This article presents a parameter-uniform computational approach for singularly perturbed convection–diffusion problems arising in catalytic surface reactions. We compute the evolution of reactant concentration at different time levels in reactive flow systems. The interplay between convection and diffusion causes sharp variations in reactant concentration, leading to the formation of twin boundary layers. These layers are accurately resolved by combining uniform time discretization via the implicit Euler method with exponential B-spline collocation on a layer-adapted spatial mesh. The stability and convergence of the scheme are rigorously analysed, and numerical experiments demonstrate that it accurately captures reactant distributions even for very small perturbation parameters. The numerical results are compared with existing work.
Electrocoagulation (EC) is a physicochemical process for water and wastewater treatment, although its kinetics still present difficulties. Its mathematical description remains a challenge due to the simultaneous occurrence of phenomena such as electrochemical generation, contaminant–coagulant interaction, floc formation, and phase separation. This work presents a general class of reduced nonlinear kinetic models for EC under batch operation in a perfectly mixed reactor. The model uses two variables: the contaminant concentration in the aqueous phase, x(t), and the active coagulant concentration, y(t). Contaminant removal follows an admissible kinetics g(x,y) under mild structural conditions. Coagulant generation is derived from an effective Faradaic term, which arises from a Langmuir-type site-blocking mechanism analogous to noncompetitive enzyme inhibition kinetics. The model’s solutions are shown to be positive and bounded. A physically admissible equilibrium is found to be globally stable by means of a Lyapunov function, without restricting the analysis to a specific kinetic form. Three illustrative removal kinetics are studied: mass-action, contaminant-saturating, and combined saturation, with their relative local convergence rates compared explicitly. Numerical simulations, performance indicators, and a local sensitivity analysis are carried out to complement the qualitative results. In addition, the model’s global sensitivity is studied based on variance (Sobol’ indices) and through a Bayesian identifiability study based on synthetic data. We find that the residual coagulant concentration at equilibrium is governed by a simple algebraic invariant that is independent of the kinetics. Furthermore, the combined saturation kinetics exhibits a sloppy parameter structure. In the contaminant-saturated regime examined here, the individual parameters turn out to be practically non-identifiable from finite, noisy synthetic contaminant data, even when the coagulant concentration is also observed. Finally, the model’s behavior is contrasted qualitatively and semi-quantitatively with experimental trends reported in the electrocoagulation literature. The proposed framework provides a mechanistically grounded mathematical tool for interpreting batch electrocoagulation as a dissipative kinetic system, distinguishing those features of its dynamics that are universal to the entire class of admissible kinetics from those that depend on the specific removal mechanism assumed. The model is not calibrated against experimental data and is intended as a qualitative and analytical tool rather than a predictive one.
In this work, new analytical formulae are proposed for evaluating quantum corrections to the second virial coefficient for systems interacting via attractive and repulsive hard-core Yukawa potentials, without imposing any prior assumptions on the potential’s functional form or shape. The derived expressions provide a general and unified framework for accurately describing quantum effects in interacting particle systems. The accuracy and robustness of the proposed formulae are demonstrated through analytical consistency and comparative analysis with existing theoretical results. Furthermore, the proposed formalism is employed to calculate the thermal properties of gases governed by hard-core Yukawa interactions, including temperature-dependent thermodynamic quantities. The results demonstrate the applicability and effectiveness of the present approach for modeling the thermodynamic behavior of real and model gases over a broad range of interaction parameters.
This paper presents a mixed Fourier-generalized Hermite pseudospectral method for the Fokker-Planck equation with periodic boundary conditions, encompassing both semi-discrete and fully discrete schemes. We first establish essential approximation results for the mixed Fourier-generalized Hermite interpolation, providing the mathematical foundation forsubsequent numerical analysis. Crucially, we introduce a non-standard projection operator that offers tighter error estimates than the standard interpolation operator. Based on this operator, we provide a rigorous convergence analysis of the proposed method. Finally, numerical experiments validate the theoretical findings and demonstrate the high efficiency of our approach.
This study presents a novel and efficient computational framework for numerically solving fractal-fractional Lane-Emden (FFLE) equations using the new generalized Caputo fractal-fractional derivative (NGCFFD). Lane-Emden equations arise naturally in modeling astrophysical phenomena such as stellar structure, polytropic gas spheres, and isothermal gas clouds, as well as in chemical kinetics, thermoelectric processes, and various branches of applied mathematics. The proposed technique employs shifted Legendre polynomials (SLPs) to construct an operational matrix that systematically transforms these complex fractal-fractional differential equations (FFDEs) into tractable systems of algebraic equations, applicable to both linear and nonlinear multi-order FFDEs. Six representative FFLE equations are investigated across twelve computational models with varying fractal-fractional parameters ( , , ), and the results are rigorously compared against conformable fractional (CF) and modified Riemann–Liouville (mRL) solutions available in the literature. The proposed fractal-fractional models consistently outperform the corresponding standard Caputo models. An error estimation theorem and a convergence analysis confirm that the series solutions converge to the exact solutions in the Hilbert space L^2 [0,1]. We think this is the first study to report a numerical solution of fractal-fractional Lane-Emden equations, establishing the fractal order parameter β as an essential, rather than merely supplementary, modeling component for certain classes of singular nonlinear problems.
This work presents a versatile, open-source computational framework for the quantitative analysis of equilibria in solution, with particular emphasis on problems commonly encountered in supramolecular chemistry. The approach is based on a general numerical treatment of complex equilibrium systems, overcoming the limitations of analytical solutions that are restricted to only the simplest cases. The framework is implemented through a series of MATLAB scripts that are fully transparent, easily modifiable, and freely available, thereby avoiding the “black box” character of many commercial programs and enabling users to adapt the workflow to their specific experimental systems. The capabilities and performance of the tool are illustrated through its application to a range of representative datasets, including real experimental data acquired in the course of our own research activities. We demonstrate its utility for the analysis of host–guest complexation and acid–base titrations in which complexation phenomena are involved, thereby illustrating its reliability and broad applicability to recurring problems in supramolecular chemistry. At the same time, the methodology retains general validity for the treatment of almost any equilibrium process occurring in solution. By providing detailed documentation of the underlying mathematical and computational procedures, as well as complete access to the original scripts, this work promotes reproducibility and offers a flexible platform that can be readily employed to test mechanistic hypotheses and readily extended to address new and more elaborate systems.
An approximate solution method is proposed to find the time dependence of the individual state probabilities in the master equation of the continuous time discrete state (CDS) stochastic kinetic approach. In systems where giving the number of a single type of molecule is enough to identify a state, this molecule number is assumed to show a binomial distribution whose expectation is the same as the solution of the corresponding deterministic rate equation. The approximation method is tested in ten different systems. In the first order reaction, it gives the exact solution of the stochastic master equation. In other systems, comparisons of expectations, standard deviations and distributions at selected reaction times with the exact ones at low initial molecule numbers show that the method gives reasonable results, whose precision is improved with an increase of initial molecule numbers. For the zeroth order process, a closed from for the exact solution of the master equation is also reported.
In this paper, a linearized mass-preserving Galerkin finite element method is proposed to solve the multi-dimensional coupled nonlinear Schrödinger equations. The proof of an unconditional convergence is given without any time step restrictions. Our main idea is to obtain the boundedness of the numerical approximation in certain norms, where the Sobolev embedding theorem and the inverse inequality have been used. Numerical examples are given to demonstrate our theoretical results.
In this paper we have focused on the Thomas–Fermi boundary value problems. Thomas–Fermi problems are widely used in the field of astrophysics to determine how the electrons are distributed within an atom. However, Thomas–Fermi problems are difficult to solve numerically due to their coefficient singularity and strong nonlinearity. In this proposed scheme we have implemented the use of higher derivative of the unknown function and approximated it with the help of Haar wavelet, thereby changing them into operational matrices of integration. Subsequently the governing nonlinear differential equation is modified into a system of nonlinear equations. This system is then solved using the Newton–Raphson and Newton–Krylov methods. In addition, comparisons between both the solvers are made. The regularity of the solution is ensured by properly addressing the singularity at origin. Finally comparisons with the exact solution and other known methods are made to demonstrate the effectiveness of this scheme.
This study presents an analytical investigation of substrate concentration profiles in immobilized enzyme systems exhibiting hysteresis behavior under enzyme flow calorimetry conditions. A mathematical model is formulated using the ordinary differential equations that incorporate nonlinear substrate inhibition kinetics together with diffusion parameters. Closed-form analytical expressions for the dimensionless substrate concentration at steady state are derived for planar, cylindrical, and spherical particle geometries using the Hosoya Polynomial Approximation Method (HPAM). The solution is expressed in terms of a quadratic polynomial basis constructed from path-graph Hosoya polynomials, whose coefficients are determined by satisfying the governing nonlinear boundary value problem at selected collocation points. The accuracy of HPAM is rigorously assessed by comparing its predictions against numerical solutions obtained via the fourth-order Runge–Kutta (RK4) method and against results from the Homotopy perturbation method (HPM). For all six parameter cases examined, HPAM achieves an average percentage deviation from the RK4 numerical solution as low as 0.0004
This study proposes a multi-order fractional chemostat model with a Monod functional response to investigate the influence of memory effects on microbial growth dynamics. The classical integer-order formulation is generalized using Caputo fractional derivatives to capture the non-local hereditary characteristics frequently observed in biochemical systems. Fundamental qualitative properties, including the positivity and uniform boundedness of solutions, are rigorously established to ensure biological feasibility. Furthermore, a detailed local stability analysis is performed; the local asymptotic stability criteria for both the washout and coexistence equilibria are mathematically derived via Matignon’s criterion, establishing the parameter thresholds governing microbial extinction and persistence. To simulate the system, efficient numerical approximations are developed using the high-order Toufik–Atangana scheme and the fractional Adams–Bashforth–Moulton predictor–corrector method. Their computational performance is comparatively analyzed, demonstrating that the Toufik–Atangana scheme reduces mathematical complexity through efficient history-weighting loops. Numerical simulations reveal that fractional memory significantly modifies both transient and steady-state behaviors. Specifically, memory effects induce smoother state transitions and slower convergence toward equilibrium states compared to the classical integer-order system. These findings highlight the value of fractional-order modeling in capturing realistic microbial dynamics and provide an effective computational framework for evaluating non-linear bioreactor and ecological processes with memory.
In this work, a shifted Legendre orthogonal polynomial based on the collocation method is implemented for obtaining computational solutions of fractional order multi-delay differential equations (FOMDDEs). The proposed numerical technique is applied to both linear and non-linear FOMDDEs. These FOMDDEs encompass multi-term integer and fractional order derivatives in both delayed and non-delayed components, with the fractional derivative regarded in the Caputo sense. The proposed method transforms the entire problem into a system of linear or non-linear algebraic equations, which is then solved numerically. Error analysis and approximation bounds are also discussed. To validate the implemented numerical scheme, various examples from the literature are compared, demonstrating the accuracy of the method even when using only a few terms of the Legendre polynomials. The numerical results show that the shifted Legendre collocation technique (SLCT) is both simple and efficient for solving a wide range of FOMDDEs.
This study presents a bibliometric analysis on the use of artificial neural networks in chemical reactions, focusing on processes involving esters. Global trends, authors, institutions, countries, and methodological approaches were examined between 2020 and 2025. Bibliometric mapping was conducted using VOSviewer and RStudio. The initial search identified 3227 publications, of which only 38 (1.18
This paper presents a novel numerical technique based on the Hosoya Wavelet Collocation Method (HWCM) for solving fractional-order nonlinear differential models of CO2 emission. The Hosoya polynomials, generated through a recursive relation, are utilized to construct Hosoya wavelet basis functions. By introducing a new operational matrix of fractional integration for the Hosoya basis, the Caputo fractional derivatives in the model are transformed into an algebraic system. The proposed method offers higher accuracy and numerical stability compared with the classical Chebyshev Wavelet Collocation Method (CWCM). Numerical experiments confirm that HWCM efficiently approximates the atmospheric CO2 dynamics with minimal computational effort.
This paper develops a high-order numerical scheme for the variable-order time-fractional generalized Burgers equation (VOTFGBE). The temporal discretization is based on the L2– 1_σ formula for the Caputo fractional derivative, providing second-order accuracy, while spatial derivatives are approximated using a Radial Basis Function–Hermite Finite Difference (RBF-HFD) method with fourth-order convergence. A rigorous Fourier analysis is carried out to establish the unconditional stability of the fully discrete scheme. The proposed method is validated through a series of numerical experiments and benchmark comparisons with existing techniques. The results show that the proposed scheme possesses satisfactory numerical accuracy, and its computational performance is reflected by measured CPU runtime.
Non-additive behavior in multicomponent systems is commonly described using empirical indices that lack direct thermodynamic interpretation. In this work, a general thermodynamic criterion is developed for identifying and quantifying non-additive behavior in systems with coupled equilibria. The formulation is based on a measurable deviation parameter defined as the ratio between real and pairwise-additive reference states and leads to the definition of an irreducible contribution to excess Gibbs free energy. A thermodynamic relation is derived showing that deviations from pairwise-additive behavior can be quantified through a measurable deviation functional and its corresponding Gibbs-energy contribution. The formulation is derived from general thermodynamic considerations and does not depend on the detailed chemical identity of the interacting species, provided that the real and reference states are defined consistently. The implications of the criterion are illustrated using a representative multicomponent system involving coupled homogeneous and heterogeneous equilibria. The analysis shows that non-additive behavior becomes pronounced near phase boundaries and can be expressed as an irreducible Gibbs-energy contribution. This relation allows non-additivity in complex chemical systems to be quantified from equilibrium observables.
This paper studies the dynamics of a one-dimensional temperature-phytoplankton model. We introduce spatial diffusion and time-delay terms to describe the diffusion of temperature and phytoplankton and the growth cycle of phytoplankton, respectively. First, in the delay-free case with diffusion only, the thermal inertia λ and diffusion coefficient d_1 are chosen as bifurcation parameters for Hopf and Turing bifurcations, and the resulting Turing–Hopf bifurcation is identified. Second, when time delay is included, the delay τ and d_1 serve as bifurcation parameters; based on the analysis of Hopf and Turing bifurcations, the parameter-induced Turing–Hopf bifurcation is identified. For both cases, the corresponding normal forms and bifurcation diagrams of the Turing–Hopf bifurcation are obtained using center manifold theory. Finally, numerical simulations reveal typical spatiotemporal patterns, including spatially homogeneous periodic solutions, spatially inhomogeneous steady-state solutions, and spatially inhomogeneous quasi-periodic solutions. The results demonstrate that the coupling of diffusion and delay significantly enriches the system’s spatiotemporal dynamics.
This paper investigates the temporal and spatiotemporal dynamics of a generalized Degn–Harrison chemical reaction model with nonlinear interaction rate k u^pv/1+s u^q. The kinetic orders p and q separate the nonlinear activation in the u-channel from the inhibitory saturation in the denominator. Under positive parameter assumptions, we establish global-in-time existence, positivity and finite-time a priori bounds for the no-flux reaction–diffusion problem, and we derive boundedness and permanence estimates for the temporal kinetics. For the kinetic subsystem, the unique positive equilibrium is obtained explicitly, and its local stability is characterized completely by the trace and determinant of the Jacobian. A conditional global asymptotic stability result is proved in the monotone-rate subcase. At the loss of temporal stability, an explicit Hopf threshold is derived, the transversality condition is verified, and the first Lyapunov coefficient is evaluated through Kuznetsov’s multilinear normal-form formula. After diffusion is restored, the Turing instability conditions are written in terms of the Neumann spectrum and the kinetic Jacobian coefficients, giving a computable instability band and critical wave number. High-resolution numerical experiments show Hopf bifurcation diagrams, stable limit cycles, square-root Hopf scaling, dispersion curves, stationary Turing patterns, and transient breathing spatiotemporal states. The comparison with the classical choice (p,q)=(1,2) demonstrates that stronger inhibitory kinetics can shift the Hopf threshold and open a genuine Turing window that is absent for the corresponding classical parameter regime.
This study presents a comprehensive analysis of a quantum Stirling engine whose working substance is a two-level atom coupled to a single-mode optical cavity within the Jaynes–Cummings model. The non-extensive Tsallis entropy formalism, parameterized by q , is employed to investigate the influence of strong quantum correlations and non-Markovian effects. The performance metrics, including work output (W), thermodynamic efficiency (η), and the coefficient of performance (ϵ) for refrigeration, are rigorously evaluated as functions of the atom–field coupling ratio g2/g1, the bath temperature gradient Th/Tc, and the non-extensivity parameter q. It is demonstrated that a reduction in q, signifying stronger non-extensivity, systematically enhances all performance indicators. The engine efficiency and work output are found to increase with both the coupling strength and the temperature difference, revealing a synergistic quantum-thermodynamic effect. Notably, the coefficient of performance for refrigeration increases with the temperature gradient, a distinctive departure from classical behavior. Furthermore, the thermodynamic phase diagram of operational modes is shown to simplify under increased non-extensivity, with dissipative modes being suppressed in favor of stable engine and refrigerator operation.
Lambda functions constitute a Laguerre-type basis that supports highly accurate atomic structure calculations. While 150-term quadruple-precision expansions yield 30-digit Hartree–Fock energies for atoms up to the third period, heavier atoms require larger expansions, placing the computational bottleneck on two-electron integral evaluation. The algebraic framework developed by Zamastil et al. [18] for Coulomb Sturmians and the su(1,1)-based recurrence formalism for Lambda functions of McCoy and Caprio [26] were extended to formulate two-electron integrals over Lambda functions and to implement a quadruple-precision computer program. However, when the number of expansion terms exceeds 100, deterioration in numerical accuracy became significant. This problem was overcome by substituting multiple-precision arithmetic for quadruple-precision arithmetic in the numerically unstable step. The revised implementation achieves a significant speedup over our previous Gauss–Laguerre quadrature-based program and has enabled a 250-term Hartree–Fock calculation for radon with 25-digit accuracy. The theoretical derivation and computational results are presented.
