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This document discusses recent trends in mathematical models and solitons used to analyze real-world problems. The focus is on the application of fractional calculus in modeling phenomena such as anomalous diffusion, non-Markovian processes, and random walk. The document provides summaries of 15 papers accepted for publication in a special issue, covering topics such as digestive system modeling, COVID-19 pandemic modeling, control systems for motors, symmetry in pine wilt disease, soliton solutions for various equations, and epidemic modeling. The papers employ various mathematical techniques and provide insights into different aspects of real-world problems. [Extracted from the article]

Grain-oriented silicon steel (GO FeSi) laminations are vital components for efficient energy conversion in electromagnetic devices. While traditionally optimized for power frequencies of 50/60 Hz, the pursuit of higher frequency operation (f ≥ 200 Hz) promises enhanced power density. This paper introduces a model for estimating GO FeSi laminations' magnetic behavior under these elevated operational frequencies. The proposed model combines the Maxwell diffusion equation and a material law derived from a fractional differential equation, capturing the viscoelastic characteristics of the magnetization process. Remarkably, the model's dynamical contribution, characterized by only two parameters, achieves a notable 4.8% Euclidean relative distance error across the frequency spectrum from 50 Hz to 1 kHz. The paper's initial section offers an exhaustive description of the model, featuring comprehensive comparisons between simulated and measured data. Subsequently, a methodology is presented for the localized segregation of magnetic losses into three conventional categories: hysteresis, classical, and excess, delineated across various tested frequencies. Further leveraging the model's predictive capabilities, the study extends to investigating the very high-frequency regime, elucidating the spatial distribution of loss contributions. The application of proportional–iterative learning control facilitates the model's adaptation to standard characterization conditions, employing sinusoidal imposed flux density. The paper deliberates on the implications of GO FeSi behavior under extreme operational conditions, offering insights and reflections essential for understanding and optimizing magnetic core performance in high-frequency applications. [ABSTRACT FROM AUTHOR]

This paper focuses on the analysis of a coupled system governed by a Caputo-fractional derivative with q-integral-coupled boundary conditions. This system is particularly relevant in modeling multi-atomic systems, including scenarios involving adsorbed atoms or clusters on crystalline surfaces, surface–atom scattering, and atomic friction. To investigate this system, we introduce an operator that exhibits fixed points corresponding to the solutions of the problem, effectively transforming the system into an equivalent fixed-point problem. We established the necessary conditions for the existence and uniqueness of solutions using the Leray–Schauder nonlinear alternative and the Banach contraction mapping principle, respectively. Stability results in the Ulam sense for the coupled system are also discussed, along with a sensitivity analysis of the range parameters. To support the validity of their findings, we provide illustrative examples. Overall, the paper offers a thorough examination and analysis of the considered coupled system, making important contributions to the understanding of multi-atomic systems and their mathematical modeling. [ABSTRACT FROM AUTHOR]

Conditions for the existence and uniqueness of mild solutions for a system of semilinear impulsive differential equations with infinite fractional Brownian movements and the Wiener process are established. Our approach is based on a novel application of Burton and Kirk's fixed point theorem in extended Banach spaces. This paper aims to extend current results to a differential-inclusions scenario. The motivation of this paper for impulsive neutral differential equations is to investigate the existence of solutions for impulsive neutral differential equations with fractional Brownian motion and a Wiener process (topics that have not been considered and are the main focus of this paper). [ABSTRACT FROM AUTHOR]

After the initiation of Jachymski's contraction principle via digraph, the area of metric fixed point theory has attracted much attention. A number of outcomes on fixed points in the context of graph metric space employing various types of contractions have been investigated. The aim of this paper is to investigate some fixed point theorems for a class of nonlinear contractions in a metric space endued with a transitive digraph. The outcomes presented herewith improve, extend and enrich several existing results. Employing our findings, we describe the existence and uniqueness of a singular fractional boundary value problem. [ABSTRACT FROM AUTHOR]

In this paper, we present a highly efficient analytical method that combines the Laplace transform and the residual power series approach to approximate solutions of nonlinear time-fractional partial differential equations (PDEs). First, we derive the analytical method for a general form of fractional partial differential equations. Then, we apply the proposed method to find approximate solutions to the time-fractional coupled Berger equations, the time-fractional coupled Korteweg–de Vries equations and time-fractional Whitham–Broer–Kaup equations. Secondly, we extend the proposed method to solve the two-dimensional time-fractional coupled Navier–Stokes equations. The proposed method is validated through various test problems, measuring quality and efficiency using error norms E 2 and E ∞ , and compared to existing methods. [ABSTRACT FROM AUTHOR]

In this paper, we present an efficient method for solving a class of higher order fractional differential equations with general boundary conditions. The convergence of the numerical method is proved and an error estimate is given. Finally, eight numerical examples, both linear and nonlinear, are presented to demonstrate the accuracy of our method. The proposed method introduces suitable base functions to calculate the approximate solutions and only requires us to deal with the linear or nonlinear systems. Thus, our method is convenient to implement. Furthermore, the numerical results show that the proposed method performs better compared to the existing ones. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of at least two weak solutions to a class of singular two-phase problems with variable exponents involving a ψ-Hilfer fractional operator and Dirichlet-type boundary conditions when the term source is dependent on one parameter. Here, we use the fiber method and the Nehari manifold to prove our results. [ABSTRACT FROM AUTHOR]

The two-parameter Mittag–Leffler function E α , β is of fundamental importance in fractional calculus, and it appears frequently in the solutions of fractional differential and integral equations. However, the expense of calculating this function often prompts efforts to devise accurate approximations that are more cost-effective. When α > 1 , the monotonicity property is largely lost, resulting in the emergence of roots and oscillations. As a result, current rational approximants constructed mainly for α ∈ (0 , 1) often fail to capture this oscillatory behavior. In this paper, we develop computationally efficient rational approximants for E α , β (− t) , t ≥ 0 , with α ∈ (1 , 2) . This process involves decomposing the Mittag–Leffler function with real roots into a weighted root-free Mittag–Leffler function and a polynomial. This provides approximants valid over extended intervals. These approximants are then extended to the matrix Mittag–Leffler function, and different implementation strategies are discussed, including using partial fraction decomposition. Numerical experiments are conducted to illustrate the performance of the proposed approximants. [ABSTRACT FROM AUTHOR]

Ordinary differential equations have recently been extended to fractional equations that are transformed using fractional differential equations. These fractional equations are believed to have high accuracy and low computational cost compared to ordinary differential equations. For the first time, this paper focuses on extending the nonlinear heat equations to a fractional order in a Caputo order. A new perturbation iteration algorithm (PIA) of the fractional order is applied to solve the nonlinear heat equations. Solving numerical problems that involve fractional differential equations can be challenging due to their inherent complexity and high computational cost. To overcome these challenges, there is a need to develop numerical schemes such as the PIA method. This method can provide approximate solutions to problems that involve classical fractional derivatives. The results obtained from this algorithm are compared with those obtained from the perturbation iteration method (PIM), the variational iteration method (VIM), and the Bezier curve method (BCM). All solutions are tested with numerical simulations. The study found that the new PIA algorithm performs better than the PIM, VIM, and BCM, achieving high accuracy and low computational cost. One significant advantage of this algorithm is that the solutions obtained have established that the fractional values of alpha, specifically α , significantly influencing the accuracy of the outcome and the associated computational cost. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of multi-term implicit fractional differential equation with boundary conditions. The application of the Schauder fixed point theorem and the Banach fixed point theorem allows us to establish the criterion for a solution that exists for the given equation, and the solution is unique. Afterwards, we give the criteria of Ulam–Hyers stability and Ulam–Hyers–Rassias stability. Additionally, we present an example to illustrate the practical application and effectiveness of the results. [ABSTRACT FROM AUTHOR]

In [[10]], the authors investigate certain image formulas related to the product of Srivastava polynomials and extended Wright functions using fractional integral and differential operators, including Marichev-Saigo-Maeda, Lavoie-Trottier, and Oberhettinger. The paper [[13]] investigates the Dirichlet and modified Dirichlet average of the I R i function, i.e., the Mittag-Leffler-type function. Among the numerous applications of the theory of fractional calculus in almost all applied sciences, applications in numerical analysis and various fields of physics and engineering stand out. [Extracted from the article]

The Ebola virus continues to be the world's biggest cause of mortality, especially in developing countries, despite the availability of safe and effective immunization. In this paper, we construct a fractional-order Ebola virus model to check the dynamical transmission of the disease as it is impacted by immunization, learning, prompt identification, sanitation regulations, isolation, and mobility limitations with a constant proportional Caputo (CPC) operator. The existence and uniqueness of the proposed model's solutions are discussed using the results of fixed-point theory. The stability results for the fractional model are presented using Ulam–Hyers stability principles. This paper assesses the hybrid fractional operator by applying methods to invert proportional Caputo operators, calculate CPC eigenfunctions, and simulate fractional differential equations computationally. The Laplace–Adomian decomposition method is used to simulate a set of fractional differential equations. A sustainable and unique approach is applied to build numerical and analytic solutions to the model that closely satisfy the theoretical approach to the problem. The tools in this model appear to be fairly powerful, capable of generating the theoretical conditions predicted by the Ebola virus model. The analysis-based research given here will aid future analysis and the development of a control strategy to counteract the impact of the Ebola virus in a community. [ABSTRACT FROM AUTHOR]

This paper presents a groundbreaking numerical technique for solving nonlinear time fractional differential equations, combining the conformable continuity equation (CCE) with the Non-Polynomial Spline (NPS) interpolation to address complex mathematical challenges. By employing conformable descriptions of fractional derivatives within the CCE framework, our method ensures enhanced accuracy and robustness when dealing with fractional order equations. To validate our approach's applicability and effectiveness, we conduct a comprehensive set of numerical examples and assess stability using the Fourier method. The proposed technique demonstrates unconditional stability within specific parameter ranges, ensuring reliable performance across diverse scenarios. The convergence order analysis reveals its efficiency in handling complex mathematical models. Graphical comparisons with analytical solutions substantiate the accuracy and efficacy of our approach, establishing it as a powerful tool for solving nonlinear time-fractional differential equations. We further demonstrate its broad applicability by testing it on the Burgers–Fisher equations and comparing it with existing approaches, highlighting its superiority in biology, ecology, physics, and other fields. Moreover, meticulous evaluations of accuracy and efficiency using ( L 2 and L ∞ ) norm errors reinforce its robustness and suitability for real-world applications. In conclusion, this paper presents a novel numerical technique for nonlinear time fractional differential equations, with the CCE and NPS methods' unique combination driving its effectiveness and broad applicability in computational mathematics, scientific research, and engineering endeavors. [ABSTRACT FROM AUTHOR]

In this paper, the tempered Ψ –Riemann–Liouville fractional derivative and the tempered Ψ –Caputo fractional derivative of order n − 1 < α < n ∈ N are introduced for C n − 1 –functions. A nonlinear version of the second Henry–Gronwall inequality for integral inequalities with the tempered Ψ –Hilfer fractional integral is derived. By using this inequality, an existence and uniqueness result and a sufficient condition for the non-existence of blow-up solutions of nonlinear tempered Ψ –Caputo fractional differential equations are proved. Illustrative examples are given. [ABSTRACT FROM AUTHOR]

In recent years, various complex systems and real-world phenomena have been shown to include memory and hereditary properties that change with respect to time, space, or other variables. Consequently, fractional partial differential equations containing variable-order fractional operators have been extensively resorted for modeling such phenomena accurately. In this paper, we consider the two-dimensional fractional cable equation with the Caputo variable-order fractional derivative in the time direction, which is preferable for describing neuronal dynamics in biological systems. A point-wise scheme, namely, the Crank–Nicolson finite difference method, along with a group-wise scheme referred to as the explicit decoupled group method are proposed to solve the problem under consideration. The stability and convergence analyses of the numerical schemes are provided with complete details. To demonstrate the validity of the proposed methods, numerical simulations with results represented in tabular and graphical forms are given. A quantitative analysis based on the CPU timing, iteration counting, and maximum absolute error indicates that the explicit decoupled group method is more efficient than the Crank–Nicolson finite difference scheme for solving the variable-order fractional equation. [ABSTRACT FROM AUTHOR]

This paper aims to establish the existence and uniqueness of mild solutions to abstract sequential fractional differential equations. The approach employed involves the utilization of resolvent operators and the fixed-point theorem. Additionally, we investigate a specific example concerning a partial differential equation incorporating the Caputo fractional derivative. [ABSTRACT FROM AUTHOR]

In this paper, a new class of coupled hybrid systems of proportional sequential ψ -Hilfer fractional differential equations, subjected to nonlocal boundary conditions were investigated. Based on a generalization of the Krasnosel'ski i ˘ 's fixed point theorem due to Burton, sufficient conditions were established for the existence of solutions. A numerical example was constructed illustrating the main theoretical result. For special cases of the parameters involved in the system many new results were covered. The obtained result is new and significantly contributes to existing results in the literature on coupled systems of proportional sequential ψ -Hilfer fractional differential equations. [ABSTRACT FROM AUTHOR]

In this study, we explore fractional partial differential equations as a more generalized version of classical partial differential equations. These fractional equations have shown promise in providing improved descriptions of certain phenomena under specific circumstances. The main focus of this paper comprises the development, analysis, and application of two explicit finite difference schemes to solve an initial boundary value problem involving a fuzzy time fractional convection–diffusion equation with a fractional order in the range of 0 ≤   ξ   ≤   1 . The uniqueness of this problem lies in its consideration of fuzziness within both the initial and boundary conditions. To handle the uncertainty, we propose a computational mechanism based on the double parametric form of fuzzy numbers, effectively converting the problem from an uncertain format to a crisp one. To assess the stability of our proposed schemes, we employ the von Neumann method and find that they demonstrate unconditional stability. To illustrate the feasibility and practicality of our approach, we apply the developed scheme to a specific example. [ABSTRACT FROM AUTHOR]

Over the past few years, many scholars began to study averaging principles for fractional stochastic differential equations since they can provide an approximate analytical method to reduce such systems. However, in the most previous studies, there is a misunderstanding of the standard form of fractional stochastic differential equations, which consequently causes the wrong estimation of the convergence rate. In this note, we take fractional stochastic differential equations with Lévy noise as an example to clarify these two issues. The corrections herein have no effect on the main proofs except the two points mentioned above. The innovation of this paper lies in three aspects: (i) the standard form of the fractional stochastic differential equations is derived under natural time scale; (ii) it is first proved that the convergence interval and rate are related to the fractional order; and (iii) the presented results contain and improve some well known research achievements. [ABSTRACT FROM AUTHOR]

This paper deals with a nonlocal fractional coupled system of (k , ψ) -Hilfer fractional differential equations, which involve, in boundary conditions, (k , ψ) -Hilfer fractional derivatives and (k , ψ) -Riemann–Liouville fractional integrals. The existence and uniqueness of solutions are established for the considered coupled system by using standard tools from fixed point theory. More precisely, Banach and Krasnosel'skiĭ's fixed-point theorems are used, along with Leray–Schauder alternative. The obtained results are illustrated by constructed numerical examples. [ABSTRACT FROM AUTHOR]

This paper is dedicated to studying the existence results of solutions to the nonlinear coupled system of Hilfer fractional differential equations and inclusions, with multi-strip and multi-point mixed boundary conditions. Through tools such as the Leray-Schauder alternative and the nonlinear alternative of Leray-Schauder type, continuous and measurable selection theorems, along with Leray-Schauder degree theory, the main results can be obtained. The Hilfer fractional differential system has practical implications for specific physical phenomena. Examples are provided to clarify the application of our main results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an initial value problem for a nonlinear fractional differential equation on an infinite interval. The differential operator is taken in the Hadamard sense and the nonlinear term involves two lower-order fractional derivatives of the unknown function. In order to establish the global existence criteria, we first verify that there exists a unique positive solution to an integral equation based on a class of new integral inequality. Next, we construct a locally convex space, which is metrizable and complete. On this space, applying Schäuder's fixed point theorem, we obtain the existence of at least one solution to the initial value problem. [ABSTRACT FROM AUTHOR]

In this paper, we improved a mathematical model of monkeypox disease with a time delay to a crossover model by incorporating variable-order and fractional differential equations, along with stochastic fractional derivatives, in three different time intervals. The stability and positivity of the solutions for the proposed model are discussed. Two numerical methods are constructed to study the behavior of the proposed models. These methods are the nonstandard modified Euler Maruyama technique and the nonstandard Caputo proportional constant Adams-Bashfourth fifth step method. Many numerical experiments were conducted to verify the efficiency of the methods and support the theoretical results. This study's originality is the use of fresh data simulation techniques and different solution methodologies. [ABSTRACT FROM AUTHOR]

In this paper, the fractional Benjamin–Ono differential equation with a Riemann–Liouville fractional derivative is considered using the Lie symmetry analysis method. Two symmetries admitted by the equation are obtained. Then, the equation is reduced to a fractional ordinary differential equation with an Erdélyi–Kober fractional derivative by one of the symmetries. Finally, conservation laws for the equations are constructed using the new conservation theorem. [ABSTRACT FROM AUTHOR]

The current values of many populations depend on the past values of the population. In many cases, this dependence is caused by the time certain processes take. This dependence on the past can be introduced into mathematical models by adding delays. For example, the growth rate of a population depends on the population τ time units ago, where τ is the maturation time. For an epidemic, there is a time τ between the contact of an infected individual and a susceptible one, and the time the susceptible individual actually becomes infected. This time τ is also a delay. So, the number of infected individuals depends on the population at the time τ units ago. A second way of introducing this dependence on past values is to use non-local operators in the description of the model. Fractional derivatives have commonly been used to provide non-local effects. In population growth models, it can also be done by introducing a new compartment, the immature population, and in epidemic models, by introducing an additional exposed population. In this paper, we study and compare these methods of adding dependence on past values. For models of processes that involve delays, all three methods include dependence on past values, but fractional-order models do not justify the form of the dependence. Simulations show that for the models studied, the fractional differential equation method produces similar results to those obtained by explicitly incorporating the delay, but only for specific values of the fractional derivative order, which is an extra parameter. But in all three methods, the results are improved compared to using ordinary differential equations. [ABSTRACT FROM AUTHOR]

This paper introduces fractional operators in the complex domain as generalizations for the Srivastava–Owa operators. Some properties for the above operators are also provided. We discuss the convexity and starlikeness of the generalized Libera integral operator. A condition for the convexity and starlikeness of the solutions of fractional differential equations is provided. Finally, a fractional differential equation is converted into an ordinary differential equation by wave transformation; illustrative examples are provided to clarify the solution within the complex domain. [ABSTRACT FROM AUTHOR]

We investigated the blow-up of the weak solution to a class of fractional nonlinear stochastic differential equations driven by multiplicative noise in this paper. The a priori estimates and Galerkin method were applied to demonstrate the existence and uniqueness of the weak solution. Underlying the hypotheses of the nonlinear function and the initial data, for finite time, we prove that the solution does not blow up. Additionally, under further assumptions, we verified that the presence of multiplicative noise can delay the blow-up of the solution to infinity. [ABSTRACT FROM AUTHOR]

This paper solves a generalized class of first-order fractional ordinary differential equations (1st-order FODEs) by means of Riemann–Liouville fractional derivative (RLFD). The principal incentive of this paper is to generalize some existing results in the literature. An effective approach is applied to solve non-homogeneous fractional differential systems containing 2 n periodic terms. The exact solutions are determined explicitly in a straightforward manner. The solutions are expressed in terms of entire functions with fractional order arguments. Features of the current solutions are discussed and analyzed. In addition, the existing solutions in the literature are recovered as special cases of our results. [ABSTRACT FROM AUTHOR]

The focus of this paper is to conduct a comprehensive analysis of the advancements made in the understanding of Interpolative contraction, building upon the ideas initially introduced by Karapinar in 2018. In this paper, we develop the notion of Interpolative contraction mappings to the case of non-linear Kannan Interpolative, Riech Rus Ćirić interpolative and Hardy–Roger Interpolative contraction mappings based on controlled function, and prove some fixed point results in the context of controlled metric space, thereby enhancing the current understanding of this particular analysis. Furthermore, we provide a concrete example that illustrates the underlying drive for the investigations presented in this context. An application of the proposed non-linear Interpolative-contractions to the Liouville–Caputo fractional derivatives and fractional differential equations is provided in this paper. [ABSTRACT FROM AUTHOR]

The paper focuses on exploring the existence and uniqueness of a specific solution to a class of Caputo impulsive fractional differential equations with boundary value conditions on Banach space, referred to as (ω , ρ) -BVP solution. The proof of the main results of this study involves the application of the Banach contraction mapping principle and Schaefer's fixed point theorem. Furthermore, we provide the necessary conditions for the convexity of the set of solutions of the analyzed impulsive fractional differential boundary value problem. To enhance the comprehension and practical application of our findings, we conclude the paper by presenting two illustrative examples that demonstrate the applicability of the obtained results. [ABSTRACT FROM AUTHOR]

Fractional differential equations are used to construct mathematical models and can describe the characteristics of real systems. In this paper, the parameter estimation problem of a fractional Wiener system is studied by designing linear filters which can obtain smaller tunable parameters and maintain the stability of the parameters in any case. To improve the identification performance of the stochastic gradient algorithm, this paper derives two modified stochastic gradient algorithms for the fractional nonlinear Wiener systems with colored noise. By introducing the forgetting factor, a forgetting factor stochastic gradient algorithm is deduced to improve the convergence rate. To achieve more efficient and accurate algorithms, we propose a multi-innovation forgetting factor stochastic gradient algorithm by means of the multi-innovation theory, which expands the scalar innovation into the innovation vector. To test the developed algorithms, a fractional-order dynamic photovoltaic model is employed in the simulation, and the dynamic elements of this photovoltaic model are estimated using the modified algorithms. Concurrently, a numerical example is given, and the simulation results verify the feasibility and effectiveness of the proposed procedures. [ABSTRACT FROM AUTHOR]

The main purpose of this paper was to consider new sandwich pairs and investigate the existence of a solution for a new class of fractional differential equations with p-Laplacian via variational methods in ψ -fractional space H p α , β ; ψ (Ω) . The results obtained in this paper are the first to make use of the theory of ψ -Hilfer fractional operators with p-Laplacian. [ABSTRACT FROM AUTHOR]

The state feedback controller design for a class of Generalized Proportional Fractional Order (GPFO) Nonlinear Systems is presented in this paper. The design is based on the combination of the One-Sided Lipschitz (OSL) system class with GPFO modeling. The main contribution of this study is that, to the best of the authors' knowledge, this work presents the first state feedback control design for GPFO systems. The suggested state feedback controller is intended to ensure the system's generalized Mittag Leffler (GML) stability and to deliver optimal performance. The findings of this paper show that the proposed strategy is effective in stabilizing Generalized Proportional Fractional Order Nonlinear Systems. A numerical example is presented to demonstrate the usefulness of the stated theoretical conclusions. [ABSTRACT FROM AUTHOR]

In this paper, we present an application of the renormalization group method developed by Chen, Goldenfeld and Oono for a stochastic differential equation in a space of Hilbert space-valued generalized random variables with multiplicative noise. The driving process is a real-valued fractional white noise with a Hurst parameter greater than 1 / 2 . The stochastic integration is understood in the Wick–Itô–Skorohod sense. This article is a generalization of results of Glatt-Holtz and Ziane, which were for the systems with white noise. We firstly demonstrate the process of formulating the renormalization group equation and the asymptotic solution. Then, we give rigorous proof of the consistency of the approximate solution. In addition, some numerical comparisons are given to illustrate the validity of our results. [ABSTRACT FROM AUTHOR]

In this paper, we present a more general approach based on a Picard integral scheme for nonlinear partial differential equations with a variable time-fractional derivative of order α (x , t) ∈ (1 , 2) and space-fractional order s ∈ (0 , 1) , where v = u ′ (t) is introduced as the new unknown function and u is recovered using the quadrature. In order to get rid of the constraints of traditional plans considering the half-time situation, integration by parts and the regularity process are introduced on the variable v. The convergence order can reach O (τ 2 + h 2) , different from the common L 1 , 2 − α schemes with convergence rate O (τ 2 , 3 − α (x , t)) under the infinite norm. In each integer time step, the stability, solvability and convergence of this scheme are proved. Several error results and convergence rates are calculated using numerical simulations to evidence the theoretical values of the proposed method. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence and multiplicity of monotone positive solutions for a class of nonlinear fractional differential equation with a disturbance parameter in the integral boundary conditions on the infinite interval. By using Guo–Krasnosel'skii fixed-point theorem and the analytic technique, we divide the range of parameter for the existence of at least two, one and no positive solutions for the problem. In the end, an example is given to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, we present a fractional version of the Sakiadis flow described by a nonlinear two-point fractional boundary value problem on a semi-infinite interval, in terms of the Caputo derivative. We derive the fractional Sakiadis model by substituting, in the classical Prandtl boundary layer equations, the second derivative with a fractional-order derivative by the Caputo operator. By using the Lie symmetry analysis, we reduce the fractional partial differential equations to a fractional ordinary differential equation, and, then, a finite difference method on quasi-uniform grids, with a suitable variation of the classical L1 approximation formula for the Caputo fractional derivative, is proposed. Finally, highly accurate numerical solutions are reported. [ABSTRACT FROM AUTHOR]

In this paper, we derive an explicit formula of solutions to Hilfer linear fractional integro-differential equations with a variable coefficient in a weighted space, and obtain the existence and uniqueness of solutions for fractional kinetic equations and fractional integro-differential equations with a generalized Mittag–Leffler function. An example is given to illustrate the result obtained. [ABSTRACT FROM AUTHOR]

In this article, we are concerned with a very general integral boundary value problem of Riemann–Liouville derivatives. We will study the problem in Banach space. To be more specific, we are interested in proving the existence of a solution to our problem via the measure of noncompactness and Mönch fixed-point theorem. Our study in Banach space contains two nonlinear terms and two different orders of derivatives, ς and τ , such that ς ∈ 1 , 2 and τ ∈ 0 , ς . Our paper ends with a conclusion. [ABSTRACT FROM AUTHOR]

This paper presents the construction of a family of radial functions aimed at emulating the behavior of the radial basis function known as thin plate spline (TPS). Additionally, a method is proposed for applying fractional derivatives, both partially and fully, to these functions for use in interpolation problems. Furthermore, a technique is employed to precondition the matrices generated in the presented problems through Q R decomposition. Similarly, a method is introduced to define two different types of abelian groups for any fractional operator defined in the interval [ 0 , 1) , among which the Riemann–Liouville fractional integral, Riemann–Liouville fractional derivative, and Caputo fractional derivative are worth mentioning. Finally, a form of radial interpolant is suggested for application in solving fractional differential equations using the asymmetric collocation method, and examples of its implementation in differential operators utilizing the aforementioned fractional operators are shown. [ABSTRACT FROM AUTHOR]

An initial value problem for nonlinear fractional differential equations with a tempered Caputo fractional derivative of variable order with respect to another function is studied. The absence of semigroup properties of the considered variable order fractional derivative leads to difficulties in the study of the existence of corresponding differential equations. In this paper, we introduce approximate piecewise constant approximation of the variable order of the considered fractional derivative and approximate solutions of the given initial value problem. Then, we investigate the existence and the Ulam-type stability of the approximate solution of the variable order Ψ -tempered Caputo fractional differential equation. As a partial case of our results, we obtain results for Ulam-type stability for differential equations with a piecewise constant order of the Ψ -tempered Caputo fractional derivative. [ABSTRACT FROM AUTHOR]

This paper introduces a novel concept of the impulsive delayed Mittag–Leffler-type vector function, an extension of the Mittag–Leffler matrix function. It is essential to seek explicit formulas for the solutions to linear impulsive fractional differential delay equations. Based on explicit formulas of the solutions, the finite-time stability results of impulsive fractional differential delay equations are presented. Finally, we present four examples to illustrate the validity of our theoretical results. [ABSTRACT FROM AUTHOR]

The results for a new modeling integral boundary value problem (IBVP) using Caputo-Hadamard impulsive fractional integro-differential equations (C-HIFI-DE) with Banach space are investigated, along with the existence and uniqueness of solutions. The Krasnoselskii fixed-point theorem (KFPT) and the Banach contraction principle (BCP) serve as the basis of this unique strategy, and are used to achieve the desired results. We develop the illustrated examples at the end of the paper to support the validity of the theoretical statements. [ABSTRACT FROM AUTHOR]

This paper applies two different types of Riemann–Liouville derivatives to solve fractional differential equations of second order. Basically, the properties of the Riemann–Liouville fractional derivative depend mainly on the lower bound of the integral involved in the Riemann–Liouville fractional definition. The Riemann–Liouville fractional derivative of first type considers the lower bound as a zero while the second type applies negative infinity as a lower bound. Due to the differences in properties of the two operators, two different solutions are obtained for the present two classes of fractional differential equations under appropriate initial conditions. It is shown that the zeroth lower bound implies implicit solutions in terms of the Mittag–Leffler functions while explicit solutions are derived when negative infinity is taken as a lower bound. Such explicit solutions are obtained for the current two classes in terms of trigonometric and hyperbolic functions. Some theoretical results are introduced to facilitate the solutions procedures. Moreover, the characteristics of the obtained solutions are discussed and interpreted. [ABSTRACT FROM AUTHOR]

In this paper, the existence of a unique solution is established for a coupled system of Langevin fractional problems of ψ -Caputo fractional derivatives with generalized slit-strip-type integral boundary conditions and impulses using the Banach contraction principle. We also find at least one solution to the aforementioned system using some assumptions and Schaefer's fixed point theorem. After that, Ulam–Hyers stability is discussed. Finally, to provide additional support for the main results, pertinent examples are presented. [ABSTRACT FROM AUTHOR]

This paper proposes a novel approach for analyzing the stability of polynomial fractional-order systems using the frequency-distributed fractional integrator model. There are two types of frequency and temporal stabilization methods for fractional-order systems that global and semi-global stability conditions attain using the sum-of-squares (SOS) method. Substantiation conditions of global and asymptotical stability are complicated for fractional polynomial systems. According to recent studies on nonlinear fractional-order systems, this paper concentrates on polynomial fractional-order systems with any degree of nonlinearity. Global stability conditions are obtained for polynomial fractional-order systems (PFD) via the sum-of-squares approach and the frequency technique employed. This method can be effective in nonlinear systems where the linear matrix inequality (LMI) approach is incapable of response. This paper proposes to solve non-convex SOS-designed equations and design framework key ideas to avoid conservative problems. A Lyapunov polynomial function is determined to address the problem of PFD stabilization conditions and stability established using sufficiently expressed conditions. The main goal of this article is to present an analytical method based on the optimization method for fractional order models in the form of frequency response. This method can convert it into an optimization problem, and by changing the solution of the optimization problem, the stability of the fractional-order system can be improved. [ABSTRACT FROM AUTHOR]

In recent times, fractional calculus has gained popularity in various types of engineering applications. Very often, the mathematical model describing a given phenomenon consists of a differential equation with a fractional derivative. As numerous studies present, the use of the fractional derivative instead of the classical derivative allows for more accurate modeling of some processes. A numerical solution of anomalous heat conduction equation with Riemann-Liouville fractional derivative over space is presented in this paper. First, a differential scheme is provided to solve the direct problem. Then, the inverse problem is considered, which consists in identifying model parameters such as: thermal conductivity, order of derivative and heat transfer. Data on the basis of which the inverse problem is solved are the temperature values on the right boundary of the considered space. To solve the problem a functional describing the error of the solution is created. By determining the minimum of this functional, unknown parameters of the model are identified. In order to find a solution, selected heuristic algorithms are presented and compared. The following meta-heuristic algorithms are described and used in the paper: Ant Colony Optimization (ACO) for continous function, Butterfly Optimization Algorithm (BOA), Dynamic Butterfly Optimization Algorithm (DBOA) and Aquila Optimize (AO). The accuracy of the presented algorithms is illustrated by examples. [ABSTRACT FROM AUTHOR]

In this paper, we define an operator function as a series of operators corresponding to the Taylor series representing the function of the complex variable. In previous papers, we considered the case when a function has a decomposition in the Laurent series with the infinite principal part and finite regular part. Our central challenge is to improve this result having considered as a regular part an entire function satisfying the special condition of the growth regularity. As an application, we consider an opportunity to broaden the conditions imposed upon the second term not containing the time variable of the evolution equation in the abstract Hilbert space. [ABSTRACT FROM AUTHOR]

This paper explores the social dynamics of processes in complex systems involving humans by focusing on user activity in online media outlets. The R/S analysis showed that the time series of the processes under consideration are fractal and anti-persistent (they have a short-term memory and a Hurst exponent significantly less than 0.5). Following statistical processing, the observed data showed that there is a small amount of asymmetry in the distribution of user activity change amplitudes in news comments; the amplitude distribution is almost symmetrical, but there is a heavy tail as the probability plots lie above the normal probability plot. The fractality of the time series for the observed processes could be due to the variables describing them (the time and level of a series), which are characterized by fractional variables of measurement. Therefore, when figuring out how to approximate functions to determine the probability density of their parameters, it is advisable to use fractional differential equations, such as those of the diffusion type. This paper describes the development of such a model and uses the observed data to analyze and compare the modeling results. [ABSTRACT FROM AUTHOR]