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This paper is dedicated to the study of optical soliton solutions for the perturbed Fokas–Lenells equation with conformable derivative using the Kudryashov auxiliary equation method. The studied optical solutions include a class of categories, comprising dark, mixed dark-bright, multi bell-shaped, bell-shaped, and wave optical solutions. Furthermore, we analyzed the magnitude of the perturbed conformable Fokas–Lenells equation by investigating the impact of the conformable parameter and the effect of the time parameter on the novel optical solutions. It can be claimed that the current optical soliton solutions are novel and have not existed in the literature. The results obtained illustrate that the proposed method is robust, efficient, and readily applicable for constructing new solutions to a wide range of nonlinear fractional partial differential equations. The results of this study are expected to shed light on the field of soliton theory in nonlinear optics and mathematical physics. [ABSTRACT FROM AUTHOR]

In this paper, we consider a predictor-corrector method for solving fractional differential equations (FDEs). We construct a 4th-order Adams-Moulton formula suitable for FDEs and develop a higher-order predictor-corrector technique based on this 4th-order formula within the traditional framework. In addition to the traditional predictor structure, which uses all lower-order schemes as predictors for FDEs, we propose an enhanced predictor that employs only one higher-order Adams-Bashforth type predictor. This new approach aims to complete a higher-order predictor-corrector technique by constructing the Adams-Bashforth formula for FDEs. The convergence orders of the proposed schemes are theoretically proved through error analysis, and numerically demonstrated through several experiments. The numerical results show that the higher-order proposed scheme achieves a higher convergence order compared to existing techniques. Furthermore, the enhanced predictor technique enhances the overall convergence order compared to the traditional predictor technique. [ABSTRACT FROM AUTHOR]

This article investigates the solvability and controllability of stochastic non-instantaneous impulsive Hilfer fractional switched differential equations with deviated arguments and fractional Brownian motion (fBm) in the finite-dimensional space. The first part focuses on analyzing the existence and uniqueness of solutions using the Banach fixed-point theorem. In the second part, controllability results are established for the considered system. This study introduces a new class of control functions designed to govern the system at the termination of time intervals and on each impulsive event, incorporating stochastic noise. This approach leads to comprehensive controllability outcomes, often termed as total controllability results. The theoretical results are primarily established using fixed-point theorem, fractional calculus, Laplace transform, stochastic analysis, and Mittag-Leffler function. A numerical example is provided to validate the obtained theoretical results. [ABSTRACT FROM AUTHOR]

We consider the application of the generalized convolution quadrature (gCQ) to approximate the solution of an important class of sectorial problems. The gCQ is a generalization of Lubich's convolution quadrature (CQ) that allows for variable steps. The available stability and convergence theory for the gCQ requires non realistic regularity assumptions on the data, which do not hold in many applications of interest, such as the approximation of subdiffusion equations. It is well known that for non smooth enough data the original CQ, with uniform steps, presents an order reduction close to the singularity. We generalize the analysis of the gCQ to data satisfying realistic regularity assumptions and provide sufficient conditions for stability and convergence on arbitrary sequences of time points. We consider the particular case of graded meshes and show how to choose them optimally, according to the behaviour of the data. An important advantage of the gCQ method is that it allows for a fast and memory reduced implementation. We describe how the fast and oblivious gCQ can be implemented and illustrate our theoretical results with several numerical experiments. [ABSTRACT FROM AUTHOR]

The major objective of this study is to create solitary pattern solutions to the generalised KdV–Zakharov–Kuznetsov equation and fractional-order Triki–Biswas equation using an iterative procedure known as the residual power series method (RPSM). The efficiency of the suggested technique has been demonstrated by example considerations and the results have been graphically demonstrated. This article also discusses the numerical solution of the technique presented. The current method has been demonstrated to be reliable, highly effective and straightforward and can be used in numerous forms of fractional differential equations developed and in scientific domains. [ABSTRACT FROM AUTHOR]

We employ the Laplace Residual Power Series Method to approximate analytical solutions for differential equations and neutrosophic differential equations with associated parameters, including non-homogeneous equations and fractional formulas in partial differential equations (PDEs). This approach showcases the method's simplicity, effectiveness, and robustness in deriving analytical series solutions for PDEs that involve associated parameters, especially in the context of fractional differential equations. Several practical uses of LRPSM with an emphasis on non-homogeneous and partial differential equations and neutrosophic equations with fractions (PDEs). These applications are significant in a variety of scientific and engineering domains that simulate complicated dynamic system such as anomalous diffusion in physics, viscoelastic material modeling in engineering and signal processing. [ABSTRACT FROM AUTHOR]

In this paper, we consider different versions of the classical Hopf's boundary lemma in the setting of the fractional $ p- $Laplacian for $ p \geq 2 $. We start by providing for a new proof to a Hopf's lemma based on comparison principles. Afterwards, we give a Hopf's result for sign-changing potential describing the behavior of the fractional normal derivative of solutions around boundary points. The main contribution here is that we do not need to impose a global condition on the sign of the solution. Applications of the main results to boundary point lemmas and a discussion of non-local non-linear overdetermined problems are also provided. [ABSTRACT FROM AUTHOR]

This article establishes a sufficient criteria for the existence of an optimal pair and a mild-solution of a non-autonomous fractional Sobolev-type integro-differential equation (FSIDE) with a generalized nonlocal condition. The fixed-point technique, tools from fractional calculus and the semigroup-theory played a valuable role in deriving the results. It is imperative to mention the distinguish attributes of this work that the developed results don't assume (i) that the generated semigroup is compact; (ii) that the linear operators - S (t) are bounded; and (iii) that the operator D is strongly continuous. We also provide an example to demonstrate the established results. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of solutions to a class of weakly dissipative fractional hyperbolic equations equations on R N driven by colored and white noise. The equation has two significant features: the weak dissipativeness due to the fractional dispersive term, and the superlinear growth of the noise. Under certain conditions, the existence, uniqueness and temperedness of pullback random attractors are proved in H s (R N) × H s (R N) for all s ∈ (0 , 1) and N ∈ N when the nonlinear draft term satisfies a fractional subcritical (N, s)-growth condition. In addition, the upper semi-continuity(almost surely and in probability) of the random attractors of the equation with additive colored noise is established as the correlation time of the colored noise shrinks to zero. The idea of uniform tail-estimates and the method of spectral decomposition are combined to prove the pullback asymptotic compactness of the solutions in order to overcome the lack of compact Sobolev embeddings on R N and the weakly dissipative structure of the equation. The results of the paper are even new when the random equation reduces to a deterministic one, and our methods are applicable when the fractional Laplace operators are replaced by the standard one. [ABSTRACT FROM AUTHOR]

This study investigates a coupled model of high-order fractional differential equations (FDEs) using a modified Atangana–Baleanu–Caputo (mABC) operator, incorporating impulsive effects and a novel class of initial value conditions. We establish existence and data dependence theorems for the modified ABC fractional coupled model through Perov’s fixed point theorem. Our approach enhances the modeling of engineering challenges across various fields, including mechanical, electrical, civil, and aerospace engineering. A numerical example is provided to demonstrate the robustness of our findings. [ABSTRACT FROM AUTHOR]

A novel idea, physics informed neural networks introduced a few years back to solve forward and inverse problems for differential equations using the physics information that lies inside them. The central pillar of the physics informed neural networks is automatic differentiation, which is based on the chain rule of differentiation. Automatic differentiation is not applicable to non-local operators because the standard chain rule of differentiation is not valid for non-local operators. Therefore, this work presents non-local physics informed neural networks, which use standard approximation methods for non-local operators and automatic differentiation for local operators to solve differential equations containing non-local operators (forward problems) as well as learn differential equations involving non-local operators (inverse problems). In this work, we consider the Caputo fractional derivative, Volterra integral, and Itô integral as non-local operators. Moreover, we demonstrate the efficiency of the non-local physics informed neural networks with different test examples like the time-fractional diffusion equation in one and two dimensions, the time-fractional Burgers' equation (both equations involving Caputo fractional derivative as non-local operators), the fractional integro-differential equation (Caputo fractional derivative and Volterra integral as non-local operators), and the stochastic fractional integro-differential equation (Caputo fractional derivative, Itô integral, and Volterra integral as non-local operators). Furthermore, for the non-smooth solution, we use the approximation method on non-uniform mesh for non-local operators and compare the results with the approximation method on uniform mesh. We also discuss the error analysis and convergence of the proposed non-local physics informed neural networks. Finally, we take real-world data, which is described by the differential equation containing non-local operators, and show the effectiveness of non-local physics-informed neural networks in addressing practical applications. [ABSTRACT FROM AUTHOR]

In this paper, we consider the nonlinear cauchy problem 휗ρ + (−Δ)휃 2휗 = λȷ0/ρ1−αe휗 + μȷ0/ρ1−α|휗|p−1휗,x ∈ ℝN,ρ > 0,휗(x, 0) = 휗0(x), x ∈ ℝN, where J0/ραf(x) := − 1 Γ(α)∫0ρ f(s) (ρ−s)1−αds and 0 < 휃 ≤ 2, 0 < α < 1. We will study the equation for λ = μ = 1. We demonstrate the local existence and uniqueness of the solution to this problem by the Banach fixed-point principle. Then, it is shown that local solutions experience blow-up. Finally, we present the blow-up rate when 휃 = 2. [ABSTRACT FROM AUTHOR]

By employing the third-order Chebyshev collocation technique along with relevant wavelets, we tackle a third-order singular fractional partial differential equation (PDE). We directly build the Chebyshev operation matrix of the third kind, avoiding the use of the block-pulse function or any approximations. To reduce the order of equation in this approach, we transform the higher-order PDEs into a system of PDEs. Next, we utilize the third-kind Chebyshev wavelet collocation method to convert the resulting system from the prior step into a set of algebraic equations. To demonstrate the method’s effectiveness, we apply it to the time-fractional Camassa–Holm equation and a third-order time-singular PDE. The outcomes are compared with those from several established methods to illustrate the method’s efficiency and practicality. [ABSTRACT FROM AUTHOR]

In this paper, Kudryashov and modified Kudryashov methods are implemented for the first time to compute new exact traveling wave solutions of the space-time fractional (3 + 1) -dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation and Calogero–Bogoyavlenskii–Schiff and Bogoyavlensky Konopelchenko (CBS-BK) equation. With the help of wave transformation, the aforementioned fractional differential equations are converted into nonlinear ordinary differential equations. The purpose of this paper is to devise novel exact solutions for the space-time-fractional (3 + 1) -dimensional CBS and the space-time-fractional CBS-BK equations by utilizing the Kudryashov and modified Kudryashov techniques. The solutions, thus, acquired are demonstrated in figures by choosing appropriate values for the parameters. The solutions derived take the form of various wave patterns, including the kink type, the anti-kink type and the singular kink wave solutions. The obtained solutions are indeed beneficial to analyze the dynamic behavior of fractional CBS and CBS-BK equations in describing the interesting physical phenomena and mechanisms. The obtained solutions are entirely new and can be considered as a generalization of the existing results in the ordinary derivative case. The techniques presented here are very simple, efficacious and plausible and hence can be employed to attain new exact solutions for fractional PDEs. [ABSTRACT FROM AUTHOR]

Determining the unknown order of the fractional derivative in differential equations simulating various processes is an important task of modern applied mathematics. In the last decade, this problem has been actively studied by specialists. A number of interesting results with a certain applied significance were obtained. This paper provides a short overview of the most interesting works in this direction. Next, we consider the problem of determining the order of the fractional derivative in the subdiffusion equation, provided that the elliptic operator included in this equation has at least one negative eigenvalue. An asymptotic formula is obtained according to which, knowing the solution at least at one point of the domain under consideration, the required order can be calculated. [ABSTRACT FROM AUTHOR]

A numerical method based on fractional B-spline is developed and examined to approximate the solution of fractional multiterm pantograph equations. The method is applied to a uniform partition, and the midpoints of the partition are used as extra points to construct a uniquely solvable system. Convergence analysis of the method is discussed via Green's function approach, and an error bound dependent on the regularity of the exact solution is obtained. Additionally, to demonstrate the efficiency and accuracy of the proposed method, several examples are solved. The results indicate that the practical orders of convergence obtained by our proposed method are in good agreement with the theoretical results. [ABSTRACT FROM AUTHOR]

This work uses the modified Extended Direct Algebraic Method (mEDAM) with conformable derivatives to obtain accurate solutions for the diffusion-reaction equation with cubic nonlinearity and the nonlinear fractional generalised density-independent DR problem. We use fractional derivatives in the conformable sense to achieve precise polynomial-form solutions by converting the equations into autonomous two-dimensional plane systems. The technique solves nonlinear ordinary differential equations (ODEs) produced by a fractional transformation to produce travelling wave solutions. Through the visualisation of various solution profiles, as periodic, shock, kink, anti-kink, and soliton waves, numerical simulations offer important insights into the dynamics of nonlinear fractional partial differential equations. This work builds on earlier research and broadens our understanding of complex events. [ABSTRACT FROM AUTHOR]

This paper proposes a numerical technique to solve the time-fractional generalized Kawahara differential equation (TFGKDE). Certain shifted Lucas polynomials are utilized as basis functions. We first establish some new formulas concerned with the introduced polynomials and then tackle the equation using a suitable collocation procedure. The integer and fractional derivatives of the shifted polynomials are used with the typical collocation method to convert the equation with its governing conditions into a system of algebraic equations. The convergence and error analysis of the proposed double expansion are rigorously investigated, demonstrating its accuracy and efficiency. Illustrative examples are provided to validate the effectiveness and applicability of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, the use of a class of fractional-order dynamical systems with discontinuous right-hand side defined with Caputo's derivative is considered. The existence of the solutions is analyzed. For this purpose, differential inclusions theory is used to transform, via the Filippov regularization, the discontinuous right-hand side into a set-valued function. Next, via Cellina's Theorem, the obtained set-valued differential inclusion of fractional order can be restarted as a single-valued continuous differential equation of fractional order, to which the existing numerical schemes for fractional differential equations can be applied. In this way, the delicate problem of integrating discontinuous problems of fractional order, as well as integer order, is solved by transforming the discontinuous problem into a continuous one. Also, it is noted that even the numerical methods for fractional-order differential equations can be applied abruptly to the discontinuous problem, without considering the underlying discontinuity, so the results could be incorrect. The technical example of a single-degree-of-freedom preloaded compliance system of fractional order is presented. [ABSTRACT FROM AUTHOR]

In this article, we define a new class of noncommuting self mappings known as the S-operator pair. Also, we provide the existence and uniqueness of common fixed point results involving the S-operator pair satisfying the (F , φ , ψ , Z) -contractive condition in m-metric spaces, which unifies and generalizes most of the existing relevant fixed point theorems. Furthermore, the variables in the m-metric space are symmetric, which is significant for solving nonlinear problems in operator theory. In addition, examples are provided in order to illustrate the concepts and results presented herein. It has been demonstrated that the results can be applied to prove the existence of a solution to a system of integral equations, a nonlinear fractional differential equation and an ordinary differential equation for damped forced oscillations. Also, in the end, the satellite web coupling problem is solved. [ABSTRACT FROM AUTHOR]

This paper introduces a new concept of random Sehgal contraction in the setting of random cone metric spaces. We explore the modern advancements of traditional fixed-point theorems in a random setting, elaborating on the Sehgal–Guseman fixed-point theorem within the realm of random cone metric spaces. A significant aspect of our research is the interplay between symmetry and randomness; while symmetry provides a framework for understanding structural properties, randomness introduces complexity, which can lead to unexpected behaviors. Our research provides a deeper understanding of the classical results and incorporates a detailed example to illustrate our findings. In addition, major random fixed-point results are also established, which could be applied to nonlinear random fractional differential equations (FDEs) and integral equations as well as to random boundary value problems (BVPs) related to homogeneous random transverse bars. [ABSTRACT FROM AUTHOR]

This paper addresses the questions of well-posedness to fractional m-Laplacian reaction diffusion equation with singular potential term and logarithmic nonlinearity: x - 2 s ∂ t u + (- Δ) m s u + (- Δ) s ∂ t u = u | u | - 2 R (u) , where R (u) = u r ln (| u |) . Guided by the made assumptions, we arrive at the conclusions of the local and global solvability of solutions within the framework of Galerkin approximation. In addition, this study considers weak solutions' asymptotic stability and explosion in finite time. Significantly, we not only figure out the relationship between the non-local fractional operator and singular potential term, but generalize and improve earlier results in the literature. [ABSTRACT FROM AUTHOR]

We propose an alternative definition of a mixed partial derivative in the Caputo sense for functions of two variables defined on the rectangle P = [ 0 , a ] × [ 0 , b ] ( a > 0 , b > 0 ). We give an integral representation of functions possessing such a derivative. Moreover, we study the existence and uniqueness of a solution, as well as the Ulam–Hyers type stability of a fractional counterpart of a nonlinear continuous Goursat-Darboux system described by the introduced Caputo derivative. This paper is a continuation of our paper [R. Kamocki, C. Obczyński, On the single partial Caputo derivatives for functions of two variables, Periodica Mathematica Hungarica 87(2), (2023), 324–339]. [ABSTRACT FROM AUTHOR]

The solution and source term of nonlinear fractional differential equations (NFDEs) with initial values generally have the initial singularity. As is known that numerical methods for NFDEs usually occur the phenomenon of order reduction due to the existence of initial singularity. In this paper, an improved fractional predictor-corrector (PC) method is developed for NFDEs based on the technique of variable transformation. This improved fractional PC method can achieve the optimal convergence order, i.e., the 1 + α order convergence rate for fractional order α ∈ (0 , 1) , of the classical fractional PC method under the high smoothness requirement on the solution and source term. Furthermore, the detailed error analysis also exhibits the relationship between the convergence rate of the improved fractional PC method and the regularities of the solution and source term. Finally, the theoretical error estimate is verified through numerical experiments. [ABSTRACT FROM AUTHOR]

In this work, we consider an inverse problem of determining the coefficient at the lower term of a fractional diffusion equation. The direct problem is the initial-boundary problem for this equation with non-local initial and homogeneous Dirichlet conditions. To determine the unknown coefficient, an overdetermination condition of the integral form is specified with respect to the solution of the direct problem. Using Green's function for an ordinary fractional differential equation with a non-local boundary condition and the Fourier method, the inverse problem is reduced to an equivalent problem. Further, by using the fixed-point argument in suitable Sobolev spaces, the global theorems of existence and uniqueness for the solution of the inverse problem are obtained. [ABSTRACT FROM AUTHOR]

This paper studies the properties of weak solutions to a class of space-time fractional parabolic-elliptic Keller-Segel equations with logistic source terms in R n , n ≥ 2 . The global existence and L ∞ -bound of weak solutions are established. We mainly divide the damping coefficient into two cases: (i) b > 1 - α n , for any initial value and birth rate; (ii) 0 < b ≤ 1 - α n , for small initial value and small birth rate. The existence result is obtained by verifying the existence of a solution to the constructed regularization equation and incorporate the generalized compactness criterion of time fractional partial differential equation. At the same time, we get the L ∞ -bound of weak solutions by establishing the fractional differential inequality and using the Moser iterative method. Furthermore, we prove the uniqueness of weak solutions by using the hyper-contractive estimates when the damping coefficient is strong. [ABSTRACT FROM AUTHOR]

In this paper, an important property of fractional order operators involving discontinuous functions is discussed, First, a pioneering work of impulsive fractional differential equations is recalled to illuminate the incorrectness of notation t k C D t q . Second, a class of piecewise-defined equations with Caputo fractional derivative is contrastively investigated, and it is revealed that the additivity of integration on intervals for integer-order integral does not hold for fractional integrals, not to mention fractional derivatives. Third, by utilizing the Heaviside step function, an interesting property of fractional integral involving piecewise-defined functions is correspondingly presented. Finally, illustrative examples are given for validation of the derived results, which may lead to a new way to reconsider the dynamic behavior of fractional hybrid systems, as well as discontinuous control design for fractional systems. [ABSTRACT FROM AUTHOR]

This article describes the production of biogas in a cellulose-based substrate using a multifractional dynamic model. The objective is to give more precise depiction of the nonlinear characteristics of the chemical reactions involved in anaerobic digestion. In addition well-posedness and consistency, we present the sensitivity analysis used to determine which system equations follow non-integer order dynamics. We illustrate the efficacy of the model with numerical simulations that compare experimental data with the conventional model. The multifractional model's outputs are in good agreement with the biogas production process's overall response, which may lead to more effective control strategies. [ABSTRACT FROM AUTHOR]

In this paper the resolvent family { S α , β (t) } t ≥ 0 ⊂ L (X , Y) generated by an almost sectorial operator A, where α , β > 0 , X, Y are complex Banach spaces and its Laplace transform satisfies S ^ α , β (z) = z α - β (z α - A) - 1 is studied. This family of operators allows to write the solution to an abstract initial value problem of time fractional type of order 1 < α < 2 as a variation of constants formula. Estimates of the norm ‖ S α , β (t) ‖ , as well as the continuity and compactness of S α , β (t) , for t > 0 , are shown. Moreover, the Hölder regularity of its solutions is also studied. [ABSTRACT FROM AUTHOR]

This study investigates nonlinear Caputo-type fractional differential equations with iterated delays, focusing on the neutral type. Initially formulated by D. Bainov and the second author of the current paper between 1972 and 1978, these superneutral equations have been extensively studied in scholarly inquiry. The present research seeks to reinvigorate interest in such delays within sophisticated frameworks of differential equations, particularly those involving fractional calculus. The primary objectives are to thoroughly examine neutral-type fractional differential equations with iterated delays and provide novel insights into their existence and uniqueness by applying Bielecki's and Chebyshev's norms for solution constraints analysis. Additionally, this work establishes Hyers–Ulam–Mittag–Leffler stability for these equations. [ABSTRACT FROM AUTHOR]

In this paper, we study nonlinear systems of fractional differential equations with a Caputo fractional derivative with respect to another function (CFDF) and we define the strict stability of the zero solution of the considered nonlinear system. As an auxiliary system, we consider a system of two scalar fractional equations with CFDF and define a strict stability in the couple. We illustrate both definitions with several examples and, in these examples, we show that the applied function in the fractional derivative has a huge influence on the stability properties of the solutions. In addition, we use Lyapunov functions and their CFDF to obtain several sufficient conditions for strict stability. [ABSTRACT FROM AUTHOR]

Cassava is the sixth most important food crop worldwide and the third most important source of calories in the tropics. More than 800 million people depend on this plant's tubers and sometimes leaves. To protect cassava crops and the livelihoods depending on them, we developed a stochastic fractional delayed model based on stochastic fractional delay differential equations (SFDDEs) to analyze the dynamics of cassava mosaic disease, focusing on two equilibrium states, the state of being absent from cassava mosaic disease and the state of being present with cassava mosaic disease. The basic reproduction number and sensitivity of parameters were estimated to characterize the level beyond which cassava mosaic disease prevails or declines in the plants. We analyzed the stability locally and globally to determine the environment that would ensure extinction and its persistence. To support the theoretical analysis, as well as the reliable results of the model, the present study used a nonstandard finite difference (NSFD) method. This numerical method not only improves the model's accuracy but also guarantees that cassava mosaic probabilities are positive and bounded, which is essential for the accurate modeling of the cassava mosaic processes. The NSFD method was applied in all the scenarios, and it was determined that it yields adequate performance in modeling cassava mosaic disease. The ideas of the model are crucial for exploring key variables, which affect the scale of cassava mosaic and the moments of intervention. The present work is useful for discerning the mechanism of cassava mosaic disease as it presents a solid mathematical model capable of determining the stage of cassava mosaic disease. [ABSTRACT FROM AUTHOR]

In this paper, we study a posteriori error estimates of the fast L 1 − 2 scheme for time discretization of time fractional parabolic differential equations. To overcome the huge workload caused by the nonlocality of fractional derivative, a fast algorithm is applied to the construction of the L 1 − 2 scheme. Employing the numerical solution obtained by the fast L 1 − 2 scheme, a piecewise continuous function approximating the exact solution is constructed. Then, by exploring the error equations, a posteriori error estimates are obtained in different norms, which depend only on the discretization parameters and the data of the problems. Various numerical experiments for the fractional parabolic equations with smooth or nonsmooth exact solutions on different time meshes, including the frequently-used graded mesh, are carried out to verify and complement our theoretical results. Based on the obtained a posteriori error estimates, a time adaptive algorithm is proposed to reduce the computational cost substantially and provides efficient error control for the solution. • We derive a posteriori error estimates for the fast L 1 − 2 method for solving TFPDEs. • We design an adaptive fast algorithm based on these error estimates. • Theoretical analysis reveals how the error depends on the a posteriori quantities. • The a posteriori error estimators of the fast L 1 − 2 method are convergent of order 2. • The adaptive mesh can recover the convergence order of the a posteriori error estimators. [ABSTRACT FROM AUTHOR]

Fractional ordinary differential equations are crucial for modeling various phenomena in engineering and physics. This study aims to enhance solution methodologies by combining the Laplace transform with the reproducing kernel Hilbert space method (RKHSM). This integration leads to a more effective approach compared to the classical RKHSM. We apply the Laplace-reproducing kernel Hilbert space method (L-RKHSM) to develop novel numerical solutions for nonlinear fractional Riccati differential equations. The L-RKHSM systematically produces both approximate and analytic solutions in series form. We present detailed results for four illustrative examples, showcasing the superior performance of the L-RKHSM over traditional methods. This innovative approach not only advances our understanding of nonlinear fractional ordinary differential equations but also demonstrates its effectiveness through significantly improved outcomes in various applications. [ABSTRACT FROM AUTHOR]

The Mittag-Leffler (ML) function exists in the study of special functions. It may be used to solve a variety of fractional differential equations (FDEs) including fractional Laplace and fractional Poisson equations. This paper introduces the generalization of ML function, hereafter referred to as the g -generalized ML (p , s , k) -function, which extends the applicability of fractional calculus and complex analysis to a broader range of problems. The basic properties of the g -generalized ML (p , s , k) -function including its convergence, differentiability, and connection with the Wright hypergeometric and Fox H-functions, which is essential in various mathematical and scientific contexts are presented. The paper then delves into the utilization of the g -generalized ML (p , s , k) -function in integral transforms, with a focus on the Laplace and Fourier transforms. How the g -generalized ML (p , s , k) -function serves as a valuable tool to solve FDEs, including those that arise in engineering, physics, and biology are highlighted. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the study of the system of fractional differential equations$ \begin{align*} ^cD_{0+}^{\alpha} X(t) & = \Gamma(\alpha)(t+1)^{k_1}X^{a}(t)Y^{q}(t), \quad 0<\alpha<1, \quad t>0, \\ ^cD_{0+}^{\beta} Y(t) & = \Gamma(\beta)(t+1)^{k_2}Y^{b}(t)X^{p}(t), \quad 0<\beta<1, \quad t>0, \\ \label{eq:4-frac sys_sigma} \end{align*} $subject to$ \begin{equation*} X(0) = X_{0}>0,\quad Y(0) = Y_{0}>0, \label{eq3:initial_value} \end{equation*} $where $ \Gamma(\sigma) $ stands for the Gamma function, $ ^cD_{0+}^{\alpha} $ stands for the Caputo fractional derivative, and $ a,b,p,q, k_1, $ and $ k_2 $ are real numbers that will be specified later. We present sufficient conditions for the non-existence of global solutions. Furthermore, we present the asymptotic growth of blowing-up solutions near the blow up time, and the graphical profile of the blowing-up solutions. [ABSTRACT FROM AUTHOR]

We establish embeddings on a class of Sobolev spaces with potential weights on unbounded domains. Our results provide embeddings into weighted Lebesgue spaces $ L^q_\theta $ with radial power weights and establish the existence and non-existence of the maximizers for their Trudinger-Moser type inequalities. We also sharpen the maximal integrability by 'removing' necessary terms from the exponential series while maintaining the continuity of the embedding. [ABSTRACT FROM AUTHOR]