Fractional Differential Equations Research Articles

Computational precision in time fractional PDEs: Euler wavelets and novel numerical techniques

This paper presents innovative numerical methodologies designed to solve challenging time fractional partial differential equations, with a focus on the Burgers’, Fisher–KPP, and nonlinear Schrödinger equations. By employing advanced wavelet techniques integrated with fractional calculus, we achieve highly accurate solutions, surpassing conventional methods with minimal absolute error in numerical simulations. A thorough series of numerical experiments validates the robustness and effectiveness of our approach across various parameter regimes and initial conditions. The results underscore significant advancements in the computational modeling of complex physical phenomena governed by time fractional dynamics and offering a powerful tool for a wide range of applications in science and engineering.

A Novel Fractional Model and Its Application in Network Security Situation Assessment

The evaluation process of the Fractional Order Model is as follows. To address the commonly observed issue of low accuracy in traditional situational assessment methods, a novel evaluation algorithm model, the fractional-order BP neural network optimized by the chaotic sparrow search algorithm (TESA-FBP), is proposed. The fractional-order BP neural network, by incorporating fractional calculus, demonstrates enhanced dynamic response characteristics and historical dependency, showing exceptional potential for handling complex nonlinear problems, particularly in the field of network security situational awareness. However, the performance of this network is highly dependent on the precise selection of network parameters, including the fractional order and initial values of the weights. Traditional optimization methods often suffer from slow convergence, a tendency to be trapped in local optima, and insufficient optimization accuracy, which significantly limits the practical effectiveness of the fractional-order BP neural network. By introducing cubic chaotic mapping to generate an initial population with high randomness and global coverage capability, the exploration ability of the sparrow search algorithm in the search space is effectively enhanced, reducing the risk of falling into local optima. Additionally, the Estimation of Distribution Algorithm (EDA) constructs a probabilistic model to guide the population toward the globally optimal region, further improving the efficiency and accuracy of the search process. The organic combination of these three approaches not only leverages their respective strengths, but also significantly improves the training performance of the fractional-order BP neural network in complex environments, enhancing its generalization ability and stability. Ultimately, in the network security situational awareness system, this integration markedly enhances the prediction accuracy and response speed.

Special functions with general kernel: Properties and applications to fractional partial differential equations

Abstract In this paper, we reconstruct the gamma and beta functions using a general kernel function in their integral representations. We also reconstruct the Gauss and confluent hypergeometric functions using the beta function with general kernel in their series representations. The general kernel function we use here can be chosen as any special function such as exponential function, Mittag-Leffler function, Wright function, Fox-Wright function, Kummer function or M-series. Using different choices of this general kernel function, various of the generalized gamma, beta, Gauss hypergeometric and confluent hypergeometric functions in the literature can be obtained. In this paper, we first obtain the integral representations, functional relations, summation, derivative and transformation formulas and double Laplace transforms of the special functions we construct. Furthermore, we compute the solutions of some fractional partial differential equations involving special functions with general kernel via the double Laplace transform and graph some of the solutions for specific values. Finally, we obtain the incomplete beta function with general kernel by defining the beta distribution with general kernel.

Topological degree method for a new class of Φ-Hilfer fractional differential Langevin equation

The aim of this article is to study the existence and uniqueness of the solution of a new class of fractional Langevin Φ-Hilfer differential equations. The proposed study is based on some fundamental definitions of fractional calculus and topological degree theory. We obtained the existence result by employing the topological degree method for condensing maps, and by making use of Banach’s fixed point theorem we deal with the uniqueness result. In order to illustrate our theoretical finding, we provide an example.

Existence and approximate controllability for the Hilfer fractional neutral stochastic hemivariational inequality with Rosenblatt process

ABSTRACT The approximate controllability of the Hilfer fractional neutral stochastic system with Rosenblatt process is discussed in this article along with the hemivariational inequalities. Initially, we demonstrate the existence of mild solution using fractional calculus, semigroup theory, stochastic analysis, Hilfer fractional derivatives, multivalued maps, and the fixed point theorem. Moreover, we prove the conditions under which the Hilfer fractional system is approximately controllable. Finally, we provide our key findings using an example. Abbreviation: FDEs: Fractional differential equations; R–L: Riemann–Liouville; HF: Hilfer fractional; HFD: Hilfer fractional derivative; SDEs: Stochastic differential equations; HVI: Hemivariational Inequality; HFNSHVI: Hilfer fractional neutral stochastic hemi-variational inequality

Designing a switching law for Mittag-Leffler stability in nonlinear singular fractional-order systems and its applications in synchronization

In this work, the stability and synchronization issue of switched singular continuous-time fractional-order systems with nonlinear perturbation are examined. Using the fixed-point principle and S-procedure lemma, a sufficient condition for the existence and uniqueness of the solution to the switched singular fractional-order system is first stated. Next, using the Lyapunov functional method in combination with some techniques related to singular systems and fractional calculus, a switching rule for regularity, impulse-free, and Mittag-Leffler stability is developed based on the formation of a partition of the stability state regions in convex cones. For synchronizing switched fractional singular dynamical systems, we propose a state feedback controller that ensures regularity, impulse-free, and Mittag-Leffler stable in the error closed-loop system. Finally, the ease of use and computational convenience of our proposed methods are illustrated by two numerical examples and a practical example about DC motor controlling an inverted pendulum accompanied by simulation results.

Diverse soliton wave profile analysis in ion-acoustic wave through an improved analytical approach

In engineering and applied sciences, several physical phenomena can be more precisely characterized by employing nonlinear fractional partial differential equations. The primary goal of this research is to examine the traveling wave solution in closed form for the nonlinear acoustic wave propagation model known as the time fractional simplified modified Camassa–Holm equation, which is used to explain the unidirectional propagation of shallow-water waves with non-hydrostatic pressure and explains the dispersion properties of numerous phenomena like fluid flow, control theory, liquid drop patterning in plasma, acoustics, fusion, and fission processes, etc. The utmost potential approach, namely the new auxiliary equation technique, is applied for analyzing the time nonlinear fractional simplified modified Camassa-Holm equation in the logic of the newest established truncated M-fractional derivative. The fractional partial differential equations have been transformed to the ordinary differential equation using the complex wave transformation in the sense of truncated M-fractional derivative. A variety of soliton solutions, including anti-kink, single soliton, anti-bell, bell, kink, multiple soliton, double soliton, singular-kink, compacton shape, periodic shape, and so many, are displayed in the diagram of 3D and contour plots by taking into account a number of various parameters. It is essential to point out that all derived outcomes are directly compared to the original solutions to certify their exactness. Results show that the used scheme is capable, simple, and straightforward and can be useful to a variety of complex phenomena. The acquired results are unique for the model equation and could be applied to the analysis of several nonlinear study fields.

Novel approach for solving fractional partial differential equations using conformable Elzaki Transform

AbstractThis article discusses three different kinds of fractional partial differential equations (FPDEs) and how to solve them. First, a reformulation of the conformable fractional Elzaki Transform (ET) method is shown. The proposed methodology is then applied to the solution of three different types of FPDEs. The effectiveness and dependability of the proposed strategy are found to be validated by numerical simulations. We believe the provided strategy is very successful and straightforward to adapt of many types of FPDEs. This method is one of the completely new approaches for solving FPDEs because of its exceptional correctness and speed, which set it apart from other approaches.

A three-stage framework for accurate detection of high-speed train body paint film defects

Detecting paint film defects on high-speed train (HST) body is a crucial step towards zero-defect manufacturing (ZDM). The effectiveness of defect detection in industrial environments, however, is significantly impacted by factors such as high reflection noise, weak defect features, small defect size, and various defect scales, resulting in unsatisfactory detection accuracy. To address these challenges, this paper proposes a technical framework for the accurate detection of paint film defects on HST body, encompassing three stages. A reflection detection removal method that takes into account the reflection transition region as well as the preservation of defect features is at first used to reduce reflection noise interference. Then a lightness-weighted adaptive image enhancement algorithm based on the fractional order differentiation is utilized to enhance the features of paint film defects while avoiding the introduction of reflection noise. Finally, an improved YOLOv5 algorithm, incorporating attention mechanism feature extraction and multi-scale feature fusion, is employed to detect and classify paint film defects. Experimental results demonstrate that the proposed framework effectively reduces the reflection in paint film images, and enhances the contrast between paint film defects and the background, with a mean average precision (mAP) for defect detection reaching up to 90.13%. This work serves as a valuable reference for the automated detection of defects arising in the painting process of HST body.

On a Generalized Class of Non-Singular Kernel Operators and Their Singular Kernel Extensions: Useful Modeling Insights

Abstract Some possible definitions of fractional derivative operators with non-singular analytic kernels have been introduced. In this paper, we propose a new generalized class of fractional derivative operators of Caputo-type with non-singular analytic kernels which includes some known operators as special cases. We demonstrate a relationship between the fractional derivative operators of the proposed generalized class and the Riemann-Liouville fractional integral operator. We also, using this relationship, introduce the corresponding fractional integral operators. Then, mainly, we provide extensions to the fractional derivative operators of the proposed generalized class that display integrable singular kernels. The extended fractional derivative operators provide useful insights regarding the modeling issue so that the initialization problem can be overcome. Finally, we discuss some basic properties of the proposed operators that are expected to be widely used in fractional calculus.

Conformable Double Laplace Transform Method (CDLTM) and Homotopy Perturbation Method (HPM) for Solving Conformable Fractional Partial Differential Equations

In the present article, the method which was obtained from a combination of the conformable fractional double Laplace transform method (CFDLTM) and the homotopy perturbation method (HPM) was successfully applied to solve linear and nonlinear conformable fractional partial differential equations (CFPDEs). We included three examples to help our presented technique. Moreover, the results show that the proposed method is efficient, dependable, and easy to use for certain problems in PDEs compared with existing methods. The solution graphs show close contact between the exact and CFDLTM solutions. The outcome obtained by the conformable fractional double Laplace transform method is symmetrical to the gain using the double Laplace transform.

Positive Solutions to a System of Coupled Hadamard Fractional Boundary Value Problems

We explore the existence, uniqueness, and multiplicity of positive solutions to a system of Hadamard fractional differential equations that contain fractional integral terms. Defined on a finite interval, this system is subject to general coupled nonlocal boundary conditions encompassing Riemann–Stieltjes integrals and Hadamard fractional derivatives. To establish the main results, we employ several fixed-point theorems, namely the Banach contraction mapping principle, the Schauder fixed-point theorem, the Leggett–Williams fixed-point theorem, and the Guo–Krasnosel’skii fixed-point theorem.

A novel numerical method to solve fractional ordinary differential equations with proportional Caputo derivatives

In this paper, we develop a novel numerical scheme, namely ‘NPCM-PCDE,’ to integrate fractional ordinary differential equations with proportional Caputo derivatives of the type pc D α u(t) = f 1(t, u(t)), t ≥ 0, 0 < α < 1 involving a non-linear operator f 1. A new method is developed using a natural discretization of the proportional Caputo derivative and the decomposition method to decompose the non-linear operator f 1. The error and stability analyses for the proposed method are provided. Some illustrated examples are given to compare the solution curves graphically with the exact solution and to prove the utility and efficiency of the method. The proposed NPCM-PCDE is found to be efficient, easy to implement, convergent, and stable.

Existence results for some fractional order coupled systems with impulses and nonlocal conditions on the half line

Abstract. In this paper, we deal with initial value problems for coupled systems of nonlinear fractional differential equations, subject to coupled nonlocal initial and impulsive conditions on the half line. Global existence-uniqueness results are obtained under weak conditions allowing the reaction part of the problem to increase indefinitely with time. Our approach relies mainly to some fixed point theorem of Perov’s type in generalized gauge spaces. The obtained results improve, generalize and complement many existing results in the literature. An example illustrating our main finding is also given. Mathematics Subject Classification (2010): 26A33, 93C23, 35E15, 47H10. Keywords: Fractional differential equation, coupled systems, generalized spaces in Perov’s sens, nonlocal initial conditions, impulses.

Monotone iterative technique for a sequential delta-Caputo fractional differential equations with nonlinear boundary conditions

Abstract. In this article, we discuss the existence of extremal solutions for a class of nonlinear sequential δ–Caputo fractional differential equations involving nonlinear boundary conditions. Our results are founded on advanced functional analysis methods. To be more specific, we use the monotone iterative approach in conjunction with the upper and lower solution method to create adequate requirements for the existence of extremal solutions. As an application, we give an example to illustrate our results. Mathematics Subject Classification (2010): 34A08, 26A33. Keywords: Sequential δ–Caputo derivative, nonlinear boundary conditions, monotone iterative technique, upper and lower solutions.

Some operators of fractional calculus and their applications regarding various complex functions analytic in certain domains

Abstract. In this academic research note, some familiar operators prearranged by fractional-order calculus will first be introduced and various characteristic properties of those operators will next be propounded. Through the instrumentality of various earlier results associating with both those operators and some complex- exponential forms, and also in the light of certain special information in [1], [20], [17] and [38], an extensive result together with a variety of its implications consisting of several exponential type inequalities will then be determined. A number of its possible implications will extra be pointed out. Mathematics Subject Classification (2010): 26A33, 30A10, 34A40, 35A30, 41A58, 30C45, 30C55, 30C80, 33D15, 26E05, 33E20. Keywords: Complex plane, domains, regular functions, complex exponential, series expansions, fractional calculus, operators of fractional calculus, exponential type inequalities, differential inequalities.

Analysis of the composition of Brazilian black soybean seeds and mathematical modelling of intermittent drying and extraction of antioxidant compounds using fractional calculus

AbstractThe present study aims to analyze the composition of black soybean seeds, evaluate the intermittent drying of black soybean seeds by fractional calculus, optimize the extraction conditions of the antioxidant compounds by varying the solvent ratio and the extraction time, fit traditional models of extraction kinetics, and compare with the fractional order model. Regarding the oil content, it is analyzed that black soybeans have a high lipid content (18.86%), an oil source of interest for research and industrial applications. Regarding drying, it was found that the first order model cannot be used to describe the kinetics of intermittent drying for black soybean seeds and the best kinetic fits were obtained with the Page and fractional order models, which can be applied in simulations of drying and dryer designs. Regarding the extraction process, ethanol/water 45/55 (v/v) solvent proportion obtained the highest antioxidant compound content during 1 h of extraction. So and MacDonald's and hyperbolic kinetic models presented the best fitting to experimental data. About the fractional order model, it is found that for extraction conditions with more significant amounts of water in the ethanolic solution, the α value obtained is less than 1, resulting in the phenomenon of subdiffusion. For the extraction condition with a low amount of water in the ethanolic solution, the α value was greater than 1, depicting a superdiffusion process.

On α-Fractional Bregman Divergence to study α-Fractional Minty’s Lemma

In this paper fractional variational inequality problems (FVIP) and dual fractional variational inequality problems (DFVIP), Fractional minimization problems are defined with the help of fractional Caputo differential operator and Reimann-Liouville differential operator. The equivalence theorem of Caputo FVIP and fractional minimization problem is established. The equivalence between FVIP and DFVIP is studied with the help of fractional Bregman divergence to establish the fractional Minty’s lemma. Introduction: The theory of variational inequality problems is one of the best tool to solve the equilibrium and non-equilibrium problems arises in the branches of Applied Mathematics, Applied Physics, Engineering, Medical, Financial and many more. On the other hand, fractional calculus manages to study various types of dynamic problems arises in real and complex analysis It is certain for the development of fractional calculus because the fractional derivative has global correlation, which can reflect the historical process of the systematic function.With regard to the definition of fractional derivatives, many mathematicians studied fractional derivatives from different aspects and advanced different definitions for them, for example Riemann-Liouville (RL) derivative, Caputo derivative and other derivatives. Conclusions: The concept of -fractionally differential inequality problem is defined and studied its existence theorem with the help of -fractionally convex function. The concept of -fractionally Bregman divergence is developed and using it the existence of -fractionally minimization problem and -fractionally differential inequality problem are studied. Equivalence theorem in between -fractionally differential inequality problem and -fractionally dual differential inequality problem is studied to establish the Minty’s lemma in fractional calculus.

Non-Local Problems for the Fractional Order Diffusion Equation and the Degenerate Hyperbolic Equation

This research explores nonlocal problems associated with fractional diffusion equations and degenerate hyperbolic equations featuring singular coefficients in their lower-order terms. The uniqueness of the solution is established using the energy integral method, while the existence of the solution is equivalently reduced to solving Volterra integral equations of the second kind and a fractional differential equation. The study focuses on a mixed domain where the parabolic section aligns with the upper half-plane, and the hyperbolic section is bounded by two characteristics of the equation under consideration and a segment of the x-axis. By utilizing the solution representation of the fractional-order diffusion equation, a primary functional relationship is derived between the traces of the sought function on the x-axis segment from the parabolic part of the mixed domain. An explicit solution form for the modified Cauchy problem in the hyperbolic section of the mixed domain is presented. This solution, combined with the problem’s boundary condition, yields a fundamental functional relationship between the traces of the unknown function, mapped to the interval of the equation’s degeneration line. Through the conjugation condition of the problem, an equation with fractional derivatives is obtained by eliminating one unknown function from two functional relationships. The solution to this equation is explicitly formulated. For a specific solution of the proposed problem, visualizations are provided for various orders of the fractional derivative. The analysis demonstrates that the derivative order influences both the intensity of the diffusion (or subdiffusion) process and the shape of the wave front.

Effects of fractional derivative and Wiener process on approximate boundary controllability of differential inclusion

One of the core concepts of contemporary control theory is the idea that a dynamical system can be controlled. Several abstract settings have been developed to describe the distributed control systems in a domain in which the control is acted through the boundary. In this manuscript, we investigate the boundary approximate controllability (ABC) of stochastic differential inclusion (SDI) with Hilfer fractional derivative (HFD) and nonlocal condition by implementing the principle of stochastic analysis, the fixed-point theorem, fractional calculus and multi-valued map. Moreover, an example is offered to define the primary results.