This paper investigates the existence, uniqueness, and Hyers–Ulam stability of a new class of higher-order coupled stochastic non-instantaneous impulsive Hilfer fractional switched differential equations with deviated arguments and Poisson jumps in finite-dimensional spaces. The well-posedness of the system is established using the method of integral contractors under suitable regularity assumptions and a weakened Lipschitz-type condition on the nonlinear operators. The proposed framework introduces a unified class of coupled Hilfer fractional stochastic switching systems that incorporates non-instantaneous impulses, switching dynamics, fractional integral initial conditions, and both continuous and jump-type stochastic perturbations. This structure provides a mathematically consistent and physically realistic model for hybrid dynamical systems with memory and random disturbances. The analysis employs techniques from fractional calculus, stochastic analysis, Laplace transforms, and Mittag–Leffler function theory. A numerical example and an application to electromechanical coupling and seeker stabilisation are presented to illustrate the effectiveness and practical relevance of the theoretical results.

Solution of nonlinear variational problems for generalized fractional viscoelastic models

ABSTRACT: In this paper, the existence of solutions and the stability results are derived for impulsive Hilfer fractional integrodifferential stochastic systems (IDSSs) with Poisson jumps in R n space. The main results are obtained by using fractional calculus, stochastic analysis approach, and measure of non-compactness (MNC) via M o ¨ nch fixed point technique for the first time in the literature to the finite dimensional space. For the stability result, boundedness properties of Mittag-Leffler (M-L) function is effectively used. Two numerical examples are given for verification of theoretical results.

A new structure of Hermite polynomials associated with the Mittag–Leffler function is introduced in this paper. A generating function in terms of Mittag–Leffler function is obtained. Several differential recurrence relations are established. Pure recurrence relations are derived without the use of differential recurrence relations. Properties of the gamma function are used to obtain integral representations and integral duplication formulae.

Abstract Simulating viscoelastic fluids at high Weissenberg numbers is challenging due to numerical instabilities, especially in flows with singularities. The natural stress formulation (NSF) is a robust technique designed to overcome these issues. Separately, the generalized Phan-Thien and Tanner (gPTT) model offers enhanced rheological flexibility by using the Mittag-Leffler function. This work develops and validates an in-house, finite-difference NSF-gPTT solver. The method is first validated against the traditional HiG-Flow solver in channel flow and 1:4 sudden expansion geometries, showing excellent agreement for velocity/stress profiles and vortex reattachment lengths. We then apply the framework to the L-shaped channel benchmark, which features a re-entrant corner. The NSF-gPTT solver remains stable and accurate up to a Weissenberg number ( Wi ) of 100. The results reveal a counter-intuitive decrease in the peak of the first normal stress difference ( $$N_1$$ N 1 ) at the corner with increasing Wi , a direct result of the gPTT model’s shear-thinning. Furthermore, we demonstrate the NSF’s stability by showing that at $$Wi=100$$ W i = 100 , the internal conformation tensor trace grows to $$\approx 100$$ ≈ 100 , while the elastic stress trace remains small ( $$\approx 0.48$$ ≈ 0.48 ). This study demonstrates that the NSF-gPTT formulation is a stable and powerful tool for probing complex viscoelastic phenomena in high- Wi regimes.

Subdiffusion on graphs is often modeled by time-fractional diffusion equations; yet, its structural and dynamical consequences remain unclear. We show that subdiffusive transport on graphs is a memory-driven process generated by a random time change that compresses operational time, produces long-tailed waiting times, and breaks Markovianity while preserving linearity and mass conservation. While the subordination representation and complete monotonicity properties of the Mittag-Leffler function are classical, we develop a graph-based synthesis in which Mittag-Leffler dynamics admit an exact convex, mass-preserving representation as a superposition of Laplacian semigroups evaluated at rescaled times. This perspective reveals fractional diffusion as ordinary diffusion acting across multiple intrinsic time scales and enables new structural and dynamical interpretations of graphs. This framework uncovers heterogeneous, vertex-dependent memory effects and induces transport biases absent in classical diffusion, including algebraic relaxation, degree-dependent waiting times, and early-time asymmetries between sources and neighbors. These features define a subdiffusive geometry on graphs, enabling the recovery of global shortest paths, in contrast to the graph exploration of diffusive geometry, while simultaneously favoring high-degree regions. Finally, we show that time-fractional diffusion can be interpreted as a singular limit of multi-rate diffusion, in an appropriate asymptotic sense.

In this paper, we study a nonlocal initial boundary value problem for an abstract time-fractional diffusion-wave equation involving an unbounded, positive, self-adjoint operator on a Hilbert space. The existence of a mild solution is investigated using the eigenfunction decomposition method. By expanding the solution in terms of the operators eigenfunctions, the problem is reduced to a system of ordinary fractional differential equations with nonlocal conditions. The solutions of these equations are expressed in terms of the Mittag-Leffler function. By examining the zeros of the denominator, we identify the appropriate interval of definition, which excludes the right endpoint, ensuring the correctness of the solution. The solution to the original problem is expressed as a series expansion in terms of the eigenfunctions of an abstract operator. By applying estimation techniques in the corresponding Hilbert spaces, we establish the existence, uniqueness, and regularity of the solution.

The Mittag-Leffler function holds significant importance in fractional calculus due to its extensive applications in addressing challenges across science, engineering, biology, hydrology, and earth sciences. Notably, the closed-form solution of a time-fractional model naturally emerges as the Mittag-Leffler function (MLF), necessitating precise and efficient computations. Consequently, numerical approximations are essential for accurately calculating the Mittag-Leffler function. In this study, we develop a straightforward yet precise real pole rational approximation for the Mittag-Leffler function. We demonstrate first-order convergence and L-acceptability, which aid in mitigating unwanted oscillations. Additionally, we create an effective and precise first-order generalized exponential time differencing scheme to solve the time-fractional reaction–diffusion equations. We obtain and prove the convergence result using Grönwall-type inequality. Several numerical experiments are conducted to confirm the efficiency and accuracy of the proposed numerical scheme compared with exact solutions. The computational efficiency of the proposed method is compared with another existing first-order numerical technique. Furthermore, our proposed scheme is crucial for developing higher-order predictor–corrector schemes for solving time-fractional models.

Generalized incommensurate fractional differential systems (GIFDSs) unify classical fractional frameworks via weight functions, capturing non-uniform multicomponent system dynamics. This paper fills a critical research gap by analyzing GIFDSs for both commensurate and incommensurate weight functions. For commensurate weights (wi(t)=w(t)), classical IFDS equivalence is established via state transformation. Linear homogeneous mild solutions are derived using the incommensurate Mittag–Leffler function. Existence and uniqueness of nonlinear solutions are proved under continuity and Lipschitz assumptions. Hyers–Ulam stability is verified for linear non-homogeneous systems. For incommensurate weights (distinct wi(t)), a novel framework is developed: by the integral bound lemma and Picard iteration, local existence (existence on [a,t1]) is established, then it is extended to the full interval. The global uniqueness is obtained by Gronwall-type inequality via combined substitution. These results are applied to Hopfield Neural Networks, showing that one-layer HNNs with tanh or sigmoid activations admit unique mild solutions under GIFDS dynamics.

Progress has been made in developing sigmoids derived from the Mittag-Leffler function. Generally, the Mittag-Leffler functions of one and two parameters, as well as the Atangana-Baleanu formulation, are used to substitute the traditional exponential in the Verhulst model. Sigmoid formulations have been successfully demonstrated using numerous subcases of the Mittag-Leffler function. Maple and Mathematica computing have enabled us to successfully visualize the results and identify emerging computational problems.

Abstract In the present article, we first introduce the problem of the diffusive logistic equation with memory in Bessel potential spaces, discussing the parameters $$\alpha $$ and $$\widetilde{\eta }$$ and, in particular, their influence on the memory term of the model. Next, we present a result via a lemma that provides an estimate for the integral of a Mittag-Leffler function in terms of the memory effect. Based on this, using the Banach Fixed Point Theorem, Gronwall’s inequality, and the lemma estimating the Mittag-Leffler function, we investigate the existence, uniqueness, regularity, and continuous dependence of weak solutions to the diffusive logistic equation.

This article aims to provide a practical and reliable approach to solving fractional M-derivative partial differential equations with nine parameters that involve the Mittag-Leffler function. Several theorems have been developed to describe and express the M-derivative triple Laplace transform. Furthermore, these defined concepts and theorems are demonstrated by applying them to fractional partial differential equations. This proposed transformation appears to efficiently enable finding solutions to partial differential equations with M-derivatives that match mathematical, engineering, and physical models.

ABSTRACT We study a semilinear time‐fractional diffusion equation in one spatial dimension, where the time derivative is understood in the sense of Caputo. The problem is complemented with suitable boundary conditions, possibly of moving type. We establish the existence, uniqueness, and regularity of weak solutions under natural assumptions on the data and nonlinearities. The analysis relies on fractional calculus tools, including estimates for the Mittag–Leffler function and a fractional Grönwall inequality. Extensions to nonlinear boundary conditions and moving interfaces are also discussed.

This work aims to investigate the nonlinear time-fractional impulsive Navier-Stokes equation in Hilbert space driven by fractional Brownian motion. To start with, the non-linear stochastic partial differential equation is remodeled by using the Helmholtz-Hodge projection operator, stochastic calculus, and the Stokes operator. The existence of mild solution is obtained through the application of Mittag-Leffler functions, Krasnoselskii’s fixed point theorem, and stochastic analysis. The approximate controllability result is obtained for the presented system under suitable assumptions. Finally, a suitable application for turbulence control, which is an aircraft model in automobile engineering is presented and validated the obtained theoretical results.

The Mittag-Leffler Type Poisson (MLTP) distribution emerges out of the Mittag-Leffler (ML) function, which has vast application in engineering, physics, mathematics, earth sciences, etc. The present paper develops new distributions of order statistics and record statistics for the MLTP distribution along with their key properties. The diverse shapes of the MLTP distribution for different values of the parameters disclose its wide application for different types of data sets. We also present two characterizations of the under study distribution by following the conditional distribution and the Rao-Rubin property, respectively. Certain interesting results are derived by the distributions of sum, product, and ratio for two MLTP random variables. The resulting distributions turn out to be well-known binomial-like expressions. The model parameters are estimated through the maximum likelihood method, while its practical utility is evidenced by its superior performance over several existing distributions using a medical dataset.

This paper investigates inverse problems for the space-time fractional sub-diffusion equation driven by a fractional Brownian sheet with a temporally singular source term. The main contribution lies in the simultaneous recovery of multiple parameters (including the temporal singular source term μ, fractional order α, diffusion coefficient a, stochastic source term g 2 ( t ) , and terminal time T from a single lateral boundary measurement in partially unknown media. First, we introduce new fractional integral operators and use the Sobolev–dual Sobolev framework to regularize the singular source term μ from a negative-order space to L 2 [ 0 , T ] . Subsequently, the expectation characteristics of boundary measurement data are utilized to uniquely recover μ, α and a. Building on this foundation, through equation decoupling techniques and comprehensive application of variance analysis and generalized Gronwall inequality, we rigorously prove the uniqueness of the stochastic source term g 2 ( t ) . Finally, under the assumption of a known fractional order α, the uniqueness of the terminal time T is demonstrated based on the representation of the mild solution and asymptotic properties of the Mittag–Leffler function.

This study presents an extended fractional-order mathematical model for blood flow through a stenosed artery under the combined effects of a magnetic field, porous medium, chemical reaction, and nanoparticle diffusion. The study aims to provide a more accurate and physiologically relevant representation of nanoparticle transport in pathological arterial flow conditions. The governing nonlinear equations for momentum and mass transfer were formulated and solved using a semi-analytical approach involving modified Bessel and Mittag–Leffler functions. Model validation through comparison with existing results showed excellent agreement, confirming the reliability of the proposed formulation. The parametric analysis revealed that increasing the chemical reaction parameter and Schmidt number reduced nanoparticle concentration, whereas a higher fractional order enhanced mass transport by weakening memory effects. The study concludes that the fractional-order framework offers an improved description of nanoparticle transport in stenosed arterial blood flow and contributes to the advancement of mathematical modeling for physiologically realistic hemodynamic analysis.

This paper addresses the stabilization problem for a class of fractional-order memristive reaction–diffusion Cohen–Grossberg neural networks with time-varying delays under an intermittent boundary control framework. Two scenarios are considered: systems without parametric uncertainties, for which asymptotic stability is established, and systems with uncertainties, for which robust asymptotic stability is ensured. By constructing appropriate Lyapunov functionals and employing Wirtinger-type inequalities, the fractional Razumikhin approach, and key properties of the Mittag–Leffler function, sufficient stability conditions are derived in terms of linear matrix inequalities with reduced conservatism. Furthermore, the effects of time-varying delays and control activation intervals on the stabilization performance are systematically investigated. The effectiveness and advantages of the proposed control methodology are validated through numerical simulations.

The long-term prestress relaxation of soil anchors embedded in saturated clay is a critical issue affecting the safety of geotechnical structures such as slopes and foundation pits. Traditional integer-order constitutive models are often unable to accurately describe the nonlinear and time-dependent relaxation behavior observed in such anchorage systems. Based on fractional calculus theory, this study establishes a constitutive model for the coupled stress relaxation behavior of soil anchors and saturated clay. The Riemann–Liouville fractional derivative and the two-parameter Mittag-Leffler function are introduced to represent the material memory effect and continuous relaxation characteristics. To achieve reliable parameter identification, a hybrid optimization strategy combining the Adaptive Hybrid Differential Evolution (AHDE) algorithm and the Levenberg–Marquardt (L-M) method is proposed. The proposed model and identification approach are validated using field monitoring data from soil anchors in a slope engineering project at the Guangxi Friendship Pass Port. The results show that the proposed model can accurately reproduce the entire stress relaxation process, with a coefficient of determination of R2 = 0.9517. Parameter sensitivity analysis further clarifies the influence of key parameters, including the fractional order and viscosity coefficient. The proposed approach provides a systematic theoretical framework and practical reference for the analysis and prediction of long-term prestress relaxation in soil anchorage systems.

Wick Mittag-Leffler function and Wick positivity