A novel distributed-order Cattaneo–Christov model is proposed to effectively characterize non-classical heat conduction processes with memory effect and time–space relaxation behaviors originating from distributed-order fractional derivatives. A fractional physics-informed neural networks (fPINN) algorithm is employed to address both the forward and inverse problems of the distributed-order heat conduction model. For the forward problem, we propose an SfPINN algorithm that incorporates a squared loss term and employs an adaptive updating strategy for the loss-term weights. First, the boundary conditions are embedded into the network output such that they are automatically satisfied. In addition, we design a two-stage training strategy to enhance computational efficiency: in the first stage, the squared loss term associated with the initial condition is incorporated into the loss function; in the second stage, the squared residual term of the governing equation is introduced into the loss function. Numerical results show that the proposed algorithm outperforms the standard fPINN method in both solution accuracy and training iteration speed. For the inverse problem, the numerical results demonstrate that as the iteration number increases, the estimated parameter values progressively converge to their true values and finally stabilize.

This paper introduces convex contractions of order two in complete suprametric spaces and establishes a conditional fixed point theorem for such mappings. The suprametric setting produces a nonlinear Picard recurrence with a quadratic term, requiring explicit orbit-smallness and diameter conditions to ensure convergence. Under these hypotheses, we prove the existence and uniqueness of a fixed point and the geometric convergence of the Picard sequence, recovering Istrăţescu’s classical theorem when the suprametric parameter is zero. Examples are provided to illustrate both the role and applicability of the conditions. The result is further applied to fractional Volterra–Fredholm integro-differential equations and fractional discrete-time neural networks, yielding existence, uniqueness, iterative convergence, and Mittag-Leffler stability of solutions.

This study extends the discrete version of the Susceptible–Infected–Recovered (SIR) model to the non-autonomous case in which the rates of infection and cure are time-dependent. This generalization addresses a critical open problem in the literature. We develop a non-standard finite difference approach that maintains essential biological properties, such as population conservation and non-negativity, even under parameter variation. The main novelty here is the derivation of an exact semi-analytical solution to the non-autonomous nonlinear system. We also leverage the exact solution to develop an optimal control problem, which illustrates the potential for rigorous optimization of treatment strategies to minimize infection costs. The numerical simulations verify the theoretical results, showing not only the stability of the model when parameters change seasonally, as well as the efficiency of the optimal controls, but also the predictive power of the non-autonomous approach compared to classical autonomous models.

This paper addresses the existence of mild solutions and the approximate controllability of a class of higher-order Hilfer fractional semi-linear neutral stochastic differential equations with non-instantaneous impulses in Hilbert spaces. The system is driven by both fractional Brownian motion and Poisson jumps, thereby capturing long-range dependence as well as random discontinuities. By combining techniques from fractional calculus, stochastic analysis, and operator theory, we establish sufficient conditions for the existence of mild solutions. The analysis is carried out through the construction of suitable solution operator families and the application of Sadovskii’s fixed point theorem in an appropriate phase space framework. In addition, we investigate the controllability properties of the system and derive criteria ensuring approximate controllability of the underlying fractional neutral dynamics. The proposed approach relies on the structural properties of the higher-order Hilfer fractional derivative, estimates for stochastic integrals with respect to fractional Brownian motion, and compactness arguments adapted to non-instantaneous impulsive effects. The inclusion of Poisson jumps and neutral terms introduces significant analytical difficulties, which are overcome using refined resolvent operator techniques and fractional power estimates. An illustrative example is presented to demonstrate the applicability of the theoretical results. The results obtained generalize and unify several recent developments in the theory of fractional stochastic systems and provide a flexible framework for analyzing controlled dynamical models with memory, randomness, and impulsive behavior.

We develop a high-order space-time spectral method for nonlinear convection–diffusion equations with a Riemann–Liouville time-fractional derivative and a spectrally defined space-fractional Laplacian. The spatial discretization uses a Fourier spectral method that diagonalizes the fractional Laplacian under periodic boundary conditions. The temporal discretization employs a Petrov–Galerkin method based on generalized Jacobi functions which capture the initial singularity exactly. The nonlinear convection term is treated pseudo-spectrally, and the resulting algebraic system is solved with a damped Newton iteration. Rigorous error analysis proves exponential convergence in both space and time. Numerical experiments for various fractional orders confirm the spectral accuracy. Simulations of the fractional Burgers equation demonstrate that increasing the viscosity enhances diffusion and stabilizes the solution, while a nonlinear coefficient that significantly exceeds the viscosity leads to error growth over long time intervals. The method provides an efficient and accurate tool for simulating anomalous transport phenomena.

This paper introduces a novel memristor-based hyperchaotic system in which the integer-order derivatives are replaced by a variable-order fractal-fractional operator. The dynamical properties of the system, including equilibrium points, Lyapunov exponents, bifurcation diagrams with respect to the variable orders, and the Kaplan–Yorke dimension, are analyzed. A synchronization scheme based on active control is designed for the master–slave configuration, and global Mittag–Leffler stability of the error dynamics is established using a suitable variable-order Lyapunov function. The synchronized states are then applied to an image encryption algorithm. Numerical simulations, security analyses, and NIST randomness tests demonstrate the effectiveness and enhanced performance of the proposed framework compared to existing fixed-order and classical fractional-order methods.

This paper investigates and characterizes Smarandache space curves, an important class of curves, using the Caputo fractional Frenet frame. The Frenet frame and fractional curvature functions have been calculated for these fractional Smarandache curves. To demonstrate the theoretical results obtained, an example of a fractional Smarandache curve derived from a helical curve is considered, and the curvatures of this curve are explicitly calculated. Finally, to show the effect of the fractional order parameter on the geometric behavior, a graphical analysis of the curvatures obtained for different fractional orders is presented.

We study fractional and complex powers of a fixed directional derivative in Rd, defined via a Marchaud-type singular integral representation. Under explicit convergence assumptions, this yields a pointwise nonlocal realization along rays. We then formulate a Ramanujan–Hardy approach to fractional directional differentiation based on analytic interpolation of the directional jet at a point. This construction is local in jet space and is governed by Hardy’s formulation of Ramanujan’s Master Theorem. We emphasize that the resulting Ramanujan–Hardy derivative is defined through a Hardy-admissible interpolant of the directional jet. As an application, we investigate fractional directional derivatives of the Newtonian kernel in dimension d≥3. After a justified regularization and reduction to a Marchaud-type integral, we obtain a one-dimensional integral representation and a zonal harmonic description of the resulting function. This leads to a fractional Maxwell–Gegenbauer identity for 0<ℜ(s)<1, expressing the fractional directional derivative of ∥x∥2−d in terms of Gegenbauer functions of complex degree. In this way, the classical Maxwell multipole formula appears as the integer-order case of a continuous analytic family. Moreover, the fractional operator preserves the main structural properties of the Newtonian kernel, including homogeneity, rotational invariance, and harmonicity away from the origin. The paper thus connects Mellin analysis, Ramanujan’s Master Theorem, fractional calculus, and harmonic analysis on the sphere, while clarifying the distinction between Marchaud and jet-interpolation constructions of fractional directional operators.

This article proposes a novel approach for dealing with the time-fractional Navier–Stokes equations via the natural generalized Laplace transform decomposition method (NGLTDM). This hybrid method utilizes both the natural generalized Laplace transform (NGLT) and a decomposition method. The method is correct because the series solutions become more accurate when more terms are added. We establish precise theorems that verify the existence of solutions and the convergence of the series. The analysis shows that the suggested method is more general than the Homotopy Perturbation Method (HPM) and the Adomian Decomposition Method (ADM). Also, this approach can be applied to handle difficult fluid dynamics problems governed by the Navier–Stokes equations. This study enhances analytical methodologies for fractional-order flow models.

This study models trachoma transmission in Cameroon using a deterministic approach with integer and fractional-order derivatives, incorporating direct, fly-mediated, and environmental transmission routes. Fitting disease data from 1990–2019, the model forecasts trachoma prevalence until 2035. The research confirms the solution existence and uniqueness, calculates the basic reproduction number R0λ where λ∈(0,1] represents the fractional-order parameter, and analyzes equilibrium stability. A stable trachoma-free equilibrium exists when R0λ<1, while an endemic equilibrium is proven stable for R0λ>1 under specific conditions. Calibration of a fractional model with Cameroon data yielded an R0 of 1.169 (indicating endemicity) and identified an optimal fractional order of λ=0.98. By calculating the strength number, we found that another epidemic wave could occur in 50 years. Global sensitivity analysis highlighted key parameters affecting trachoma dynamics. A numerical scheme of the model based on the Adams–Bashforth–Moulton method is constructed and its stability demonstrated. It is then used to perform several numerical simulations, first to validate the theoretical results obtained, and then to compare the different models (statistical and deterministic). The conclusion is reached that the disease will persist in the population (R0>1), although the statistical model shows that it could disappear by 2030. This proves that, for trachoma dynamics in Cameroon, it is advisable to use a deterministic model.