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In this paper, we examine the time-dependent two-dimensional cable equation of fractional order in terms of the Caputo fractional derivative. This cable equation plays a vital role in diverse areas of electrophysiology and modeling neuronal dynamics. This paper conveys a precise semi-analytical method called the q-homotopy analysis transform method to solve the fractional cable equation. The proposed method is based on the conjunction of the q-homotopy analysis method and Laplace transform. We explained the uniqueness of the solution produced by the suggested method with the help of Banach's fixed-point theory. The results obtained through the considered method are in the form of a series solution, and they converge rapidly. The obtained outcomes were in good agreement with the exact solution and are discussed through the 3D plots and graphs that express the physical representation of the considered equation. It shows that the proposed technique used here is reliable, well-organized and effective in analyzing the considered non-homogeneous fractional differential equations arising in various branches of science and engineering. [ABSTRACT FROM AUTHOR]

This study explores solutions for a mathematical equation called the time-space fractional phi-four equation using two methods: the Sardar-subequation method and the modified extended auxiliary equation method. The phi-four equation is connected to the Klein–Gordon model and is important in different scientific areas like biology and nuclear physics. Understanding its solutions is crucial. By using a specific wave transformation, the equation is changed into a simpler form for analysis. The methods proposed give a variety of solutions, such as Kink, bright singular, dark, combo dark bright, periodic, and singular periodic solutions. Each solution we find using these methods has specific rules that determine when it's correct. We carefully choose specific values for the parameters to help us understand more about the solutions. This helps us see the detailed features of the solutions and improves our understanding of how the model behaves in the real world. These methods create a strong framework for studying solitons, which are specific types of mathematical solutions. The study compares the outcomes of these methods with earlier ones to get a complete understanding. Graphical illustrations are used to visually represent some of these solutions, helping us grasp their characteristics. Visual representations in two- and three-dimensional figures add originality to the findings. Importantly, these methods can be applied to solve similar problems with fractional derivatives in various scientific contexts. In summary, this research not only deepens our understanding of the phi-four equation but also introduces powerful methods with broad applications in fractional differential equations. [ABSTRACT FROM AUTHOR]

In this paper, Lie symmetry analysis method is applied to time fractional coupled (2 + 1)-dimensional nonlinear Schrödinger equations. We obtain all the Lie symmetries and reduce the (2 + 1)-dimensional fractional partial differential equations with Riemann-Liouville fractional derivative to (1 + 1)-dimensional counterparts with Erdélyi-Kober fractional derivative. Then we obtain the power series solutions and prove their convergence. In addition, the conservation laws for the governing equations are constructed by using the generalization of the Noether operators and the new conservation theorem. [ABSTRACT FROM AUTHOR]

In this paper, an attempt has been made to establish the sufficient conditions for the existence and multiplicity of positive solutions by using the fixed point index theory and the Avery–Peterson fixed point theorem respectively for the following class of nonlinear singular fractional differential equation D 0 + β ⁢ ( ϕ p ⁢ ( D 0 + α ⁢ u ⁢ ( t ) ) ) + f ⁢ ( t , u ⁢ ( t ) , … , u ( n - 2 ) ⁢ ( t ) ) = 0 , t ∈ ( 0 , 1 ) , D_{0^{+}}^{\beta}(\phi_{p}(D_{0^{+}}^{\alpha}u(t)))+f(t,u(t),\dots,u^{(n-2)}(t% ))=0,\quad t\in(0,1), with the boundary conditions u ( k ) ⁢ ( 0 ) = 0 , 0 ≤ k ≤ n - 2 , D 0 + α ⁢ u ⁢ ( 0 ) = 0 , D 0 + α - 1 ⁢ u ⁢ ( 1 ) = ∫ 0 1 D 0 + α - 1 ⁢ u ⁢ ( t ) ⁢ 푑 A ⁢ ( t ) , u^{(k)}(0)=0,\quad 0\leq k\leq n-2,\qquad D_{0^{+}}^{\alpha}u(0)=0,\qquad D_{0% ^{+}}^{\alpha-1}u(1)=\int_{0}^{1}D_{0^{+}}^{\alpha-1}u(t)\,dA(t), where 0 < β ≤ 1 {0<\beta\leq 1} , n - 1 < α ≤ n {n-1<\alpha\leq n} , n ≥ 3 {n\geq 3} , A : [ 0 , 1 ] → ℝ {A:[0,1]\to\mathbb{R}} is a function of bounded variation, ϕ p ⁢ ( s ) = | s | p - 2 ⁢ s {\phi_{p}(s)=|s|^{p-2}s} for p > 1 {p>1} and ϕ q ⁢ ( s ) {\phi_{q}(s)} is the inverse of ϕ p ⁢ ( s ) {\phi_{p}(s)} , where p , q {p,q} satisfies the relation 1 p + 1 q = 1 {\frac{1}{p}+\frac{1}{q}=1} , f may have singularity at t = 0 {t=0} . [ABSTRACT FROM AUTHOR]

In this paper, Lie symmetry analysis method is applied to the ( 2 + 1 ) (2+1) -dimensional time fractional Kadomtsev–Petviashvili (KP) system, which is an important model in mathematical physics. We obtain all the Lie symmetries admitted by the KP system and use them to reduce the ( 2 + 1 ) (2+1) -dimensional fractional partial differential equations with Riemann–Liouville fractional derivative to some ( 1 + 1 ) (1+1) -dimensional fractional partial differential equations with Erdélyi–Kober fractional derivative or Riemann–Liouville fractional derivative, thereby getting some exact solutions of the reduced equations. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the system studied. [ABSTRACT FROM AUTHOR]

The notion of a generalized Riemann–Liouville fractional integral is introduced, and its domain, range, and properties are studied. The new notion and properties provide new insight and understanding into the classical Riemann–Liouville fractional integral and its properties. Based on the generalized Riemann–Liouville fractional integral, equivalences between linear first ‐order fractional differential equations (FDEs) and integral equations are established. These equivalence results are applied to obtain solutions of Abel type integral equations and to study the existence and uniqueness of generalized normal solutions of initial value problems for nonlinear first‐order FDEs. [ABSTRACT FROM AUTHOR]

This paper investigates the nonlinear conformable Schrödinger equation with nonlocal nonlinearities and Kudryashov’s arbitrary refractive index. The study employs the Uniform Method (UM) and the Extended Hyperbolic Function Method (EHFM) to derive various optical soliton solutions for this present conformable equation. To demonstrate the importance of the novel solutions, the paper presents two-dimensional, three-dimensional, and contour plots, illustrating kink-type, wave, bright, bell-shaped, and mixed dark–bright soliton solutions. Further, the impact of the fractional order parameter, temporal parameter, and nonlinearity coefficient on these optical solutions is analyzed, providing valuable insights into the conformable nonlinear Schrödinger model. The behavior of these optical solutions is analyzed through illustrative graphs that account for variations in the conformable order derivative, temporal parameter, and nonlinearity coefficient. The methodologies applied in this research show potential for broader application to various nonlinear Schrödinger equations in fields such as applied mathematics and nonlinear optics. The research suggests that these methods could be applied in the future to investigate other differential equations with fractional and integer orders across different fields of applied sciences. This study expands our understanding of nonlinear optics and has potential practical implications in various fields like optical signal processing, laser technology, and telecommunications. [ABSTRACT FROM AUTHOR]

In this paper, we study the initial value problem for the fractional differential equation with multiple deviating arguments. By using Krasnoselskii's fixed point theorem, the conditions of solvability of the problem are obtained. Furthermore, we establish Ulam–Hyers and generalized Ulam–Hyers stability of the fractional functional differential problem. Finally, two examples are presented to illustrate our results, one is with a pantograph-type equation and the other is numerical. [ABSTRACT FROM AUTHOR]

This paper investigates the qualitative properties of the solutions for neutral implicit stochastic Hilfer fractional differential equations involving Lévy noise with retarded and advanced arguments. The existence property of the solution of the aforementioned equation is demonstrated by the Mónch condition, and the uniqueness is demonstrated by the remarkable fixed point of Banach. In addition, we examine the Hyers–Ulam (HU) stability of the presented mathematical models. To substantiate our theoretical conclusions, a real-world example is included to illustrate their practical application. [ABSTRACT FROM AUTHOR]

The Chebyshev cardinal functions based on the Lobatto grid are introduced and used for the first time to solve the fractional delay differential equations. The presented algorithm is based on the collocation method, which is applied to solve the corresponding Volterra integral equation of the given equation. In the employed method, the derivative and fractional integral operators are expressed in the Chebyshev cardinal functions, which reduce the computational load. The method is characterized by its simplicity, adherence to boundary conditions, and high accuracy. An exact analysis has been provided to demonstrate the convergence of the scheme, and illustrative examples validate our investigation. [ABSTRACT FROM AUTHOR]

The purpose of this article concerns a type of nonlinear boundary value problem of fractional differential equations with the Caputo derivative. Using the fixed point index theory of condensing mapping in cones, the existence results of positive solutions for such a problem are obtained. [ABSTRACT FROM AUTHOR]

We consider the fully nonlocal diffusion equations with nonnegative L1$L^1$‐data. Based on the approximation and energy methods, we prove the existence and uniqueness of nonnegative entropy solutions for such problems. In particular, our results are valid for the time‐space fractional Laplacian equations. [ABSTRACT FROM AUTHOR]

This research study's primary goal is to create an efficient wavelet collocation technique to resolve a kind of nonlinear fractional order systems of ordinary differential equations that arise in the modeling of autocatalytic chemical reaction problems. Here, we created the functional matrix of integration for the Fibonacci wavelets. The Fibonacci wavelet collocation method is employed to find the numerical solution of the system of nonlinear coupled ordinary differential equations of both integer and fractional order. The nonlinear Brusselator system is transformed into an algebraic equation system using the operational matrices of fractional derivative and collocation technique. These algebraic equations are treated by the Newton–Raphson method, and obtained unknown coefficient values are substituted in the approximation. We demonstrate our method's computational effectiveness and accuracy using different model constraints in the numerical examples. The effectiveness and consistency of the developed strategy's performance are shown in graphs and tables. Comparisons with existing methods available in the literature demonstrate the high accuracy and robustness of the developed Fibonacci wavelet collocation method. Mathematical software Mathematica has been used to perform all calculations. [ABSTRACT FROM AUTHOR]

In this paper, we propose an explicit two-grid spectral deferred correction method for solving the nonlinear fractional pantograph differential equations. We design a partition including the global and local grids, which reduces the interaction between the subintervals caused by the delay term. We also analyze the numerical errors of the suggested approach for the prediction step and the correction step, respectively. Numerical experiments confirm the theoretical expectations. [ABSTRACT FROM AUTHOR]

For solving the semilinear fractional differential equations with the nonsmooth force term, we construct a class of Runge–Kutta-based convolution quadrature. Moreover, we analyse the convergence of the proposed scheme. In addition, we employ the fast Runge–Kutta approximation to reduce the calculation cost. Finally, we give some numerical experiments to verify the theoretical results. [ABSTRACT FROM AUTHOR]

The aim of the present work is to discuss the fractional mass-spring system with damping and driving force, considering a simple modification to the fractional derivatives with a non-singular kernel of the Atangana–Baleanu and Caputo–Fabrizio types. We introduce two novel modified fractional derivatives that offer advantages when the fractional differential equations involve higher-order fractional derivatives of order 1 + α or α + 1 , with 0 < α < 1 . Previous definitions of fractional derivatives with non-singular kernel do not have a unique definition, leading to significant inconsistencies. One of the main results of the present work is that the proposed modifications provide a unique result for the fractional-order derivatives 1 + α and α + 1 . Additionally, we apply these two novel fractional derivatives to the fractional mass-spring system with damping and driving force. In the case of the modified Caputo–Fabrizio fractional derivative, novel analytical solutions have been constructed, showing interesting oscillating time evolution with a transient term not previously reported. This transient term features an initial nonzero oscillating return away from the equilibrium position. For the modified Atangana–Baleanu fractional derivative, the numerical solutions also exhibit this nonzero oscillating return away from the equilibrium position. These results are not present when using the Caputo singular kernel derivative, as demonstrated in the comparison figures reported here. [ABSTRACT FROM AUTHOR]

In this paper we consider two classes of nonlinear partial differential equations with the fractional Laplacian, namely $$\begin{align*} \left(-\Delta\right)^{\frac{\alpha}{2}}\left(u^{m} \right) &=u\left|u\right|^{q-1}+w\left(x\right), \quad x\in \mathbb{R}^{N},\\ &\qquad 1\leq m (− Δ) α 2 (u m) = u | u | q − 1 + w (x) , x ∈ R N , 1 ≤ m < q , 0 < α ≤ 2 , N > α and $$\begin{align*} &\frac{\partial^{k}u}{\partial t^{k}}+\left(-\Delta\right)^{\frac{\alpha}{2}}u=u^{q}, \quad \left(x,t\right)\in \mathbb{R}^{N}\times\left(0,+\infty\right),\\ &\qquad k\geq 1, \ 0 ∂ k u ∂ t k + (− Δ) α 2 u = u q , (x , t) ∈ R N × (0 , + ∞) , k ≥ 1 , 0 < α ≤ 2 , N > α. Solutions defined for all $ x\in \mathbb {R}^{N} $ x ∈ R N of the first equation are referred to as entire solutions, while solutions defined for all $ \left (x,t\right)\in \mathbb {R}^{N}\times \left [0,+\infty \right) $ (x , t) ∈ R N × [ 0 , + ∞) of the second equation are referred to as global solutions. Several existence and nonexistence theorems are established over different ranges of q, and thus the respective relations between the existence, nonexistence of solutions for these equations and the index q in the nonlinear terms are obtained. It is illustrated that our results are sharp in cases of m = 1 and k = 1 respectively. In addition, we prove the positivity, symmetry and odevity of solutions we constructed for the first equation with m = 1 associated with the inhomogeneous term w. [ABSTRACT FROM AUTHOR]

We consider the numerical stability problem for fractional delay Schrödinger equations involving a Caputo fractional derivative in time, which is developed by Galerkin finite element method (FEM) in space and fractional linear multistep methods (FLMMs) in time. Through rigorous analyses of the characteristic equation yielded by the Laplace transform, we first present an α -dependent coefficient criterion to ensure the stability of spatially semidiscrete Galerkin FEM and extend the stability property to all convergent spatially semidiscrete methods. Secondly, by introducing a decoupling technique after Z transform, we prove the stability of FLMMs generated by both A-stable and A( β )-stable linear multistep methods, without any restriction on the time step size. The stability results are formulated by the fractional exponent, the principal eigenvalue of Dirichlet Laplacian, and the mesh size, but are not related to the delay and time step size. However, for a general spatial region, the principal eigenvalue of Dirichlet Laplacian is always unavailable. In order to provide an effective method for stability detection, when the stability condition is violated, we prove that the fractional trapezoidal rule is an effective method to detect stability because it can not only maintain the stable behavior of the semidiscrete solution, but also the unstable behavior. Extensive numerical experiments for fractional delay Schrödinger equations confirm the long-time decay behaviors of the fully discrete numerical solutions. [ABSTRACT FROM AUTHOR]

Multi-term weakly singular Volterra integral equations arise in numerous scientific and engineering applications modeled by fractional differential equations. Numerical analysis of these types of integral equations with non-smooth solutions is one of the open problems due to singularities at the initial time leading to the destruction of the accuracy of the approximate solutions. The spectral approximations based on classical orthogonal polynomials have a low convergence rate compared with the exact non-smooth solutions. We theoretically investigate the regularity of the solution of multi-term weakly singular Volterra integral equations, which ensures the non-smooth property of the solutions. This non-smooth representation is significant in constructing highly accurate numerical schemes based on fractional-order approximate solutions. The advantage of the proposed method is that there is no need to solve an algebraic system of equations, as the unknown coefficients can be specified by a simple recursive algorithm. This makes the numerical method for solving the problem efficient and of low computational cost. The L ∞ -convergence analysis of the proposed method is discussed. Some numerical experiments are provided to demonstrate the efficiency and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

This work establishes the presence of two non-negative solutions of some $ \psi- $Hilfer FBVPs involving the $ p- $Laplacian operator by using an iterative method. Then we prove with examples the validity of our results. [ABSTRACT FROM AUTHOR]

This paper deals with the effect of fractional order on the fractional Euler-Bernoulli beams model. This model is based on the elastic curve that can be approximated using tangent lines and the osculating circle. We introduce a general form of the fractional Green's function for the highest-order term 1 < α ≤ 2 , which gives a unique solution for the geometrical fractional bending beams with an inhomogeneous fractional differential equation. The analytical closed-form responses were obtained by inverse problem technique and integration over Green's function kernel. The theory and properties of fractional Green's function method are described by generalizing classical theory to the initial and two-point fractional boundary value problems. Static analysis of the numerical examples, such as determinate and indeterminate beams with typical loads and beam property boundary conditions, are presented and discussed. The results were verified using the direct integration method. The fractional conventional ratio indicates the beam's deformation deviation from its classical state. The analysis results reveal that changing the fractional parameter significantly impacts the deflection behavior of flexural beams due to the beam's characteristics and the fractional order. The behavior of the fractional beam is identical to the classical state when the fractional parameter is an integer number. [ABSTRACT FROM AUTHOR]

In this paper, we study a new class of coupled systems of nonlinear generalized fractional differential equations complemented with coupled nonlocal Riemann–Stieltjes and generalized fractional integral boundary conditions. The nonlinearities also include the lower order generalized fractional derivatives of the unknown functions. We apply the Banach contraction mapping principle and Leray–Schauder alternative to derive the desired results. An illustrative example is also discussed. The results presented in this work are novel in the given configuration and yield some new results as special cases (for details, see the Conclusion section). [ABSTRACT FROM AUTHOR]

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In this paper, we prove the existence of multiple solutions in the Bessel Potential space for a new class of nonlinear fractional Schr¨odinger-Poisson systems involving the distributional Riesz fractional derivative. To reach our goal, we use the symmetric mountain pass theorem under some suitable assumptions on nonlinearity f(x, u) and potential V (x). [ABSTRACT FROM AUTHOR]

We prove the Hölder–Dini regularity for a class of fractional Fokker–Planck–Kolmogorov equations. Then, by applying the Itô–Tanaka trick, we obtain the unique strong solvability of time inhomogeneous stochastic differential equations with drift coefficients in Lt∞(풞 b,dβ,ρ) for β = 1 − α/2 and driven by α-stable processes. [ABSTRACT FROM AUTHOR]

Nonlinear fractional-order partial differential equations play an important role in science and engineering by illustrating a variety of nonlinear processes. The nonlinear space-time fractional combined Korteweg-de Vries and modified Korteweg-de Vries equation is a very significant fractional partial differential equation and is used to simulate shallow water surface wave phenomena, pulse waves in large arteries, ion acoustic waves in plasmas, and atmospheric dust-acoustic solitary waves. The improved Bernoulli sub-equation function method and the new generalized (G′/G)-expansion method are two noteworthy approaches that have been used to analyze and extract solutions to the above-mentioned equation that include various types of traveling waves as well as soliton solutions via beta-derivative. Through the utilization of a wave transformation, the fractional-order equation is transformed into a nonlinear ordinary differential equation. The exponential function, trigonometric function, rational function, and hyperbolic trigonometric function solutions with arbitrary constants have been used to articulate the obtained solutions. By utilizing the aforementioned approaches, several standard waveforms have been recognized including multiple periodic types, kink shapes, bell-shaped, single solitons, and other types of solitons. Mathematica software has been used to illustrate the wave profiles through 3D and contour plots, providing a clearer physical sketch based on diverse values of free parameters. The suggested methods have demonstrated their reliability in establishing more generalized wave solutions and exhibit computational efficiency, making them suitable for soliton solutions. [ABSTRACT FROM AUTHOR]

This paper introduces a novel numerical technique for solving fractional stochastic differential equations with neutral delays. The method employs a stepwise collocation scheme with Jacobi poly-fractonomials to consider unknown stochastic processes. For this purpose, the delay differential equations are transformed into augmented ones without delays. This transformation makes it possible to use a collocation scheme improved with Jacobi poly-fractonomials to solve the changed equations repeatedly. At each iteration, a system of nonlinear equations is generated. Next, the convergence properties of the proposed method are rigorously analyzed. Afterward, the practical utility of the proposed numerical technique is validated through a series of test examples. These examples illustrate the method's capability to produce accurate and efficient solutions. [ABSTRACT FROM AUTHOR]

The nonlinear fractional Langevin equation is an important equation used in various areas of physics and mathematics. In this article, we focus on the singular-hybrid case of the generalized Langevin equations and prove the existence of a solution using fixed point theory. We also provide numerical and graphical simulations to demonstrate the convergence and existence of a solution for this singular problem. Finally, we present numerical examples in the form of tables and graphs to examine the accuracy of the method. [ABSTRACT FROM AUTHOR]

This paper presents a novel spectral algorithm for the numerical solution of multi-dimensional fractional-order telegraph equations, a critical model used to capture the combined effects of diffusion and wave propagation. The core innovation of this work is the application of Jacobi-Romanovski polynomials as the basis functions for spectral discretization. These polynomials offer unique advantages, including the ability to handle nonstandard domains and boundary conditions, making them particularly suitable for partial differential equation (PDE) applications. A comprehensive error analysis is conducted, providing deep insights into the convergence rates and factors affecting the accuracy of the numerical solutions. Extensive numerical experiments further demonstrate the superior performance of the proposed spectral algorithm in solving a wide range of multi-dimensional fractional-order telegraph equation models. The results show a significant improvement in accuracy and computational efficiency compared to traditional numerical methods, such as finite difference or finite element techniques. This research advances the field of computational science by offering a robust, efficient, and versatile numerical framework for the precise solution of complex multi-dimensional PDEs. [ABSTRACT FROM AUTHOR]

The truncated M-fractional Kuralay (TMFK)-II equation is prevalent in the exploration of specific complex nonlinear wave phenomena. Such types of wave phenomena are more applicable in science and engineering. These equations could potentially provide insights into understanding the intricate dynamics of optical phenomena, encompassing solitons, nonlinear effects, and wave interactions. This study aims to uncover a diverse range of soliton solutions to the model, spanning trigonometric, hyperbolic, exponential, and rational expressions. These solutions are unveiled through the application of extended hyperbolic functions and improved F-expansion techniques, representing the primary objective of this research. The three-dimensional (3D) and two-dimensional (2D) combined charts are plotted for some of these solutions. The impact of the fractional parameters and time variations is also illustrated. Moreover, the models are converted into a planar dynamical system using a Galilean transformation, and the analysis of bifurcation is examined. This research underscores the versatility of the aforementioned techniques for exploring complex nonlinear phenomena across various engineering and scientific disciplines. Finally, the findings of this study hold significant implications for advancing our understanding and analysis of nonlinear wave dynamics in diverse physical systems. [ABSTRACT FROM AUTHOR]

Building on previous work in fractional calculus, this paper introduces new definitions for the (ρ, k, φ)‐proportional integral and (ρ, k, φ)‐proportional H fractional derivative. This new approach retains the semigroup properties of traditional fractional integrals. A significant advantage of this fractional calculus is its compatibility with the majority of existing studies on fractional differential equations. Furthermore, we delve into the properties of the generalized fractional integrals and derivatives. We discuss, for instance, the mapping properties of the (ρ, k, φ)‐proportional integral. To elucidate these concepts, we introduce a set of new weighted spaces. Additionally, we explore the generalized Laplace transform of both the (ρ, k, φ)‐proportional integrals and (ρ, k, φ)‐proportional H fractional derivatives. Also, examples concerning the linear (ρ, k, φ)‐proportional H fractional equations are given to illustrate the main results. [ABSTRACT FROM AUTHOR]

This study delves into utilizing the Kudryashov auxiliary equation method on the time-fractional (3+1)-dimensional Sasa–Satsuma equation, pivotal for transmitting ultra-fast pulses in optical fibers. The main aim is to generate a variety of optical solitons for the time-fractional nonlinear (3+1)-dimensional Sasa–Satsuma model, including dark, bright, singular, and straddled solitons. These solitons are derived from exponential and hyperbolic function-type solutions, presenting novel findings not previously reported. The study demonstrates the effectiveness of the implemented method in deriving analytical solutions for fractional differential equations. The (3+1)-dimensional Sasa–Satsuma model's potential in optical fiber transmission is highlighted, offering significant insights into generating optical solitons and their applications. [ABSTRACT FROM AUTHOR]

This article focuses on the study of the existence of solution to Hadamard–Caputo fractional nonlinear differential equation with time delay. For time delay, two different cases of finite delay and infinite delay are considered. Based on the weighted function solution spaces, the existence and uniqueness of solution to the differential equations are verified by Leray–Schauder selection theorem and Banach fixed point theorem respectively. Finally, examples are given to illustrate the abstract results of this article. [ABSTRACT FROM AUTHOR]

In this paper, the approximate solutions of the time fractional Harry Dym equation with fractional derivative in the Caputo sense are obtained by using the Residual power series method (RPSM). This equation is a significant dynamical equation that occurs in a variety of physical systems. The suggested method provides good accuracy for the approximate solution when compared numerically with the exact solution. The effectiveness of the proposed method is also illustrated with the aid of numerical results. These results indicate that the RPSM is a power, useful, and applicable for determining the solutions of the time Hary Dym equation. Some of these results are illustrated by 2D and 3D graphics. Besides, the proposed method can be applied to many different differential equations due to its ease of use and reliability. [ABSTRACT FROM AUTHOR]

In this paper, we consider constructions of first derivative approximations using the generating function. The weights of the approximations contain the powers of a parameter whose modulus is less than one. The values of the initial weights are determined, and the convergence and order of the approximations are proved. The paper discusses applications of approximations of the first derivative for the numerical solution of ordinary and partial differential equations and proposes an algorithm for fast computation of the numerical solution. Proofs of the convergence and accuracy of the numerical solutions are presented and the performance of the numerical methods considered is compared with the Euler method. The main goal of constructing approximations for integer-order derivatives of this type is their application in deriving high-order approximations for fractional derivatives, whose weights have specific properties. The paper proposes the construction of an approximation for the fractional derivative and its application for numerically solving fractional differential equations. The theoretical results for the accuracy and order of the numerical methods are confirmed by the experimental results presented in the paper. [ABSTRACT FROM AUTHOR]

In this paper, using the properties of the conformable fractional difference and fractional sum, we initially establish some oscillatory and asymptotic criteria for a fifth-order fractional difference equation. Several critical inequalities, the Riccati transformation technique, and the integral technique are used in the deduction process. We provide some example to test the results. The established criteria are new results in the study of oscillation, and can be extended to other types of high-order fractional difference equations as well as fractional differential equations with more complicated forms. [ABSTRACT FROM AUTHOR]

This paper explores the exact solutions of the fractional Hirota–Satsuma coupled KdV (fHScKdV) equation in the Beta fractional derivative. The logistic method is first proposed to construct analytical solutions for the fHScKdV equation. In order to better comprehend the physical structure of the solutions, three-dimensional visualizations and line graphs of the exponent function solutions are depicted with the aid of Matlab. Furthermore, the phase portraits and bifurcation behaviors of the fHScKdV model under transformation are studied. Sensitivity and chaotic behaviors are analyzed in specific conditions. The phase plots and time series map are exhibited through sensitivity analysis and perturbation factors. These investigations enhance our understanding of practical phenomena governed by the fHScKdV model, and are crucial for examining the dynamic behaviors and phase portraits of the fHScKdV system. The strategies utilized here are more direct and effective, and can be applied effortlessly to other fractional order differential equations. [ABSTRACT FROM AUTHOR]

This paper is devoted to building a general framework for constructing a solution to fractional Phi-4 differential equations using a Caputo definition with two parameters. We briefly introduce some definitions and properties of fractional calculus in two parameters and the Phi-4 equation. By investigating the homotopy analysis method, we built the solution algorithm. The two parameters of the fractional derivative gain vary the behavior of the solution, which allows the researchers to fit their data with the proper parameter. To evaluate the effectiveness and accuracy of the proposed algorithm, we compare the results with those obtained using various numerical methods in a comprehensive comparative study. [ABSTRACT FROM AUTHOR]

This study presents two novel frameworks, termed a quasi-modular b-metric space and a non-Archimedean quasi-modular b-metric space, and various topological properties are provided. Using comparison and simulation functions, this paper rigorously proves several fixed point theorems in the non-Archimedean quasi-modular b-metric space. As a useful application, it also establishes Ulam–Hyers stability for the fixed point problem. Finally, this study concludes with a unique solution to a non-linear fractional differential equation, making a substantial contribution to the discipline. [ABSTRACT FROM AUTHOR]

The method of the equivalent system of fractional integral equations is used to prove the existence results of a unique solution for initial value problems corresponding to various classes of nonlinear fractional differential equations involving the tempered Ψ –Caputo fractional derivative. These include equations with their right side depending on ordinary as well as fractional-order derivatives, or fractional integrals of the solution. [ABSTRACT FROM AUTHOR]

Symmetrical fractional differential equations have been explored through a variety of methods in recent years. In this paper, we analyze the existence and uniqueness of a class of pantograph integro-fractional stochastic differential equations (PIFSDEs) using the Banach fixed-point theorem (BFPT). Also, Gronwall inequality is used to demonstrate the Ulam–Hyers stability (UHS) of PIFSDEs. The results are illustrated by two examples. [ABSTRACT FROM AUTHOR]

Through the generalized fractional derivative, it is studied how the decay term and the fractional parameter affect the quantum system, specifically the interaction between the SU(1,1) algebraic system and a three‐level atom. By transforming the differential equations into fractional differential equations, general fractional solutions are obtained. The influence of decay and fractional parameter on phenomena such as revival and collapse, entropy squeezing, purity, and concurrence are investigated. The results demonstrate how both decay and fractal parameter affect periods of collapse and revival. It is worth noting that the decay parameter shortens the collapse periods, while an increase in the fractional parameter leads to longer collapse periods. The decay parameter also reduces the degree of entanglement between the different components of the quantum system, while increasing the fractional parameter enhances the entanglement within the quantum system. Hence, it can be concluded that the fractional parameter plays a crucial role in the observed effects on the studied properties. [ABSTRACT FROM AUTHOR]

The paper investigates fractional differential equations involving the Liouville derivative. Solution to these equations under a boundary condition inside the time interval are derived in explicit form, their uniqueness is established using integral transforms technique. [ABSTRACT FROM AUTHOR]

The Volterra integral equation associated with autonomous Caputo fractional differential equation (FDE) of order α ∈ (0 , 1) in R d was shown by the authors [4] to generate a semi-group on the space C of continuous functions f : R + → R d with the topology uniform convergence on compact subsets. It serves as a semi-dynamical system for the Caputo FDE when restricted to initial functions f(t) ≡ i d x 0 for x 0 ∈ R d . Here it is shown that this semi-dynamical system has a global Caputo attractor in C , which is closed, bounded, invariant and attracts constant initial functions, when the vector field function in the Caputo FDE satisfies a dissipativity condition as well as a local Lipschitz condition. [ABSTRACT FROM AUTHOR]

In this paper, the averaging principle for stochastic Caputo fractional differential equations with the nonlinear terms satisfying the non-Lipschitz condition is considered. The work in the article is roughly divided into three parts. Firstly, we establish a generalized Gronwall inequality with singular integral kernel which is a key part in our analysis. Secondly, we discuss the existence and uniqueness of solution. And finally, the averaging principle is considered. [ABSTRACT FROM AUTHOR]

This article proposes a predictor-corrector scheme for solving the fractional differential equations 0 C D t α y (t) = f (t , y (t)) , α > 0 with non-uniform meshes. We reduce the fractional differential equation into the Volterra integral equation. Detailed error analysis and stability analysis are investigated. The convergent order of this method on non-uniform meshes is still 3 though 0 C D t α y (t) is not smooth at t = 0 . Numerical examples are carried out to verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

A fractional order problem arising in porous media is considered. Well-posedness as well as stability are discussed. Mittag-Leffler stability is proved in case of a strong fractional damping in the displacement component and a fractional frictional one in the volume fraction component. This extends an existing result from the integer-order (second-order) case to the non-integer case. In the absence of the fractional damping in the volume fraction component, it is shown a convergence to zero and a Lyapunov uniform stability. [ABSTRACT FROM AUTHOR]

In this paper, Lie symmetry analysis method is applied to ( 2 + 1 ) {(2+1)} -dimensional time fractional cubic Schrödinger equation. We obtain all the Lie symmetries and reduce the ( 2 + 1 ) {(2+1)} -dimensional fractional partial differential equations with Riemann–Liouville fractional derivative to ( 1 + 1 ) (1+1) -dimensional counterparts with Erdélyi–Kober fractional derivative. Then we obtain the power series solutions of the reduced equations and prove their convergence. In addition, the conservation laws for the governing model are constructed by the new conservation theorem and the generalization of Noether operators. [ABSTRACT FROM AUTHOR]

In this article, we propose the fractional weak adversarial networks (f-WANs) for the stationary fractional advection dispersion equations based on their weak formulas. This enables us to handle less regular solutions for the fractional equations. To handle the non-local property of the fractional derivatives, convolutional layers and special loss functions are introduced in this neural network. Numerical experiments for both smooth and less regular solutions show the validity of f-WANs. [ABSTRACT FROM AUTHOR]

The researchers have developed numerous analytical and numerical techniques for solving fractional partial differential equations most of which provide approximate solutions. Exact solutions, however, are vitally important in a convenient conception of the qualitative properties of the concerned phenomena and processes. In this paper, the pseudo-parabolic-type equations with conformable fractional derivatives are reduced to conformable fractional nonlinear ordinary differential equations by implementing a simple wave transformation. An important benefit of the proposed transformation is that it yields analytical solutions of the conformable pseudo-parabolic type equations by applying the exponential rational function strategy. The sensitivity behaviour of the model has been mentioned thoroughly. [ABSTRACT FROM AUTHOR]