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The aim of this paper is to study the existence of self-similar solutions to the Cauchy problem for the viscous Burgers fractional equation with initial conditions in Marcinkiewcz spaces L (p , ∞) . In particular, we obtain the decay of the Mittag–Leffler family associated with the linear problem in Marcinkiewcz spaces and of its gradient. [ABSTRACT FROM AUTHOR]

For a fractional potential-KdV equation, Lie point symmetries are studied and their infinitesimal generators are investigated to construct conservation laws. Furthermore, the invariants of the obtained generators are used to reduce the studied equation to a nonlinear fractional ordinary differential equation (NFODE) and with Erdélyi–Kober fractional operator the reduced equation is solved explicitly by using the power series approach. [ABSTRACT FROM AUTHOR]

In this paper, we first investigate the Palais-Smale compactness condition of the energy functional associated to a m (ξ) -Kirchhoff-type operator in the appropriate fractional space setting. In this sense, using the Mountain Pass Theorem and the Fountain Theorem, we investigate the existence and multiplicity of weak solutions for a new class of fractional differential equations with m (ξ) -Kirchhoff-type equation. [ABSTRACT FROM AUTHOR]

In this paper, the space–time fractional Bloch–Torrey equation is discretized in space using second and fourth-order schemes. Then, it leads to a system of Caputo fractional ordinary differential equations over time. A highly accurate exponential integrator is utilized to solve systems of fractional differential equations. We rigorously demonstrate the stability and convergence of the schemes by employing the energy technique. Finally, some numerical examples are presented to validate the accuracy and efficiency of the proposed algorithms. [ABSTRACT FROM AUTHOR]

This paper presents the so-called shifted Jacobi method, an efficient numerical technique to solve second-order periodic boundary value problems with finitely many singularities involving nonlinear systems of two points. The method relies on the Jacobi polynomials used as natural basis functions in the conformable sense of fractional derivative. A study is carried out to compare the outcomes of the shifted Jacobi approach with those of other methods that are currently in use. In the same vein, a theoretical result for establishing a bound of the error generated from the proposed approximate solution is proved accordingly. The efficiency and effectiveness of the shifted Jacobi technique with conformable fractional derivative are discussed numerically. [ABSTRACT FROM AUTHOR]

The existence and uniqueness of common fixed point theorems are demonstrated in this work through the use of L -cyclic (α , β) s -contraction mapping and bipolar controlled b-dislocated metric space. In the scope of an application, we consider the potential for a unique common solution for nonintegral and nonlinear fractional differential equations. [ABSTRACT FROM AUTHOR]

Fractional‐order ray equations and fractional Duffing‐van der Pol oscillator equations are relationships utilized as a reliable means of modeling some phenomena in optics and engineering. The main motivation of this study is to introduce a new hybrid technique utilizing Chelyshkov wavelets and least squares‐support vector regression (LS‐SVR) for determining the approximate solution of fractional ray equations and fractional Duffing‐van der Pol oscillator equations (D‐v POEs). With the help of the Riemann‐Liouville operator for Chelyshkov wavelets and LS‐SVR (called Chw‐Ls‐SVR), the mentioned problems transform into systems of algebraic equations. The convergence analysis is discussed. Finally, the numerical results are proposed and compared with some schemes to display the capability of the numerical technique proposed here. [ABSTRACT FROM AUTHOR]

In this study, using Caputo and ABC derivatives, we present a mathematical analysis of two fractional models for poliomyelitis, considering the presence of vaccination (V) and a post‐paralytic class (A). The existence and uniqueness of solutions are proved. The basic reproduction number R0$$ {\mathcal{R}}_0 $$ is computed. Local and global stability of the disease‐free stationary state, depending on the threshold R0$$ {\mathcal{R}}_0 $$, is provided, along with conditions for the existence of an endemic stationary state. Moreover, we performed a sensitivity analysis to study the influence of all biological parameters on R0$$ {\mathcal{R}}_0 $$. We concluded our study with numerical simulations to illustrate the models' dynamics and to compare the trajectories of Caputo and ABC solutions. We found that the Caputo and ABC operators are both convenient for the modelization of the poliomyelitis disease. However, the ABC operator not only refined the Caputo operator by removing singularity from the kernel expression but also brought out heredity and memory in the model's characteristics. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a stochastic semilinear differential equations driven by both Brownian motion and fractional Brownian motion. Firstly, we establish an inequality for the distance between finite‐dimensional distributions of a random process at two different moments. Then, using the properties of stochastic integrals, fixed point theorems, and based on this inequality, we establish the existence and uniqueness of Besicovich almost automorphic solutions in finite‐dimensional distributions for this type of semilinear equation. Finally, we provide an example to demonstrate the effectiveness of our results. Our results are new to stochastic differential equations driven by Brownian motion or stochastic differential equations driven by fractional Brownian motion. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of positive solution to boundary value problem of fractional differential equations with p$$ p $$‐Laplacian operator. By using Avery–Peterson theorem, some new existence results of three increasing positive solutions are obtained. [ABSTRACT FROM AUTHOR]

In this article, we generalized Mittag‐Leffler‐type functions F¯A(3),F¯B(3),F¯C(3)$$ {\overline{F}}_A^{(3)},{\overline{F}}_B^{(3)},{\overline{F}}_C^{(3)} $$, and F¯D(3)$$ {\overline{F}}_D^{(3)} $$, which correspond, respectively, to the familiar Lauricella hypergeometric functions FA(3),FB(3),FC(3)$$ {F}_A^{(3)},{F}_B^{(3)},{F}_C^{(3)} $$, and FD(3)$$ {F}_D^{(3)} $$ of three variables. Initially, from the Mittag‐Leffler type function in the simplest form to the functions we are studying, necessary information about the development history, study, and importance of this and hypergeometric type functions will be introduced. Among the various properties and characteristics of these three‐variable Mittag‐Leffler‐type function F¯D(3)$$ {\overline{F}}_D^{(3)} $$, which we investigate in the article, include their relationships with other extensions and generalizations of the classical Mittag‐Leffler functions, their three‐dimensional convergence regions, their Euler‐type integral representations, their Laplace transforms, and their connections with the Riemann‐Liouville operators of fractional calculus. The link of three‐variable Mittag‐Leffler function with fractional differential equation systems involving different fractional orders is necessary on certain applications in physics.Therefore, we provide the systems of partial differential equations which are associated with them. [ABSTRACT FROM AUTHOR]

In this article, we initially provided the relationship between the RL fractional integral and the Caputo fractional derivative of different orders. Additionally, it is clear from the literature that studies into boundary value problems involving multi‐term operators have been conducted recently, and the aforementioned idea is used in the formulation of several novel models. We offer a unique coupled system of fractional delay differential equations with proper respect for the role that multi‐term operators play in the research of fractional differential equations, taking into account the newly established solution for fractional integral and derivative. We also made the assumptions that connected integral boundary conditions would be added on top of n$$ n $$‐fractional differential derivatives. The requirements for the existence and uniqueness of solutions are also developed using fixed‐point theorems. While analyzing various sorts of Ulam's stability results, the qualitative elements of the underlying model will also be examined. In the paper's final section, an example is given for purposes of demonstration. [ABSTRACT FROM AUTHOR]

This paper examines the conformable version of a recently proposed novel (3+1)-dimensional Kadomtsev–Petviashvili–Sawada–Kotera–Ramani equation. First, some basic definitions and properties of the conformable derivative are provided. To find exact solutions to this problem, the modified extended tanh-function, exp(−Φ(ξ))-expansion, and Kudryashov R function methods are applied. The results are illustrated by utilizing some of the gathered data’s physical 3D, contour, and 2D surfaces, which sheds light on how geometric patterns are physically comprehended. These solutions contribute to understanding the potential practical applications of the examined model and other nonlinear representations in the physical sciences. These techniques can provide significant solutions for various fractional differential equations. [ABSTRACT FROM AUTHOR]

In this paper, the classical one-dimensional Dirac equation is considered under the framework of fractal calculus. First, the maximal and minimal operators corresponding to the problem are defined. Then the symmetric operator is obtained, the Green’s function corresponding to the problem is constructed, and the eigenfunction expansion is given. Finally, some examples are given. [ABSTRACT FROM AUTHOR]

This study delves into the existence, uniqueness, and stability of solutions for a nonlinear coupled system incorporating mixed generalized fractional derivatives. The system is characterized by ψ-Caputo and ϕ-Riemann-Liouville fractional derivatives with mixed boundary conditions. We provide essential preliminaries and definitions, followed by a detailed analysis using fixed point theory to establish the main results. Furthermore, we discuss the Hyers-Ulam stability of the proposed system and illustrate the theoretical findings with several examples. This study extends and generalizes various results in the literature and provides new insights into the qualitative behavior of fractional differential systems. [ABSTRACT FROM AUTHOR]

This paper proposes a novel hybrid iteration process, namely the Picard–CR iteration process. We apply the proposed iteration process for the numerical reckoning of fixed points of generalized α -nonexpansive mappings. We establish weak and strong convergence results of generalized α -nonexpansive mappings. This study demonstrates the superiority of the hybrid approach in terms of convergence speed. Moreover, we numerically compare the proposed iteration process with other well-known ones from the literature. In the comparison, we consider two problems: finding a fixed point of a generalized α -nonexpansive mapping and finding roots of a complex polynomial. In the second problem, we use the so-called polynomiography in the analysis. The results showed that the proposed iteration scheme is better than other three-parameter iteration schemes from the literature. Using the proven fixed-point results, we also obtain solutions to fractional differential equations. [ABSTRACT FROM AUTHOR]

This article discusses the identification of an unknown Robin coefficient in a fractional parabolic equation using a noisy measurement of the ultimate solution over time. The problem is quite complex, involving a nonlocal, nonlinear, and ill‐posed operator. To solve this problem, the article proposes a regularized optimization approach that minimizes a least‐squares cost function. The article also examines the various conceptual and real‐world challenges associated with this problem. The article demonstrates the presence of a singular and stable solution for the optimization problem. It utilizes the Morozov discrepancy principle and the conjugate gradient method to streamline the iterative reconstruction process. The proposed method is accurate and efficient, as several numerical examples demonstrate. [ABSTRACT FROM AUTHOR]

This study investigates the fractional-order Sawada–Kotera and Rosenau–Hyman equations, which significantly model non-linear wave phenomena in various scientific fields. We employ two advanced methodologies to obtain analytical solutions: the q-homotopy Mohand transform method (q-HMTM) and the Mohand variational iteration method (MVIM). The fractional derivatives in the equations are expressed using the Caputo operator, which provides a flexible framework for analyzing the dynamics of fractional systems. By leveraging these methods, we derive diverse types of solutions, including hyperbolic, trigonometric, and rational forms, illustrating the effectiveness of the techniques in addressing complex fractional models. Numerical simulations and graphical representations are provided to validate the accuracy and applicability of derived solutions. Special attention is given to the influence of the fractional parameter on behavior of the solution behavior, highlighting its role in controlling diffusion and wave propagation. The findings underscore the potential of q-HMTM and MVIM as robust tools for solving non-linear fractional differential equations. They offer insights into their practical implications in fluid dynamics, wave mechanics, and other applications governed by fractional-order models. [ABSTRACT FROM AUTHOR]

This paper examines fractional multi-time scale stochastic functional differential equations that, in addition, are driven by fractional noises. Based on a specially crafted fixed-point principle for the so-called "local operators", we prove a Peano-type theorem on the existence of weak solutions, that is, those defined on an extended stochastic basis. To encompass all commonly used particular classes of fractional multi-time scale stochastic models, including those with random delays and impulses at random times, we consider equations with nonlinear random Volterra operators rather than functions. Some crucial properties of the associated integral operators, needed for the proofs of the main results, are studied as well. To illustrate major findings, several existence theorems, generalizing those known in the literature, are offered, with the emphasis put on the most popular examples such as ordinary stochastic differential equations driven by fractional noises, fractional stochastic differential equations with variable delays and fractional stochastic neutral differential equations. [ABSTRACT FROM AUTHOR]

In this manuscript, we investigated the existence, uniqueness, and Ulam–Hyers stability results of solutions to implicit Caputo–Hadamard fractional differential equations with noninstantaneous impulses and δ − d e r i v a t i v e initial conditions. By employing Banach's and Schaefer's fixed-point theorems, we attain the required results. Finally, we provide examples that support the main findings we arrived at. [ABSTRACT FROM AUTHOR]

The primary focus of this paper is to explore the concept of an initial condition for a noninstantaneous impulsive fractional integro-differential equation of order 0 < ϑ < 1 in an n-dimensional Euclidean space. Using the Laplace transform method, we derive the solution representation of the given dynamical system and investigate the existence and uniqueness of the mild solution through the topological degree method and Gronwall's inequality. Furthermore, we present a filter model featuring a finite impulsive response, which serves as a practical demonstration of the proposed system because it effectively captures the memory effects inherent in fractional-order systems and enhances system reliability with minimal input. Finally, the numerical computations with graphical illustrations provide concrete examples that validate the theoretical results, showcasing how the behavior of the system is impacted by changes in fractional order and emphasizing the versatility of our proposed method. [ABSTRACT FROM AUTHOR]

This paper focuses on modeling Resistor-Inductor (RL) electric circuits using a fractional Riccati initial value problem (IVP) framework. Conventional models frequently neglect the complex dynamics and memory effects intrinsic to actual RL circuits. This study aims to develop a more precise representation using a fractional-order Riccati model. We present a Jacobi collocation method combined with the Jacobi-Newton algorithm to address the fractional Riccati initial value problem. This numerical method utilizes the characteristics of Jacobi polynomials to accurately approximate solutions to the nonlinear fractional differential equation. We obtain the requisite Jacobi operational matrices for the discretization of fractional derivatives, therefore converting the initial value problem into a system of algebraic equations. The convergence and precision of the proposed algorithm are meticulously evaluated by error and residual analysis. The theoretical findings demonstrate that the method attains high-order convergence rates, dependent on suitable criteria related to the fractional-order parameters and the solution's smoothness. This study not only improves comprehension of RL circuit dynamics but also offers a solid numerical foundation for addressing intricate fractional differential equations. [ABSTRACT FROM AUTHOR]

Recent years have seen a growing interest in fractional differential equations, particularly the fractional Chaffee-Infante ( ℂ ) equation, pivotal for understanding dynamics governed by fractional orders in specific physical systems. Exploring solitary wave solutions, this study employs the extended Khater method and truncated Mittag-Leffler function properties to formulate tailored solutions for the ( ℂ ) model. Through a traveling wave ansatz, the equation transforms into a nonlinear ordinary differential equation, revealing intricate propagation patterns of solitary waves. Visual representations aid comprehension, while rigorous validation ensures solution precision, ultimately providing a comprehensive understanding of system responses to external stimuli. This study effectively integrates analytical and numerical methodologies to derive precise solitary wave solutions, with significant implications for advancing comprehension of complex phenomena in various disciplines governed by fractional-order dynamics. [ABSTRACT FROM AUTHOR]

A novel crossover model for monkeypox disease that incorporates Ψ -Caputo fractional derivatives is presented here, where we use a simple nonstandard kernel function Ψ (t). We can be obtained the Caputo and Caputo–Katugampola derivatives as special cases from the proposed derivative. The crossover dynamics model defines four alternative models: fractal fractional order, fractional order, variable order, and fractional stochastic derivatives driven by fractional Brownian motion over four time intervals. The Ψ -nonstandard finite difference method is designed to solve fractal fractional order, fractional order, and variable order mathematical models. Also, the nonstandard modified Euler Maruyama method is used to study the fractional stochastic model. A comparison between Ψ -nonstandard finite difference method and Ψ -standard finite difference method is presented. Moreover, numerous numerical tests and comparisons with real data were conducted to validate the methods' efficacy and support the theoretical conclusions. [ABSTRACT FROM AUTHOR]

ABSTRACT In this study, two improved versions related to the family of G˜$$ \tilde{G} $$‐approaches for simplicity, we will use from now onG˜≡G′G$$ \left(\mathrm{for}\ \mathrm{simplicity},\mathrm{we}\ \mathrm{will}\ \mathrm{use}\ \mathrm{from}\ \mathrm{now}\ \mathrm{on}\kern0.3em \tilde{G}\equiv \left(\frac{G^{\prime }}{G}\right)\right) $$, namely, the simple G˜$$ \tilde{G} $$‐expansion method and the generalized r+G˜$$ \left(r+\tilde{G}\right) $$‐expansion method, are applied to investigate the families of symmetric solitary wave solutions for the quintic fractional Benney–Lin equation that arises in the liquid film. The G˜$$ \tilde{G} $$‐expansion method is a transformation‐based method that has been used a lot to solve nonlinear partial differential equations and fractional partial differential equations. This method produces several solitary wave solutions to the current problem by supposing a series‐form solution. The generalized r+G˜$$ \left(r+\tilde{G}\right) $$‐expansion method, on the other hand, builds on the simple G˜$$ \tilde{G} $$‐expansion method by adding more parameters r$$ r $$ to the series‐form solution. This makes finding more families of solitary wave solutions possible and better shows how the system changes over time. These techniques identify various traveling waves, such as periodic, kink, M$$ M $$‐shaped, bell‐shaped, shock waves and others physical solutions. Some obtained solutions are graphically discussed to better visualize the wave phenomena connected to various symmetrical solitary wave solutions. The fractional Benney–Lin equation's dynamics and wave characteristics may be better understood through these graphical depictions, which makes it easier to analyze the model's behavior in detail. [ABSTRACT FROM AUTHOR]

In this study, we employed the M-truncated fractional singular manifold method to analytically address the (2+1)-dimensional M-truncated fractional Burgers equation. This approach involves reformulating the original fractional differential equation into a more tractable form through the introduction of a singular manifold. This transformation simplifies the problem and often leads to analytical solutions. We derive a general solution expressed in terms of arbitrary functions, which enables us to accommodate variations in system parameters or initial conditions. This results in a versatile expression that captures a broad spectrum of possible solutions, providing a framework for analyzing the dynamics of kink waves in the relevant fractional differential models. We also construct multiple kink wave solutions, offering analytical representations of kink wave behavior within these models. Notably, our findings revert to well-established results when the fractional order is set to one, thereby affirming the consistency of this method with existing theories and validating our approach. [ABSTRACT FROM AUTHOR]

Higher-dimensional nonlinear models can describe more complex evolutionary mechanisms. In this paper, we considered the time fractional (3 + 1) -dimensional modified extended Zakharov–Kuznetsov equation with the sense of the Riemann–Liouville fractional derivative in plasma physics. In the first place, the existence of symmetry of this studied equation through the symmetry scheme was proved. Then, the optimal system to the time fractional (3 + 1) -dimensional modified extended Zakharov–Kuznetsov equation was also constructed. Subsequently, the time fractional higher-dimensional equation was reduced into the lower-dimensional fractional differential equation with the help of the Erdélyi–Kober fractional operators. Last, some conservation laws by using a new conservation theorem were also given. These novel results provide a window for us to discover this high-dimensional nonlinear equation. [ABSTRACT FROM AUTHOR]

This paper presents a method for solving non-homogeneous linear sequential fractional differential equations (NHLSFDEs) with constant coefficients involving conformable fractional derivatives. For this purpose, the fundamental properties of the conformable derivative and fractional exponential functions are discussed. After this, we determined the particular integrals (PIs) of NHLSFDE in terms of fractional exponential functions, fractional cosine and sine functions. We have demonstrated this developed method with a few examples of NHLSFDEs. [ABSTRACT FROM AUTHOR]

In this paper, we study the boundary value problem for a singular fractional differential equation with both generalized Laplacian and positive parameter. Based on the well-known results of fixed point index on cones due to Guo-Krasnoselskii and Leggett-Williams, we establish some sufficient conditions for the existence of at least one, two and three positive solutions, respectively. Meanwhile, some corresponding examples are also presented to illustrate the main results. [ABSTRACT FROM AUTHOR]

First, we define a new class of fractional differential equations of order n -- 1 < ϑ ≤ n, (n ≥ 2). Also, we define a new Banach double sequence space m²(Δvu, ø, p) and a Hausdorff MNC on it. By using this MNC, we prove the existence of solution of infinite system of a new class of fractional differential equations of order ϑ ϵ(n -- 1, n], (n ≥ 2) in m²(Δvu, ø, p)> Finally, we give two examples to verify the usefulness of our main result. [ABSTRACT FROM AUTHOR]

Fractional order partial differential equations have a wide range of applications in image processing. The solution of partial differential equations is generally obtained using the finite difference method, which still requires improvement in terms of efficiency and effectiveness. In this work, a multi‐scale interpolative wavelet operator is constructed by means of Shannon‐Cosine wavelets with interpolation, smoothness, compact support and other excellent properties. We use the wavelet operator instead of the finite difference operator to solve fractional‐order partial differential equations. The proposed method can limit the artefacts and remove the noises appeared in the processed images effectively. In addition, for the loss of texture in the denoised image, we enhance the image by employing fractional order differential equations to improve the quality of the image. Finally, the biological sections images are taken as the examples to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Shannon‐Cosine wavelet function possesses almost all excellent characteristics such as interpolation, compact support, and smoothness. As an interpolation wavelet function, it could be applied in fractional partial differential equations effectively. However, when solving engineering problems in a finite interval, the treatment of the boundary is still not smooth enough. So, the Hermite‐Shannon‐Cosine interval wavelet is constructed using the Hermite interpolation function to achieve smoother transitions at the boundary of the interval, thereby reducing boundary effects. Based on this, a method for solving Fractional PDEs is proposed, the method's performance and its processing capability at the interval boundary are verified by taking the Fractional Fokker‐Planck equation and the Time‐Fractional Korteweg‐de Vries equation as examples. Compared with the multi‐scale Faber‐Schauder wavelet collocation method, Point‐Symmetric interval wavelet spectral method, Dynamic interval wavelet spectral method, and so forth, the experimental results show that the method performs better in terms of numerical accuracy and effectiveness. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of solutions of Hilfer-Katugampola fractional differential equations with boundary conditions. We first establish existence theory of solutions for the mentioned problem by the fixed point theory and the measure of noncompactness. Then we investigate the ε-approximate solution to our concerned problem via a generalized Gronwall inequality. Finally, as the application of abstract results, we give an example to illustrate our main results. [ABSTRACT FROM AUTHOR]

We study the stability of solutions of fractional (in the sense of Caputo) stochastic differential equations in a finite-dimensional Euclidean space Rn. We answer the question about the stochastic stability of the solution of the system x(t) = 0 in some sense. [ABSTRACT FROM AUTHOR]

In this work, we are devoted to discussing a system of fractional stochastic differential variational inequalities with Lévy jumps (SFSDVI with Lévy jumps), that comprises both parts, that is, a system of stochastic variational inequalities (SSVI) and a system of fractional stochastic differential equations(SFSDE) with Lévy jumps. Here it is noteworthy that the SFSDVI with Lévy jumps consists of both sections that possess a mutual symmetry structure. Invoking Picard's successive iteration process and projection technique, we obtain the existence of only a solution to the SFSDVI with Lévy jumps via some appropriate restrictions. In addition, the major outcomes are invoked to deduce that there is only a solution to the spatial-price equilibria system in stochastic circumstances. The main contributions of the article are listed as follows: (a) putting forward the SFSDVI with Lévy jumps that could be applied for handling different real matters arising from varied domains; (b) deriving the unique existence of solutions to the SFSDVI with Lévy jumps under a few mild assumptions; (c) providing an applicable instance for spatial-price equilibria system in stochastic circumstances affected with Lévy jumps and memory. [ABSTRACT FROM AUTHOR]

In this paper, we demonstrate that neutral fractional evolution equations with finite delay possess a stable mild solution. Our model incorporates a mixed fractional derivative that combines the Riemann–Liouville and Caputo fractional derivatives with orders 0 < α < 1 and 1 < β < 2 . We identify the infinitesimal generator of the cosine family and analyze the stability of the mild solution using both Hyers–Ulam–Rassias and Hyers–Ulam stability methodologies, ensuring robust and reliable results for fractional dynamic systems with delay. In order to guarantee that the features of invariance under transformations, such as rotations or reflections, result in the presence of fixed points that remain unchanging and represent the consistency and balance of the underlying system, fixed-point theorems employ the symmetry idea. Lastly, the results obtained are applied to a fractional order nonlinear wave equation with finite delay with respect to time. [ABSTRACT FROM AUTHOR]

This research study involves modeling Newton–Leipnik attractors within the domain of fractional variable-order (FVO) dynamics using a nonlinear and adaptable radial basis function neural network (RBFNN). The numerical solution for the FVO Newton–Leipnik system is initially obtained using a numerical scheme based on the Caputo–Fabrizio derivative with variable order. This process is carried out across a range of different control parameters. A parametric model is also constructed using RBFNN, considering various system initial values. Multiple instances of chaos are calculated using a proposed computational model within the Newton–Leipnik system with varying fractional-order functions. This investigation aims to assess and comprehend the extent of sensitivity exhibited by chaotic behavior achieved through the computation of Lyapunov exponents. The performance of the proposed computational RBFNN model is validated using the RMSE statistic. The results closely align with those obtained through numerical algorithms based on the Caputo–Fabrizio derivative, demonstrating the high accuracy of the designed network. [ABSTRACT FROM AUTHOR]

We explore the existence and uniqueness of solutions to nonlinear fractional differential equations (FDEs), defined in the sense of RL-fractional derivatives of order η ∈ (1 , 2) . The nonlinear term is assumed to have a discontinuity at zero. By employing techniques from Lebesgue spaces, including Holder's inequality, we establish uniqueness theorems for this problem, analogous to Nagumo, Krasnoselskii–Krein, and Osgood-type results. These findings provide a fundamental framework for understanding the properties of solutions to nonlinear FDEs with discontinuous nonlinearities. [ABSTRACT FROM AUTHOR]

The differential equations involving two discrete delays are helpful in modeling two different processes in one model. We provide the stability and bifurcation analysis in the fractional order delay differential equation D α x (t) = a x (t) + b x (t − τ) − b x (t − 2 τ) in the a b -plane. Various regions of stability include stable, unstable, single stable region (SSR), and stability switch (SS). In the stable region, the system is stable for all the delay values. The region SSR has a critical value of delay that bifurcates the stable and unstable behavior. Switching of stable and unstable behaviors is observed in the SS region. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence and uniqueness of a solution for the implicit two-order fractional integro-differential equation with m-point boundary conditions by applying the Banach fixed point theorem. Moreover, in the paper we establish the four different varieties of Ulam stability (Hyers–Ulam stability, generalized Hyers–Ulam stability, Hyers–Ulam-Rassias stability, and generalized Hyers–Ulam–Rassias stability) for the given problem. [ABSTRACT FROM AUTHOR]

This research employs the improved modified extended tanh-function technique to explore several solitary wave solutions to the fractional-order Fokas equation. The propagation of waves in fluid dynamics and optical systems are two examples of various natural phenomena that are effectively addressed by the fractional-order Fokas equation. The model captures a generalization of the integer derivative form by including fractional derivatives defined in the conformable sense. We use the phase portrait theory to investigate the existence of traveling wave solutions. The improved modified extended tanh-function technique is successfully applied as a reliable analytical procedure to derive several solitary wave solutions, providing an approachable structure to deal with the complexity introduced by the fractional order. The extracted solutions, which are illustrated by hyperbolic, trigonometric, and rational functions, exhibit a variety of solitary wave shapes, such as bell-shaped, kink, and anti-kink patterns. We additionally evaluate how well the employed method performs in comparison to other approaches. Furthermore, some graphical visualizations are provided to clearly demonstrate the physical behavior of the obtained solutions under various parameter values. The outcomes highlight the effectiveness and adaptability of the proposed strategy in resolving fractional nonlinear differential equations and expand our knowledge of fractional-order systems. [ABSTRACT FROM AUTHOR]

This study presents a novel and efficient iterative approach to approximating the fixed points of contraction mappings in Banach spaces, specifically approximating the solutions of nonlinear fractional differential equations of the Caputo type. We establish two theorems proving the stability and convergence of the proposed method, supported by numerical examples and graphical comparisons, which indicate a faster convergence rate compared to existing methods, including those by Agarwal, Gursoy, Thakur, Ali and Ali, and D ∗ ∗ . Additionally, a data dependence result for approximate operators using the proposed method is provided. This approach is applied to achieve the solutions for Caputo-type fractional differential equations with boundary conditions, demonstrating the efficacy of the method in practical applications. [ABSTRACT FROM AUTHOR]

The proposed study seeks to investigate various analytical and numerical techniques for solving fractional differential equations, with a particular focus on their applications in mathematical modeling and scientific research within the field of algebra. This study intends to investigate methods such as the Aboodh transform iteration method and the Aboodh residual power series method, specifically for addressing the Jaulent–Miodek system of partial differential equations. By analyzing the behavior of fractional-order differential equations and their solutions, this research seeks to contribute to a deeper understanding of complex mathematical phenomena. Furthermore, this study examines the role of the Caputo operator in fractional calculus, offering insights into its significance in modeling real-world systems within the algebraic context. Through this research, novel approaches for solving fractional differential equations are developed, offering essential tools for researchers in diverse fields of science and engineering, including algebraic applications. [ABSTRACT FROM AUTHOR]

The objective of this paper is to demonstrate the existence and uniqueness (EU) of solutions to a class of Fractional Integro-Stochastic Differential Equations (FISDEs) by utilizing the fixed-point technique (FPT) and stochastic techniques. Additionally, the paper proves the continuous dependence (CD) of solutions on the initial data. We examine the Hyers–Ulam stability (HUS) of FISDEs by applying Gronwall inequalities. Two theoretical examples are presented to demonstrate our findings. [ABSTRACT FROM AUTHOR]

This paper introduces interpolative enriched cyclic Reich–Rus–Ćirić operators in normed spaces, expanding existing contraction principles by integrating interpolation and cyclic conditions. This class of operators addresses mappings with discontinuities or non-self mappings, enhancing the applicability of fixed-point theory to more complex problems. This class of operators expands on existing cyclic contractions, including interpolative Kannan mappings, interpolative Reich–Rus–Ćirić contractions, and other known contractions in the literature. We demonstrate the existence and uniqueness of fixed points for these operators and provide an example to illustrate our findings. Moreover, we discuss the applications of our results in solving nonlinear integral equations. Furthermore, we introduce the idea of a coupled interpolative enriched cyclic Reich–Rus–Ćirić operator and establish the existence of a strongly coupled fixed-point theorem for this contraction. Finally, we provide an application to fractional differential equations to show the validity of the main result. [ABSTRACT FROM AUTHOR]

This study investigates the impact response of polymer foams commonly used in protective packaging, considering the fractal nature of their material microstructure. The research begins with static material characterization and impact tests on two low-density polyethylene foams. To capture the multiscale nature of the dynamic response behavior of two low-density foams to sustain impact loads, fractional differential equations of motion are used to qualitatively and quantitatively describe the dynamic response behavior, assuming restoring forces for each foam characterized, respectively, by a polynomial of heptic degree and by a trigonometric tangential function. A two-scale transform is employed to solve the mathematical model and predict the material's behavior under impact loads, accounting for the fractal structure of the material's molecular configuration. To assess the accuracy of the mathematical model, we performed impact tests considering eight dropping heights and two plate weights. We found good predictions from the mathematical models compared to experimental data when the fractal derivatives were between 1.86 and 1.9, depending on the cushioning material used. The accuracy of the theoretical predictions achieved using fractal calculus elucidates how to predict multiscale phenomena associated with foam heterogeneity across space, density, and average pore size, which influence the foam chain's molecular motion during impact loading conditions. [ABSTRACT FROM AUTHOR]

This study investigates the asymptotic behavior of non-oscillatory solutions to forced-perturbed fractional differential equations with the Caputo fractional derivative. The main aim is to unify the Beta Integral Lemma (Lemma 2) and the Gamma Integral Lemma (Lemma 3) into a single framework. By combining these two powerful tools, we propose new criteria that effectively characterize the asymptotic behavior of non-oscillatory solutions to the given equations. The analysis of such solutions has significant implications in the fields of oscillation and stability theory. Notably, our findings extend prior work by exploring a wider range of equations with more general functions and coefficients, thereby broadening the applicability and deepening the understanding of both asymptotic and oscillatory behaviors. Moreover, the criteria we introduce offer improvements over previous approaches, as demonstrated by the example provided, which highlights the advantages of our results in comparison to earlier methods. [ABSTRACT FROM AUTHOR]

We introduce the modified Picard–Ishikawa hybrid iterative scheme and establish some strong convergence results for the class of asymptotically generalized ϕ -pseudocontractive mappings in the intermediate sense in Banach spaces and approximate the fixed point of this class of mappings via the newly introduced iteration scheme. We construct some numerical examples to support our results. Furthermore, we apply the Picard–Ishikawa hybrid iteration scheme in solving the nonlinear Caputo type fractional differential equations. Our results generalize, extend and unify several existing results in literature. [ABSTRACT FROM AUTHOR]

In this study, the time-fractional Newell-Whitehead-Segel (NWS) equation and its different nonlinearity cases are investigated. Schemes obtained by the Newtonian linearization method are used to numerically solve different cases of the time-fractional Newell-Whitehead-Segel (NWS) equation. Stability and convergence conditions of the Newtonian linearization method have been determined for the related equation. The numerical results obtained as a result of the appropriate stability criteria are compared with the help of tables and graphs with exact solutions for different fractional values. [ABSTRACT FROM AUTHOR]

In this study, we discuss the existence and the uniqueness of the solution to Caputo-Hadamard Cauchy problems for a system of fractional differential equations, by using Perov's fixed point theorem. Finally, two examples are provided to illustrate our results. [ABSTRACT FROM AUTHOR]