This paper presents a comprehensive analysis of the existence, uniqueness, and Ulam–Hyers stability of solutions for a class of Cauchy‐type nonlinear fractional differential equations with variable order and finite delay. The motivation for this study lies in the increasing importance of variable‐order fractional calculus in modeling real‐world systems with time‐dependent memory and hereditary properties, where the order of differentiation evolves with time or state. Moreover, incorporating finite delay allows the model to capture the influence of past states on current dynamics, making it applicable to a wider range of physical and biological processes. By employing tools from fixed point theory, we derive new sufficient conditions for the existence and uniqueness of solutions using both the Banach contraction principle and Schauder’s fixed point theorem. Furthermore, we establish the Ulam–Hyers stability of the proposed system under suitable assumptions on the nonlinear term. To illustrate the validity of our theoretical findings, a numerical example is provided, demonstrating how variations in the fractional order affect the system’s behavior. The obtained results enrich the theoretical framework of variable‐order fractional calculus and extend its applicability to delayed systems. These findings may serve as a mathematical foundation for future research on more general models, including neutral or implicit fractional differential equations of variable order, which are of growing relevance in applied sciences and engineering.

This article investigates mathematical simulations of Michaelis–Menten kinetics in differential biochemical reactions by implementing fractional derivatives. It establishes numerical computations for the concentrations of enzymes, substrates, inhibitors, products, and several complex intermediates using the homotopy perturbation method (HPM), homotopy analysis method (HAM), and variational iteration method (VIM). The focus is on Caputo fractional derivatives. Numerical examples illustrate HPM, HAM, and VIM comparisons to enhance accuracy and understanding. The conclusion recaps the key findings of this biochemical reaction model involving fractional derivatives, including the relevant numerical results and graphical representations.

This paper investigates the existence and uniqueness of solutions to nonlinear Volterra integral equations of variable fractional order in Fréchet spaces. The variable‐order fractional derivative is considered in the Riemann–Liouville sense, which extends classical approaches and is central to the paper’s novelty. By employing a nonlinear alternative of the Frigon–Granas fixed‐point theorem for contraction mappings, a rigorous mathematical framework is provided, suitable for problems on semi‐infinite intervals and for functions with variable fractional order, where classical Banach space approaches may fail. Illustrative examples demonstrate the applicability of the main results. The approach highlights the flexibility and generality of the method, paving the way for future extensions such as stability analysis and numerical schemes based on the established theory.

This paper presents a novel and efficient spectral collocation framework for solving nonlinear variable‐order fractional differential equations (VO‐FDEs) involving the Atangana–Baleanu–Caputo (ABC) operator. Shifted Morgan‐Voyce polynomials (SMVPs) are employed as basic functions to construct a new operational matrix specifically adapted to the variable‐order ABC operator. This matrix substantially reduces computational complexity while maintaining high accuracy. The proposed approach converts VO‐FDEs into tractable nonlinear algebraic systems. In contrast to existing polynomial‐based techniques, the framework demonstrates enhanced robustness in handling nonlinearities and variable memory effects, delivering superior precision and stability. Numerical experiments are conducted on challenging nonlinear and multiterm VO‐FDEs. Comparative analyses confirm the advantages of the Morgan–Voyce polynomials (MVPs)‐based scheme over classical spectral methods. These findings establish the proposed method as a versatile and reliable tool for tackling complex VO‐FDEs with nonsingular memory kernels.

This study introduces a novel fractal–fractional extension of the Hodgkin–Huxley model to capture complex neuronal dynamics, with particular focus on intrinsically bursting patterns. The key innovation lies in the simultaneous incorporation of Caputo–Fabrizio operators with fractional order α for memory effects and fractal dimension τ for temporal scaling, enabling the representation of nonlocal interactions and multiscale dynamics that extend beyond the capabilities of classical models. Our numerical simulations demonstrate that the synergistic combination of ( α , τ ) parameters uniquely modulates burst duration, interburst intervals, and spike‐frequency adaptation, producing dynamical regimes inaccessible to both integer‐order models and single‐parameter fractional approaches. Lyapunov stability analysis confirms that the framework maintains biological plausibility while enabling substantially richer temporal patterns. This work establishes a comprehensive mathematical foundation for understanding multiscale neuronal behavior, with significant implications for analyzing pathological rhythms and advancing neuromodulation studies.

In the present article, a new amplitude expansion–based homotopy perturbation method (AE‐HPM) is used to study the nonlinear behavior of a damped oscillator. The traditional homotopy perturbation method is extended, considering a simple amplitude expansion to determine the solution and amplitude frequency relationship for the damped nonlinear system, which could not be solved by the traditional approach. The simplicity, efficiency, and validity of the present AE‐HPM are verified by applying it to the cubic–quintic oscillator and pendulum equation with linear damping. The analytical results obtained by the present method show that the oscillation amplitude decays exponentially with the damping parameter, while the frequency response is very much influenced by the system’s nonlinearity. The comparison of the results obtained by AE‐HPM with numerical results and He’s frequency formulation (HFF) solution shows that the present method is accurate, converges faster, and can be used in a wider range of problems, while the traditional HPM fails. Therefore, the proposed method serves as a simple and dependable analytical approach for analyzing nonlinear damped oscillatory systems.

This paper introduces a new Mittag–Leffler–Laplace memory kernel defined by and develops a unified framework for modeling heat transfer in heterogeneous media with nonlocal temporal memory. The proposed kernel combines algebraic singularity, stretched attenuation, and fractional relaxation through independent parameters, enabling precise control of heterogeneity, memory depth, and relaxation strength. A nonlocal‐in‐time heterogeneous heat equation driven by is formulated, and its well‐posedness, energy stability, and thermodynamic admissibility are established using complete monotonicity arguments. Sharp long‐time polynomial decay rates are derived via Tauberian techniques and fractional Grönwall inequalities, revealing the emergence of fractional heat dynamics as a natural asymptotic regime. Fully discrete numerical schemes are analyzed, yielding unconditional energy stability and optimal convergence rates of order . Numerical experiments confirm the theoretical decay rates and demonstrate the distinct roles of the parameters α , κ , ν , and μ in regulating thermal relaxation and heterogeneity. The proposed kernel unifies classical Fourier, Caputo, and Atangana–Baleanu heat models within a single integral formulation and provides a flexible and physically admissible tool for simulating heat transfer in complex heterogeneous systems.

This study investigates the phenomenon of bioconvection in a Darcy–Forchheimer flow of a Prandtl fluid on a stretching sheet, considering dual diffusive processes and incorporating the Cattaneo–Christov heat conduction model. For the conversion of modeled equations to dimensionless, a suitable set of variables has been incorporated. The governing equations are solved numerically using the bvp4c technique, and the effects of key parameters, including the Forchheimer parameter, Prandtl number, and bioconvection Lewis number are comprehensively analyzed. The research reveals complex interactions between buoyancy‐driven bioconvection, porous media resistance, and non‐Fourier heat conduction, providing insights into their applications in fields such as microbial remediation and geothermal reservoir engineering. It has been revealed in this work that velocity of fluid is augmented with an escalation in fluid flow parameters and elastic factor while retarded with the upsurge in porosity and inertia factors. Thermal distribution is enlarged with progression in thermophoresis, Brownian factors, and thermal source parameter, while it is retarded with escalation in Prandtl number. Concentration distribution escalates with the upsurge in Schmidt number and thermophoresis factor while retards with the escalation in Brownian, elastic, and chemical reactivity factors. Microorganisms’ profiles retard gradually with an escalation in Peclet and bioconvective Lewis numbers. The Nusselt number is enhanced by 14.50% as thermal relaxation is augmented by 0.2–0.8. Moreover, as the mass relaxation is enhanced from 1.2 to 1.8, the Sherwood number is intensified by 22.56% showing boosted mass transport in the prince of non‐Fick theory. The agreement of the current result verified with earlier literature and found an outstanding achievement.

Numerical solutions for second‐order parabolic partial differential equations (PDEs), specifically the nonlinear heat equation, are investigated with a focus on analyzing residual corrections. Initially, the Galerkin weighted residual method is employed to rigorously formulate the heat equation and derive numerical solutions using third‐degree Bernstein polynomials as basis functions. Subsequently, a proposed residual correction scheme is applied, utilizing the finite difference method to solve the error equations while adhering to the associated error boundary and initial conditions. Enhanced approximations are achieved by incorporating the computed error values derived from the error equations into the original weighted residual results. The stability and convergence of the residual correction scheme are also analyzed. Numerical results and absolute errors are compared against exact solutions and published literature for various time and space step sizes, demonstrating the effectiveness and precision of the proposed scheme in achieving high accuracy.