Fractional Differential Equations Research Articles

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.

Quantifying rheology and texture of chocolate products with oleogels using fractional calculus

Oleogels, a novel technology for structuring fats, show great potential for partially or completely replacing solid fats in chocolate products containing unhealthy saturated fatty acids. However, how to maintain similar rheological properties and sensory texture attributes is a major challenge. Given the lack of a concise and effective mathematical framework, simulating the rheological response and quantifying the textural parameters of the hybrid oleogel-dispersed chocolate products, is complicated. In this work, the rheological power law behavior of hybrid oleogel-dispersed chocolate products is accurately captured by using a fractional constitutive mathematical framework with few constitutive parameters. The sensory texture parameter firmness is quantified on the basis of creep response. The rheological response and firmness index are linked together on the basis of the fractional derivative model. This work helps provide heuristic insights into the structure-texture engineering of this food type. Such insights may, in turn, guide the formulation and design of food products.

Non-Additivity and Additivity in General Fractional Calculus and Its Physical Interpretations

Non-Additivity and Additivity in General Fractional Calculus and Its Physical Interpretations

(k,φ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\mathtt{k},\\varphi )$\\end{document}-Hilfer fractional Langevin differential equation having multipoint boundary conditions

The primary objective of this manuscript is to investigate the existence and uniqueness of solutions for the Langevin (k,φ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\mathtt{k},\\varphi )$\\end{document}-Hilfer fractional differential equation of different orders with multipoint nonlocal fractional integral boundary conditions. We consider the generalized version of the Hilfer fractional diferential equation called as (k,φ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\mathtt{k},\\varphi )$\\end{document}-Hilfer fractional differential equation. We provide some significant outcomes about (k,φ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\mathtt{k},\\varphi )$\\end{document}-Hilfer fractional Langevin differential equation that requires deriving equivalent fractional integral equation to (k,φ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\mathtt{k},\\varphi )$\\end{document}-Hilfer Langevin fractional differential equation. The existence result is established using the Krasnoselskii’s fixed-point theorem, while the uniqueness is addressed with the help of Banach contraction principle. Additionally, we investigate the different forms of Ulam stability for the solution of the mentioned problem, under specific conditions. To validate our main outcomes, we present a detailed example at the end of the manuscript.

A new representation for the solution of the Richards‐type fractional differential equation

A new representation for the solution of the Richards‐type fractional differential equation

New exploration on approximate controllability of nondensely defined Hilfer neutral-type delayed nonlinear differential inclusion system with non-instantaneous impulse

This manuscript aims to investigate the existence and approximate controllability results for integral solution of nondensely defined Hilfer neutral differential inclusion system of Sobolev-type with infinite delay and non-instantaneous impulse in a Hilbert space. By utilizing the semigroup theory of bounded linear operators, fixed point technique, fractional calculus, and the theory of multivalued maps, the principal discussions are demonstrated. The sufficient conditions for the existence and approximate controllability are demonstrated under the consideration that the linear operator A is defined nondensely and satisfies the Hille–Yosida condition. Finally, an example is given to support the validity of the study.

Multidimensional Fractional Calculus: Theory and Applications

In this paper, we introduce several new types of partial fractional derivatives in the continuous setting and the discrete setting. We analyze some classes of the abstract fractional differential equations and the abstract fractional difference equations depending on several variables, providing a great number of structural results, useful remarks and illustrative examples. Concerning some specific applications, we would like to mention here our investigation of the fractional partial differential inclusions with Riemann–Liouville and Caputo derivatives. We also establish the complex characterization theorem for the multidimensional vector-valued Laplace transform and provide certain applications.

A comprehensive study on geometric shape optical soliton solutions to the time-factional nonlinear Schrödinger-Hirota equation

In this study, we investigate the analytical soliton solutions of a fundamental model, namely the nonlinear Schrödinger-Hirota equation, in the context of beta time-fractional derivative. We adopt the (ω′/ω, 1/ω)-expansion method, which is a reliable and straightforward approach to extract fresh and general soliton solutions in terms of hyperbolic, trigonometric, and rational functions. The solitons include anti-kink, anti-bell-shaped, bell-shaped, and periodic solitons. These solitons have significant applications in various scientific fields, such as optical fiber communications, signal processing, plasma physics, and trans-oceanic data transfer. This study demonstrates the significance of fractional-order differentiation in revealing new solitons. We also provide a comprehensive comparison with existing literature in normal and anomalous dispersion regions, highlighting the uniqueness of the solutions. Moreover, the graphical representations are used to illustrate the properties and potential applications of these solitons. This research might contribute to the advancement of nonlinear optical research and technology.

Editorial for the Special Issue of “Fractional Differential and Fractional Integro-Differential Equations: Qualitative Theory, Numerical Simulations, and Symmetry Analysis”

Editorial for the Special Issue of “Fractional Differential and Fractional Integro-Differential Equations: Qualitative Theory, Numerical Simulations, and Symmetry Analysis”

Overview of fractional calculus and its computer implementation in Wolfram Mathematica

Overview of fractional calculus and its computer implementation in Wolfram Mathematica

Fractional calculus integration for improved ECG modeling: A McSharry model expansion

This study introduces a new method for modeling electrocardiogram (ECG)11- CR: Compression Ratio; FDE: Fractional Differential Equation; GA: Genetic Algorithm; IDE: Integer Differential Equation; MAE: Mean Absolute Error; MSE: Mean Square Error; NRMSE: Normalized Root Mean Square Error; PC: Predictor Corrector; PRD: Percentage Root Mean Square Difference; QS: Quality Score; RMSE: Root Mean Square Error waveforms using Fractional Differential Equations (FDEs). By incorporating fractional calculus into the well-established McSharry model, the proposed approach achieves improved representation and high precision for a wide range of ECG waveforms. The research focuses on the impact of integrating fractional derivatives into Integer Differential Equation (IDE) models, enhancing the fidelity of ECG signal modeling.To optimize the model's unknown parameters, a combination of the Predictor-Corrector method for solving FDEs and genetic algorithms for optimization is utilized. The effectiveness of the fractional-order model is assessed through distortion metrics, providing a comprehensive evaluation of the modeling quality.Comparisons show that the fractional-order model outperforms the traditional McSharry IDE model in modeling quality and compression efficiency. It improves modeling quality by 48.40 % in MSE and compression efficiency by 23.18 % when applied on five beat types of MIT/BIH arrhythmia database. The fractional-order model demonstrates enhanced flexibility while preserving essential McSharry model characteristics, with fractional orders (α) ranging from 0.96 to 0.99 across five beat types.

Neural Fractional Differential Equations: Optimising the Order of the Fractional Derivative

Neural Fractional Differential Equations (Neural FDEs) represent a neural network architecture specifically designed to fit the solution of a fractional differential equation to given data. This architecture combines an analytical component, represented by a fractional derivative, with a neural network component, forming an initial value problem. During the learning process, both the order of the derivative and the parameters of the neural network must be optimised. In this work, we investigate the non-uniqueness of the optimal order of the derivative and its interaction with the neural network component. Based on our findings, we perform a numerical analysis to examine how different initialisations and values of the order of the derivative (in the optimisation process) impact its final optimal value. Results show that the neural network on the right-hand side of the Neural FDE struggles to adjust its parameters to fit the FDE to the data dynamics for any given order of the fractional derivative. Consequently, Neural FDEs do not require a unique α value; instead, they can use a wide range of α values to fit data. This flexibility is beneficial when fitting to given data is required and the underlying physics is not known.

Advanced diffusion-weighted MRI models for preoperative prediction of lymph node metastasis in resectable gastric cancer.

To investigate the potential of six advanced diffusion-weighted imaging (DWI) models for preoperative prediction of lymph node metastasis (LNM) in resectable gastric cancer (GC). Between Nov 2022 and Nov 2023, standard MRI scans were prospectively performed in consecutive patients with endoscopic pathology-confirmed gastric adenocarcinoma who were referred for direct radical gastrectomy. Six DWI models, including fractional order calculus (FROC), continuous-time random walk (CTRW), diffusion kurtosis imaging (DKI), intravoxel incoherent motion (IVIM), the mono-exponential model (MEM) and the stretched exponential model (SEM) were computed. Surgical pathologic diagnosis of LNM was the reference standard, and patients were classified into LNM-positive or LNM-negative groups accordingly. The morphological features and quantitative parameters of the DWI models in different LNM categories were analyzed and compared. Multivariable logistic regression was used to screen significant predictors. Receiver-operating characteristic curves and the area under the curve (AUC) were plotted to evaluate the performances, the Delong test was performed to compare the AUCs. In the LNM-positive group, tumor thickness and kurtosis (DKI_K) were significantly higher, while anomalous diffusion coefficient (CTRW_D), diffusivity (DKI_D), diffusion coefficient (FROC_D), pseudodiffusion coefficient (IVIM_D*), perfusion fraction (IVIM_f), and ADC were lower compared to the LNM-negative group. Clinical tumor staging (cT) and CTRW_D were independent predictors. Their combination demonstrated a superior AUC of 0.930, significantly higher than that of individual parameters. Tumor thickness, DKI_K, CTRW_D, DKI_D, FROC_D, IVIM_D*, IVIM_f and ADC were associated with LNM status. The combination of independent predictors of cT and CTRW_D further enhanced the performance.

On the asymptotic behavior of solutions to bilinear Caputo stochastic fractional differential equations

In this paper, we focus on investigating the asymptotic behavior of solutions in a mean square sense to bilinear Caputo stochastic fractional differential equations (CSFDEs). The main tools in the proof include a variation of the constant formula for CSFDEs, the Jordan normal form of a matrix, the summation formula of Djrbashian type, and constructing a weighted norm in the associated Banach space.

Unique Solutions for Caputo Fractional Differential Equations with Several Delays Using Progressive Contractions

We take into account a nonlinear Caputo fractional-order differential equation including several variable delays. We examine whether the solutions to the Caputo fractional-order differential equation taken under consideration, which has numerous variable delays, are unique. In the present study, first, we will apply the method of progressive contractions, which belongs to T.A. Burton, to Caputo fractional-order differential equation, including multiple variable delays, which has not yet appeared in the relevant literature by this time. The significant point of the method of progressive contractions consists of a very flexible idea to discuss the uniqueness of solutions for various mathematical models. Lastly, we provide two examples to demonstrate how this paper’s primary outcome can be applied.

Existence of Positive Solutions for Hadamard-Type Fractional Boundary Value Problems at Resonance on an Infinite Interval

This paper investigates a class of resonance boundary value problems for Hadamard-type fractional differential equations on an infinite interval. Utilizing the Leggett-Williams norm-type theorem proposed by O’Regan and Zima, the existence of positive solutions is established. The main conclusions are illustrated with an example.

Fractional Calculus for Non-Discrete Signed Measures

In this paper, we suggest a first-ever construction of fractional integral and differential operators based on signed measures including a vector-valued case. The study focuses on constructing the fractional power of the Riemann–Stieltjes integral with a signed measure, using semigroup theory. The main result is a theorem that provides the exact form of a semigroup for the Riemann–Stieltjes integral with a measure having a countable number of extrema. This article provides examples of semigroups based on integral operators with signed measures and discusses the fractional powers of differential operators with partial derivatives.

Weighted estimates of certain variable kernel operators on generalized Morrey spaces with variable exponent

Let T be the singular integral operator with variable kernel and Dγ(0≤γ≤1) be the fractional differentiation operator. Simultaneously, let T⁎ and T♯ be the adjoint and pseudo-adjoint of T, T1T2 and T1∘T2 be the product and pseudo-product of T1 and T2 respectively. In this article, we show that, TDγ−DγT, (T⁎−T♯)Dγ and (T1∘T2−T1T2)Dγ on the variable exponent generalized weighted Morrey spaces are bounded.

A NEW ALGORITHM FOR GENERATING POWER SERIES SOLUTIONS FOR A BROAD CLASS OF FRACTIONAL PDES: APPLICATIONS TO INTERESTING PROBLEMS

This research offers a precise analytical solution for initial value problems of both linear and nonlinear fractional order partial differential equations. A new approach called the limit residual function method is developed and utilized to generate a series solution for the equations. The key tools of this method are the concepts of power series, residual function, and the limit at zero. The new method is characterized by the speed of determining the series solution coefficients and the limited mathematical operations used. The study examines five significant applications of fractional partial differential equations using this new method. To validate the accuracy of the results, they are compared with exact solutions in the classical case for the discussed applications. We conclude that the proposed approach is straightforward, effective, and successful in solving linear and nonlinear fractional differential equations.

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.