applications

In the present work we establish for the first time a class of (P,m)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(P,\\mathrm{m})$\\end{document}-superquadratic functions and look into its features. Using them, we come up with the Jensen and Hermite–Hadamard inequalities, as well as the fractional versions of Hermite–Hadamard inequalities with respect to Riemann–Liouville fractional integral operators. The findings are confirmed by certain numerical calculations and graphical depictions that take a few appropriate examples into account. The study is enhanced by the addition of applications of special means, moments of random variables, and modified Bessel functions of the first kind. This is achieved by considering new functions pertaining to the uniform probability density function and taking the modified Bessel functions of the first kind into consideration. The new results clearly provide extensions and improvements of the work given in the literature.

An accurate wavelets‐collocation technique for neutral delay distributed‐order fractional optimal control problems

AbstractIn this study, a new class of optimal control problems called neutral delay distributed‐order fractional optimal control problems is introduced, this problem is solved based on an efficient computational scheme. To solve the problem, we derive an exact formula for the Riemann–Liouville fractional integral operator of Genocchi wavelets based on beta functions for the first time. By taking into account this operator, collocation method, and Gauss–Legendre integration formula, the solution of fractional optimal control problems (FOCPs) under consideration is converted to a nonlinear programming one to which existing well‐developed algorithms may be applied. The mentioned scheme is applied to both FOCPs with or without delay. Error analysis associated with the proposed idea is also investigated under several mild conditions. The effectiveness of the strategy is showed by several illustrative examples, furthermore, a comparison with the previous methods highlights the preference of this scheme.

Integral inequalities of h‐superquadratic functions and their fractional perspective with applications

The purpose of this article is to provide a number of Hermite–Hadamard and Fejér type integral inequalities for a class of ‐superquadratic functions. We then develop the fractional perspective of inequalities of Hermite–Hadamard and Fejér types by use of the Riemann–Liouville fractional integral operators and bring up with few particular cases. Numerical estimations based on specific relevant cases and graphical representations validate the results. Another motivating component of the study is that it is enriched with applications of modified Bessel function of first type, special means, and moment of random variables by defining some new functions in terms of modified Bessel function and considering uniform probability density function. The results in this paper have not been initiated before in the frame of ‐superquadraticity. We are optimistic that this effort will greatly stimulate and encourage additional research.

A wavelet collocation method for fractional Black–Scholes equations by subdiffusive model

AbstractIn this investigation, we propose a numerical method based on the fractional‐order generalized Taylor wavelets (FGTW) for option pricing and the fractional Black–Scholes equations. This model studies option pricing when the underlying asset has subdiffusive dynamics. By applying the regularized beta function, we give an exact formula for the Riemann–Liouville fractional integral operator (RLFIO) of the FGTW. An error analysis of the numerical scheme for estimating solutions is performed. Finally, we conduct a variety of numerical experiments for several standard examples from the literature to assess the efficiency of the proposed method.

Analytical properties, fractal dimensions and related inequalities of [formula omitted]-Riemann–Liouville fractional integrals

In this paper, we rigorously explore (k,h)-Riemann–Liouville fractional integrals applied to continuous functions defined on closed real intervals. Firstly, we delve into the analytical properties of (k,h)-Riemann–Liouville fractional integrals for continuous functions, examining aspects like boundedness, continuity, and bounded variation. Secondly, we derive an upper bound for the upper box dimensions of (k,h)-Riemann–Liouville fractional integral graphs for continuous functions defined on real closed intervals. Finally, we present several applications of (k,h)-Riemann–Liouville fractional integral operators, including the reverse analog Hölder inequality, the reverse analog Minkowski inequality, and Hermite–Hadamard-type inequalities.

Image Edge Detection by Global Thresholding Using Riemann–Liouville Fractional Integral Operator

It is difficult to give a fractional global threshold (FGT) that works well on all images as the image contents are totally different. This paper describes an interesting use of fractional calculus in the field of digital image processing. In the proposed method, the fractional global threshold-based edge detector (FGTED) is established using the Riemann–Liouville fractional integral operator. FGTED is used to find the microedges in minimum time for any input digital images. The results demonstrate that the FGTED outperforms conventional techniques for detecting microtype edges. The image with a higher entropy was produced by the FGT value-based approach. Tables and images are used to summarize the output performance analysis of various images using structural similarity index measure, F-score (F-measure), precision and recall, signal-to-noise ratio, peak signal-to-noise ratio, and computational time. The FGTED can be used to detect very thin or microtype edges more accurately in minimum time without training or prior knowledge.

Mittag-Leffler wavelets and their applications for solving fractional optimal control problems

Herein, we design a new scheme for finding approximate solutions to fractional optimal control problems (OCPs) with and without delay. In this strategy, we introduce Mittag-Leffler wavelet functions and develop a new Riemann–Liouville fractional integral operator for these functions utilizing the hypergeometric function. The properties of the operational matrix have reflected well in the process of the numerical method and affect the accuracy of the proposed method directly. Employing the Riemann–Liouville fractional integral operator, delay operational matrix, and Galerkin method, the considered problems lead to systems of algebraic equations. An error analysis is proposed. Finally, some illustrative numerical tests are given to show the precision and validity of the suggested technique. The proposed method is very efficient for solving the OCPs with delay and without delay, and gives very accurate results.

Some Error Bounds for 2‐Point Right Radau Formula in the Setting of Fractional Calculus via s‐Convexity

In this paper, we present a new approach to construct fractional 2‐point right Radau type integral inequalities using a novel identity, for functions with s‐convex first derivatives in the second sense via Riemann–Liouville fractional integral operators. We then demonstrate the accuracy of our results through a 2D example, as well as practical applications of the integral inequalities to quadrature formulas and special means such as arithmetic and p‐logarithmic means.

Applications of Fuzzy Differential Subordination to the Subclass of Analytic Functions Involving Riemann–Liouville Fractional Integral Operator

In this research, we combine ideas from geometric function theory and fuzzy set theory. We define a new operator Dτ−λLα,ζm:A→A of analytic functions in the open unit disc Δ with the help of the Riemann–Liouville fractional integral operator, the linear combination of the Noor integral operator, and the generalized Sălăgean differential operator. Further, we use this newly defined operator Dτ−λLα,ζm together with a fuzzy set, and we next define a new class of analytic functions denoted by Rϝζ(m,α,δ). Several innovative results are found using the concept of fuzzy differential subordination for the functions belonging to this newly defined class, Rϝζ(m,α,δ). The study includes examples that demonstrate the application of the fundamental theorems and corollaries.

Riemann Liouville fractional-like integral operators, convex-like real-valued mappings and their applications over fuzzy domain

It is well-known fact that fuzzy number theory is based on the characteristic function but in the fuzzy region, the characteristic function is a membership function that depends upon the interval 01. It means that real numbers and intervals are exceptional cases of fuzzy numbers. By using this approach, another type of Riemann Liouville fractional integral operator has been introduced over a fuzzy domain which is known as Riemann Liouville fractional-like integral operator. These integrals look like Riemann Liouville fractional integral operators but classical integrals are special cases of newly defined integral operators. Some novel versions of these integrals are also obtained and explained by taking the domain's triangular and trapezoidal fuzzy numbers. We also consider the problem of computing scalar-valued. To discuss the application of new integrals, a new class of convex real-valued functions has been introduced, known as convex-like real-valued mappings. With the help of these new concepts, some applications are taken into account. By using convex-like real-valued mappings and newly proposed fractional-like integral operators, the well-known Hermite-Hadamard H.H type and related inequalities are taken into account in this work. Then by defining new differentiable real-valued functions over fuzzy regions, an identity has been obtained and with the support of this identity, certain inequalities are also acquired. Also, we clearly show the important connections of the derived outcomes with those classical integrals. Finally, some nontrivial numerical examples are also provided to verify the correctness of the presented inequalities that occur with the variation of the parameters v and β.

An efficient numerical scheme for solving a general class of fractional differential equations via fractional-order hybrid Jacobi functions

We introduce a numerical algorithm based on the hybrid of block-pulse functions and fractional-order Jacobi polynomials in order to solve fractional differential equations. The fractional derivative is described in the Caputo sense. The Riemann–Liouville fractional integral operator for these basis functions is constructed. This result together with the shifted Gauss–Chebyshev collocation points are utilized to reduce the original problem to a system of nonlinear algebraic equations. By means of solving the given system, the numerical solution of the main problem is derived. Then, the method is applied for solving the Bagley–Torvik initial and boundary value problems. After that, an error estimation is presented for the expansion of a given function based on the fractional-order basis functions. Finally, for demonstrating the precision and good performance of the new method, several numerical examples are considered and the results are compared with the exact or approximate solutions obtained by other existing techniques.

Fractional Integrals Associated with the One-Dimensional Dunkl Operator in Generalized Lizorkin Space

This paper explores the realm of fractional integral calculus in connection with the one-dimensional Dunkl operator on the space of tempered functions and Lizorkin type space. The primary objective is to construct fractional integral operators within this framework. By establishing the analogous counterparts of well-known operators, including the Riesz fractional integral, Feller fractional integral, and Riemann–Liouville fractional integral operators, we demonstrate their applicability in this setting. Moreover, we show that familiar properties of fractional integrals can be derived from the obtained results, further reinforcing their significance. This investigation sheds light on the utilization of Dunkl operators in fractional calculus and provides valuable insights into the connections between different types of fractional integrals. The findings presented in this paper contribute to the broader field of fractional calculus and advance our understanding of the study of Dunkl operators in this context.

A study of incomplete I-functions relating to certain fractional integral operators

The study discussed in this article is driven by the realization that many physical processes may be understood by using applications of fractional operators and special functions. In this study, we present and examine a fractional integral operator with an I-function in its kernel. This operator is used to solve several fractional differential equations (FDEs). FDE has a set of particular cases whose solutions represent different physical phenomena. Many mathematical physics, biology, engineering, and chemistry problems are identified and solved using FDE. Specifically, a few exciting relations involving the new fractional operator with incomplete I-function (IIF) in its kernel and classical Riemann Liouville fractional integral and derivative operators, the Hilfer fractional derivative operator, and the generalized composite fractional derivate (GCFD) operator are established. The discovery and investigation of several important exceptional cases follow this.

Application of Müntz Orthogonal Functions on the Solution of the Fractional Bagley–Torvik Equation Using Collocation Method with Error Stimate

This paper uses Müntz orthogonal functions to numerically solve the fractional Bagley–Torvik equation with initial and boundary conditions. Müntz orthogonal functions are defined on the interval 0 , 1 and have simple and distinct real roots on this interval. For the function f ∈ L 2 0 , 1 , we obtain the best unique approximation using Müntz orthogonal functions. We obtain the Riemann–Liouville fractional integral operator for Müntz orthogonal functions so that we can reduce the complexity of calculations and increase the speed of solving the problem, which can be seen in the process of running the Maple program. To solve the fractional Bagley–Torvik equation with initial and boundary conditions, we use Müntz orthogonal functions and consider simple and distinct real roots of Müntz orthogonal functions as collocation points. By using the Riemann–Liouville fractional integral operator that we define for the Müntz orthogonal functions, the process of numerically solving the fractional Bagley–Torvik equation that is solved using Müntz orthogonal functions is reduced, and finally, we reach a system of algebraic equations. By solving algebraic equations and obtaining the vector of unknowns, the fractional Bagley–Torvik equation is solved using Müntz orthogonal functions, and the error value of the method can be calculated. The low error value of this numerical solution method shows the high accuracy of this method. With the help of the Müntz functions, we obtain the error bound for the approximation of the function. We have obtained the error bounds for the numerical method using which we solved the fractional Bagley–Torvik equation with initial and boundary conditions. Finally, we have given a numerical example to show the accuracy of the solution of the method presented in this paper. The results of solving this example using Müntz orthogonal functions and comparing the results with other methods that have been used the solve this example show the higher accuracy of the method proposed in this paper.

An efficient method for the fractional electric circuits based on Fibonacci wavelet

In this article, we provide effective computational algorithms based on Fibonacci wavelet (FW) to approximate the solution of fractional order electrical circuits (ECs). The proposed computational algorithm is novel and has not been previously utilized for solving ECs problems. Firstly, we have constructed the operational matrices of fractional integration (OMFI). Secondly, we transform the given initial value problems into algebraic equations, we used the Riemann–Liouville (R–L) fractional integral operator. The proposed approach is capable of handling a wide range of fractional order dynamics in ECs. To validate the effectiveness of the method, four models of electrical circuits with fractional order parameter are considered. The numerical results are compared with exact solutions and absolute errors are calculated to demonstrate the accuracy and efficiency of the approach. The proposed method provides a valuable tool for analyzing and designing fractional order systems in electrical engineering, offering improved accuracy and capturing the intricate behavior of complex systems.

Fractional forward Kolmogorov equations in population genetics

This paper combines theoretical development and numerical methods to characterize the effect of heterogeneity in population genetics. To address heterogeneity in population genetics, we build the first fractional forward Kolmogorov equations in population genetics, where the distribution of the allele frequencies of a given set of loci in a population is a solution of forward Kolmogorov equations. This framework will be implemented, and the model will be studied computationally. To study the model, a new numerical method for solving the fractional partial differential equations is presented. The method is based upon the least squares approximation via Legendre polynomials. The Riemann–Liouville fractional integral operator for Legendre polynomials is utilized to reduce the solution of the fractional partial differential equations to a system of algebraic equations. The error bound and the stability of the method are presented. Illustrative examples are included to demonstrate the validity and applicability of the technique. Using the new numerical method, we derive the allele frequency distribution as a solution of the fractional forward Kolmogorov equations. We study the behavior of allele frequency distribution by considering the effect of evolutionary and demographic forces. This study shows that heterogeneity changes the behavior of the distribution of the allele frequencies.

SOME PROPERTIES OF k-RIEMANN-LIOUVILLE FRACTIONAL INTEGRAL OPERATOR

In this paper we will introduce some properties of k- Riemann Liouville fractional integral operator involving convolution property. The fractional derivative of k- Riemann Liouville fractional integral operator of integral transforms will be obtained. Applications of this operator will be introduced. All results of nature will be discussed as special cases

An alternating direction implicit compact finite difference scheme for the multi-term time-fractional mixed diffusion and diffusion–wave equation

An alternating direction implicit(ADI) compact finite difference scheme for the two-dimensional multi-term time-fractional mixed diffusion and diffusion–wave equation is given in this paper. By using the Riemann–Liouville fractional integral operator on both sides of the original equation, we obtain a time-fractional integro–differential equation with the order of the highest derivative being one. Using the weighted and shifted Grünwald formulas of Riemann–Liouville fractional derivative and fractional integral, and the Crank–Nicolson approximation, a temporal second-order approximation is obtained for the equivalent integrodifferential system. In the spatial direction, a fourth-order compact approximation is employed to give a fully discrete scheme. Using the splitting techniques for higher dimensional problems, we derive the fully discrete ADI Crank–Nicolson scheme. With the positive definiteness of the related time discrete coefficients and spatial operators, the stability and convergence (second-order in time and fourth-order in space) in the discrete L2-norm are proved by using energy method. Two numerical examples are given to demonstrate the efficiency and accuracy of the proposed method.