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19 pages, 287 KB

Open AccessArticle

A Generalized Nonlinear Bagley–Torvik Equation in Distributions

by Chenkuan Li, Ehsan Pourhadi and Alison Gray

Mathematics 2026, 14(10), 1766; https://doi.org/10.3390/math14101766 - 21 May 2026

Viewed by 404

This paper investigates the fractional calculus of distributions supported on

R^{+}

in the sense of L. Schwartz, based on distributional convolutions. We further study a generalized Bagley–Torvik equation involving an arbitrary number of fractional derivative terms with orders in the interval [...] Read more.

This paper investigates the fractional calculus of distributions supported on

R^{+}

in the sense of L. Schwartz, based on distributional convolutions. We further study a generalized Bagley–Torvik equation involving an arbitrary number of fractional derivative terms with orders in the interval

(0, 2)

. The existence and uniqueness of solutions for its nonlinear form are established in a space of continuous functions by applying Banach’s contraction principle, the Leray–Schauder fixed-point theorem, inverse operators, and the multivariate Mittag–Leffler function. Finally, several examples are presented, in which the values of multivariate Mittag–Leffler functions are computed to illustrate the main results. Full article

(This article belongs to the Special Issue Advanced Theoretical and Numerical Methods for Fractional Differential and Difference Equations)

21 pages, 328 KB

Open AccessArticle

Analytic Study on Φ-Hilfer Fractional Neutral-Type Functional Integro-Differential Equations with Terminal Conditions

by Ravichandran Vivek, Abdulah A. Alghamdi, Mohamed M. El-Dessoky, Dhandapani Maheswari and Natarajan Bharath

Mathematics 2026, 14(1), 182; https://doi.org/10.3390/math14010182 - 3 Jan 2026

Cited by 1 | Viewed by 538

Abstract

The current manuscript is concerned with the uniqueness and existence of a solution for a new class of

Φ

-Hilfer fractional neutral functional integro-differential equations (

Φ

-HFNFIDEs) with terminal conditions. Firstly, employing Babenko’s approach, we convert the aforesaid equation under consideration into [...] Read more.

The current manuscript is concerned with the uniqueness and existence of a solution for a new class of

Φ

-Hilfer fractional neutral functional integro-differential equations (

Φ

-HFNFIDEs) with terminal conditions. Firstly, employing Babenko’s approach, we convert the aforesaid equation under consideration into an analogous integral equation. More precisely, using the multivariate Mittag-Leffler function, Banach contraction principle, and Krasnoselskii’s fixed-point theorem, we derive some conditions that guarantee the uniqueness and the existence of the solutions. For an illustration of our results in this manuscript, two examples are provided as well. Full article

38 pages, 601 KB

Open AccessArticle

A New Laplace-Type Transform on Weighted Spaces with Applications to Hybrid Fractional Cauchy Problems

by Samten Choden, Jakgrit Sompong, Ekkarath Thailert and Sotiris K. Ntouyas

Fractal Fract. 2025, 9(11), 751; https://doi.org/10.3390/fractalfract9110751 - 20 Nov 2025

Cited by 2 | Viewed by 1031

Abstract

This paper develops a generalized Laplace transform theory within weighted function spaces tailored for the analysis of fractional differential equations involving the

ψ

-Hilfer derivative. We redefine the transform in a weighted setting, establish its fundamental properties—including linearity, convolution theorems, and action on [...] Read more.

This paper develops a generalized Laplace transform theory within weighted function spaces tailored for the analysis of fractional differential equations involving the

ψ

-Hilfer derivative. We redefine the transform in a weighted setting, establish its fundamental properties—including linearity, convolution theorems, and action on

δ_{ψ}

derivatives—and derive explicit formulas for the transforms of

ψ

-Riemann–Liouville,

ψ

-Caputo, and

ψ

-Hilfer fractional operators. The results provide a rigorous analytical foundation for solving hybrid fractional Cauchy problems that combine classical and fractional derivatives. As an application, we solve a hybrid model incorporating both

δ_{ψ}

derivatives and

ψ

-Hilfer fractional derivatives, obtaining explicit solutions in terms of multivariate Mittag-Leffler functions. The effectiveness of the method is illustrated through a capacitor charging model and a hydraulic door closer system based on a mass-damper model, demonstrating how fractional-order terms capture memory effects and non-ideal behaviors not described by classical integer-order models. Full article

(This article belongs to the Special Issue From Fractals to Fractional Calculus: Nonlinear Dynamics and Hybrid Complex Systems)

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12 pages, 268 KB

Open AccessArticle

Analysis of Delay-Type Integro-Differential Systems Described by the Φ-Hilfer Fractional Derivative

by Ravichandran Vivek, Waleed Mohammed Abdelfattah and Elsayed Mohamed Elsayed

Axioms 2025, 14(8), 629; https://doi.org/10.3390/axioms14080629 - 11 Aug 2025

Cited by 4 | Viewed by 993

Abstract

In this article, a novel type of equation, namely the

Φ

-Hilfer fractional-order integro-differential delay system (

Φ

-HFOIDDS), is proposed. Here, we study the existence and Hyers–Ulam–Mittag–Leffler (H-U-M-L) stability of the aforementioned equation which are obtained by using the multivariate Mittag–Leffler function, [...] Read more.

In this article, a novel type of equation, namely the

Φ

-Hilfer fractional-order integro-differential delay system (

Φ

-HFOIDDS), is proposed. Here, we study the existence and Hyers–Ulam–Mittag–Leffler (H-U-M-L) stability of the aforementioned equation which are obtained by using the multivariate Mittag–Leffler function, Banach contraction principle, and Picard operator method as well as generalized Gronwall inequality. Finally, we conclude this paper by constructing a suitable example to illustrate the applicability of the principal outcomes. Full article

(This article belongs to the Special Issue Dynamics, Stability, Chaos, Control and Applications of Dynamical Systems)

17 pages, 377 KB

Open AccessArticle

On the Generalized Fractional Convection–Diffusion Equation with an Initial Condition in

R^{n}

by Chenkuan Li, Reza Saadati, Safoura Rezaei Aderyani and Min-Jie Luo

Fractal Fract. 2025, 9(6), 347; https://doi.org/10.3390/fractalfract9060347 - 27 May 2025

Cited by 4 | Viewed by 1112

Abstract

Time-fractional convection–diffusion equations are significant for their ability to model complex transport phenomena that deviate from classical behavior, with numerous applications in anomalous diffusion, memory effects, and nonlocality. This paper derives, for the first time, a unique series solution to a multiple time-fractional [...] Read more.

Time-fractional convection–diffusion equations are significant for their ability to model complex transport phenomena that deviate from classical behavior, with numerous applications in anomalous diffusion, memory effects, and nonlocality. This paper derives, for the first time, a unique series solution to a multiple time-fractional convection–diffusion equation with a non-homogenous source term, based on an inverse operator, a newly-constructed space, and the multivariate Mittag–Leffler function. Several illustrative examples are provided to show the power and simplicity of our main theorems in solving certain fractional convection–diffusions equations. Additionally, we compare these results with solutions obtained using the AI model DeepSeek-R1, highlighting the effectiveness and validity of our proposed methods and main theorems. Full article

(This article belongs to the Special Issue Contemporary Methods of Fractional Order Differential and Differential-Operator Equations)

39 pages, 391 KB

Open AccessArticle

Applications of Inverse Operators to a Fractional Partial Integro-Differential Equation and Several Well-Known Differential Equations

by Chenkuan Li and Wenyuan Liao

Fractal Fract. 2025, 9(4), 200; https://doi.org/10.3390/fractalfract9040200 - 25 Mar 2025

Cited by 4 | Viewed by 1028

Abstract

This paper mainly consists of two parts: (i) We study the uniqueness, existence, and stability of a new fractional nonlinear partial integro-differential equation in

R^{n}

with three-point conditions and variable coefficients in a Banach space using inverse operators containing multi-variable functions, a [...] Read more.

This paper mainly consists of two parts: (i) We study the uniqueness, existence, and stability of a new fractional nonlinear partial integro-differential equation in

R^{n}

with three-point conditions and variable coefficients in a Banach space using inverse operators containing multi-variable functions, a generalized Mittag-Leffler function, as well as a few popular fixed-point theorems. These studies have good applications in general since uniqueness, existence and stability are key and important topics in many fields. Several examples are presented to demonstrate applications of results obtained by computing approximate values of the generalized Mittag-Leffler functions. (ii) We use the inverse operator method and newly established spaces to find analytic solutions to a number of notable partial differential equations, such as a multi-term time-fractional convection problem and a generalized time-fractional diffusion-wave equation in

R^{n}

with initial conditions only, which have never been previously considered according to the best of our knowledge. In particular, we deduce the uniform solution to the non-homogeneous wave equation in n dimensions for all

n \geq 1

, which coincides with classical results such as d’Alembert and Kirchoff’s formulas but is much easier in the computation of finding solutions without any complicated integrals on balls or spheres. Full article

(This article belongs to the Special Issue Contemporary Methods of Fractional Order Differential and Differential-Operator Equations)

33 pages, 641 KB

Open AccessArticle

Computational Representation of Fractional Inequalities Through 2D and 3D Graphs with Applications

by Muhammad Younis, Ahsan Mehmood, Muhammad Samraiz, Gauhar Rahman, Salma Haque, Ahmad Aloqaily and Nabil Mlaiki

Computation 2025, 13(2), 46; https://doi.org/10.3390/computation13020046 - 7 Feb 2025

Cited by 2 | Viewed by 1221

Abstract

The aim of this research article is to use the extended fractional operators involving the multivariate Mittag–Leffler (M-M-L) function, we provide the generalization of the Hermite–Hadamard–Fejer (H-H-F) inequalities. We relate these inequalities to previously published disparities in the literature by making appropriate substitutions. [...] Read more.

The aim of this research article is to use the extended fractional operators involving the multivariate Mittag–Leffler (M-M-L) function, we provide the generalization of the Hermite–Hadamard–Fejer (H-H-F) inequalities. We relate these inequalities to previously published disparities in the literature by making appropriate substitutions. In the last section, we analyze several inequalities related to the H-H-F inequalities, focusing on generalized h-convexity associated with extended fractional operators involving the M-M-L function. To achieve this, we derive two identities for locally differentiable functions, which allows us to provide specific estimates for the differences between the left, middle, and right terms in the H-H-F inequalities. Also, we have constructed specific inequalities and visualized them through graphical representations to facilitate their applications in analysis. The research bridges theoretical advancements with practical applications, providing high-accuracy bounds for complex systems involving fractional calculus. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications, 2nd Edition)

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11 pages, 322 KB

Open AccessBrief Report

A Matrix Mittag–Leffler Function and the Fractional Nonlinear Partial Integro-Differential Equation in ℝⁿ

by Chenkuan Li, Joshua Beaudin, Azedine Rahmoune and Walid Remili

Fractal Fract. 2023, 7(9), 651; https://doi.org/10.3390/fractalfract7090651 - 26 Aug 2023

Cited by 6 | Viewed by 1962

Abstract

In this paper, we introduce the matrix Mittag–Leffler function, which is a generalization of the multivariate Mittag–Leffler function, in order to investigate the uniqueness of the solutions to a fractional nonlinear partial integro-differential equation in

R^{n}

with a boundary condition based on [...] Read more.

In this paper, we introduce the matrix Mittag–Leffler function, which is a generalization of the multivariate Mittag–Leffler function, in order to investigate the uniqueness of the solutions to a fractional nonlinear partial integro-differential equation in

R^{n}

with a boundary condition based on Banach’s contractive principle and Babenko’s approach. In addition, we present an example demonstrating applications of the key results derived using a Python code that computes the approximate value of our matrix Mittag–Leffler function. Full article

(This article belongs to the Section General Mathematics, Analysis)

19 pages, 347 KB

Open AccessArticle

On Novel Fractional Operators Involving the Multivariate Mittag–Leffler Function

by Muhammad Samraiz, Ahsan Mehmood, Saima Naheed, Gauhar Rahman, Artion Kashuri and Kamsing Nonlaopon

Mathematics 2022, 10(21), 3991; https://doi.org/10.3390/math10213991 - 27 Oct 2022

Cited by 20 | Viewed by 1963

Abstract

The multivariate Mittag–Leffler function is introduced and used to establish fractional calculus operators. It is shown that the fractional derivative and integral operators are bounded. Some fundamental characteristics of the new fractional operators, such as the semi-group and inverse characteristics, are studied. As [...] Read more.

The multivariate Mittag–Leffler function is introduced and used to establish fractional calculus operators. It is shown that the fractional derivative and integral operators are bounded. Some fundamental characteristics of the new fractional operators, such as the semi-group and inverse characteristics, are studied. As special cases of these novel fractional operators, several fractional operators that are already well known in the literature are acquired. The generalized Laplace transform of these operators is evaluated. By involving the explored fractional operators, a kinetic differintegral equation is introduced, and its solution is obtained by using the Laplace transform. As a real-life problem, a growth model is developed and its graph is sketched. Full article

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13 pages, 308 KB

Open AccessArticle

The Grüss-Type and Some Other Related Inequalities via Fractional Integral with Respect to Multivariate Mittag-Leffler Function

by Yabin Shao, Gauhar Rahman, Yasser Elmasry, Muhammad Samraiz, Artion Kashuri and Kamsing Nonlaopon

Fractal Fract. 2022, 6(10), 546; https://doi.org/10.3390/fractalfract6100546 - 27 Sep 2022

Cited by 2 | Viewed by 2011

Abstract

In the recent era of research, the field of integral inequalities has earned more recognition due to its wide applications in diverse domains. The researchers have widely studied the integral inequalities by utilizing different approaches. In this present article, we aim to develop [...] Read more.

In the recent era of research, the field of integral inequalities has earned more recognition due to its wide applications in diverse domains. The researchers have widely studied the integral inequalities by utilizing different approaches. In this present article, we aim to develop a variety of certain new inequalities using the generalized fractional integral in the sense of multivariate Mittag-Leffler (M-L) functions, including Grüss-type and some other related inequalities. Also, we use the relationship between the Riemann-Liouville integral, the Prabhakar integral, and the generalized fractional integral to deduce specific findings. Moreover, we support our findings by presenting examples and corollaries. Full article

(This article belongs to the Special Issue Fractional Integral Inequalities and Applications)

22 pages, 398 KB

Open AccessReview

A Survey of Some Recent Developments on Higher Transcendental Functions of Analytic Number Theory and Applied Mathematics

by Hari Mohan Srivastava

Symmetry 2021, 13(12), 2294; https://doi.org/10.3390/sym13122294 - 2 Dec 2021

Cited by 99 | Viewed by 5440

Abstract

Often referred to as special functions or mathematical functions, the origin of many members of the remarkably vast family of higher transcendental functions can be traced back to such widespread areas as (for example) mathematical physics, analytic number theory and applied mathematical sciences. [...] Read more.

Often referred to as special functions or mathematical functions, the origin of many members of the remarkably vast family of higher transcendental functions can be traced back to such widespread areas as (for example) mathematical physics, analytic number theory and applied mathematical sciences. Here, in this survey-cum-expository review article, we aim at presenting a brief introductory overview and survey of some of the recent developments in the theory of several extensively studied higher transcendental functions and their potential applications. For further reading and researching by those who are interested in pursuing this subject, we have chosen to provide references to various useful monographs and textbooks on the theory and applications of higher transcendental functions. Some operators of fractional calculus, which are associated with higher transcendental functions, together with their applications, have also been considered. Many of the higher transcendental functions, especially those of the hypergeometric type, which we have investigated in this survey-cum-expository review article, are known to display a kind of symmetry in the sense that they remain invariant when the order of the numerator parameters or when the order of the denominator parameters is arbitrarily changed. Full article

13 pages, 289 KB

Open AccessArticle

Uniqueness of Solutions of the Generalized Abel Integral Equations in Banach Spaces

by Chenkuan Li and Hari M. Srivastava

Fractal Fract. 2021, 5(3), 105; https://doi.org/10.3390/fractalfract5030105 - 31 Aug 2021

Cited by 11 | Viewed by 2708

Abstract

This paper studies the uniqueness of solutions for several generalized Abel’s integral equations and a related coupled system in Banach spaces. The results derived are new and based on Babenko’s approach, Banach’s contraction principle and the multivariate Mittag–Leffler function. We also present some [...] Read more.

This paper studies the uniqueness of solutions for several generalized Abel’s integral equations and a related coupled system in Banach spaces. The results derived are new and based on Babenko’s approach, Banach’s contraction principle and the multivariate Mittag–Leffler function. We also present some examples for the illustration of our main theorems. Full article

(This article belongs to the Special Issue Operators of Fractional Calculus and Their Multidisciplinary Applications)

6 pages, 738 KB

Open AccessArticle

Certain Unified Integrals Involving a Multivariate Mittag–Leffler Function

by Shilpi Jain, Ravi P. Agarwal, Praveen Agarwal and Prakash Singh

Axioms 2021, 10(2), 81; https://doi.org/10.3390/axioms10020081 - 2 May 2021

Cited by 31 | Viewed by 3327

Abstract

A remarkably large number of unified integrals involving the Mittag–Leffler function have been presented. Here, with the same technique as Choi and Agarwal, we propose the establishment of two generalized integral formulas involving a multivariate generalized Mittag–Leffler function, which are expressed in terms [...] Read more.

A remarkably large number of unified integrals involving the Mittag–Leffler function have been presented. Here, with the same technique as Choi and Agarwal, we propose the establishment of two generalized integral formulas involving a multivariate generalized Mittag–Leffler function, which are expressed in terms of the generalized Lauricella series due to Srivastava and Daoust. We also present some interesting special cases. Full article

(This article belongs to the Special Issue Special Functions Associated with Fractional Calculus)

24 pages, 565 KB

Open AccessReview

An Overview of the Pathway Idea and Its Applications in Statistical and Physical Sciences

by Nicy Sebastian, Seema S. Nair and Dhannya P. Joseph

Axioms 2015, 4(4), 530-553; https://doi.org/10.3390/axioms4040530 - 19 Dec 2015

Cited by 9 | Viewed by 5617

Abstract

Pathway idea is a switching mechanism by which one can go from one functional form to another, and to yet another. It is shown that through a parameter α, called the pathway parameter, one can connect generalized type-1 beta family of densities, generalized [...] Read more.

Pathway idea is a switching mechanism by which one can go from one functional form to another, and to yet another. It is shown that through a parameter α, called the pathway parameter, one can connect generalized type-1 beta family of densities, generalized type-2 beta family of densities, and generalized gamma family of densities, in the scalar as well as the matrix cases, also in the real and complex domains. It is shown that when the model is applied to physical situations then the current hot topics of Tsallis statistics and superstatistics in statistical mechanics become special cases of the pathway model, and the model is capable of capturing many stable situations as well as the unstable or chaotic neighborhoods of the stable situations and transitional stages. The pathway model is shown to be connected to generalized information measures or entropies, power law, likelihood ratio criterion or λ - criterion in multivariate statistical analysis, generalized Dirichlet densities, fractional calculus, Mittag-Leffler stochastic process, Krätzel integral in applied analysis, and many other topics in different disciplines. The pathway model enables one to extend the current results on quadratic and bilinear forms, when the samples come from Gaussian populations, to wider classes of populations. Full article

(This article belongs to the Special Issue Special Functions: Fractional Calculus and the Pathway for Entropy Dedicated to Professor Dr. A.M. Mathai on the occasion of his 80th Birthday)

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