AppliedMath, Volume 6, Issue 7 (July 2026) – 18 articles

The generalized Burgers–Huxley (gBH) equation arises as a canonical model in nerve-pulse propagation (generalizing the Hodgkin–Huxley/FitzHugh–Nagumo excitable-media framework), in population dynamics with Allee-threshold reaction kinetics, and in nonlinear wave propagation in dispersive media; accurate benchmark solutions are essential for quantitative predictions in these domains. We couple the fourth-order exponential time differencing scheme ETDRK4 with a Chebyshev collocation spatial discretization and a linear boundary-lifting procedure to solve the gBH equation on a bounded interval with non-homogeneous Dirichlet data. On the canonical Ismail–Raslan–Rabboh travelling-wave benchmark the scheme attains $L^{\infty }$ errors at the level of floating-point round-off (∼10⁻¹⁹ absolute, ∼10⁻¹⁵ relative) with as few as $N=2$ collocation points and a single time step of size $\Delta t=1.0$—that is, three total nodes and one ETDRK4 advance. In strongly nonlinear regimes ($\gamma =0.1, 0.3, 0.5, 0.9$) the scheme exhibits approximately $\mathcal{O}(\Delta t^{2.45})$ temporal convergence across all four parameter values, consistent with the classical Hochbruck–Ostermann order reduction for exponential integrators on parabolic PDEs with non-homogeneous Dirichlet data. Used as a high-accuracy probe, the scheme provides a diagnostic of independent interest: the wave-speed formula of Wang, Zhu and Lu, still appearing as the exact-solution benchmark in numerical studies as recently as 2020, does not satisfy the partial differential equation. The corrected formula stated by Deng and verified symbolically by Appadu and Tijani is the unique value that makes the travelling-wave ansatz a genuine solution. We derive the residual associated with Wang’s formula in closed form, $R=\frac{\gamma A_1}{2}(A_2-A_2^{\mathrm{W}})(1-v^2)$, and show both analytically and numerically that reported errors for schemes benchmarked against Wang’s formula coincide with the analytical wave-profile gap $\frac{\gamma A_1}{2}|A_2-A_2^{\mathrm{W}}|$ rather than with true scheme accuracy. At the Ismail benchmark this gap equals $3.748\times {10}^{-7}$, which matches the N- and $\Delta t$-independent plateau observed when the scheme is measured against Wang’s profile. In the nerve-pulse and excitable-media interpretation, the two formulas correspond to action-potential propagation speeds of opposite sign at the Ismail benchmark, underscoring that the correction is not a mere algebraic curiosity but changes the qualitative physical prediction of the model. Full article

Many applications in finance and biology involve multiplicative noise geometric Brownian motion (GBM)-type stochastic differential equations (SDEs) whose drift carries a single dominant periodic component. Such structures arise in seasonal Black–Scholes option pricing, commodity derivatives with annual price cycles, and stochastic biological oscillators driven by a known frequency; the primary contribution of this paper is a high-precision numerical scheme validated on GBM-type test problems with periodic drift. This paper proposes a Strang operator splitting scheme within the Logarithmic Drift-Diffusion Splitting (LDDS) framework, which splits the SDE in y-space into a deterministic drift ODE sub-step (Step A) and an exactly solvable multiplicative diffusion sub-step (Step B). Step A employs the Periodically Fitted Adams–Bashforth–Moulton fourth-order predictor–corrector method (PABM4), which achieves zero local truncation error for trigonometric forcing terms by introducing additional shift terms and simultaneously imposing polynomial exactness conditions and trigonometric fitting conditions. When the Step A forcing belongs to the PABM4 exact function class ${\mathcal{F}}_{\omega }=\mathrm{span}\{1,t,t^2,\sin \omega t,\cos \omega t\}$, the Strang+PABM4 scheme achieves floating-point precision saturation. We investigate three test problems: the cosine-drift GBM (Test Problem 1), the polynomial–trigonometric mixed drift GBM (Test Problem 2), and a dual-frequency drift applicability test (Test Problem 3). Monte Carlo strong error experiments ($M=1000$ paths) validate that Strang+PABM4 achieves saturation at machine precision (≈${10}^{-15}$) on Test Problems 1 and 2, improving precision by ≈${10}^2\times$ over the best algebraically convergent reference. Test Problem 3 identifies the method’s applicability boundary: when the drift contains a second frequency outside ${\mathcal{F}}_{\omega }$, Strang+PABM4 degrades gracefully to order ≈ 4 without catastrophic failure. The floating-point saturation of Strang+PABM4 is contingent on the drift belonging to ${\widehat{\mathcal{F}}}_{\omega }$; when this condition is violated, the method degrades gracefully to algebraic order ≈ 4, as demonstrated in Test Problem 3. Full article

The focus of this paper is the semi-quantitative approach to risk analysis in HACCP (Hazard Analysis and Critical Control Points), where we show how alternative computations in the risk matrix, combining probability and impact, can be provided based on algebraic and many-valued logical techniques. Doing so, we further show how applying such computations requires being formal concerning underlying information structures, which in turn enables being formal concerning functional representation of mappings between information structures appearing within risk analysis in risk management. Our mathematical algebraic framework also enables a more formal treatment of the duality between threat and opportunity. Full article

Cruise ships, characterized by high density, enclosed environments and shared facilities, have become amplifiers for infectious disease outbreaks; however, predicting transmission in such settings remains challenging due to small initial case numbers. In this study, a mathematical modeling framework—including compartmental, small-world (SW), and scale-free (SF) network models—was developed to analyze the dynamics of hantavirus and norovirus transmission on recent cruise ship outbreaks. The results show that when the initial number of exposed or infected individuals is extremely small (e.g., 1–2 persons), disease spread is dominated by stochasticity, causing substantial variation between individual simulations and mean model predictions. Notably, SF networks exhibited lower transmission risk under these conditions, contradicting the theoretical expectation that when the population is large, SF networks facilitate the spread of disease. The compartmental model, while consistent with average simulation outcomes, failed to reliably predict any single outbreak event. In conclusion, for closed environments such as cruise ships with very few initial cases, stochastic variability is essential, and single-outbreak outcomes should be understood as highly contingent rather than deterministically predictable. Full article

This paper presents a procedure for constructing a matrix $A\otimes A^{*}$ of a size $3\times 3$ in max-plus algebra. The max-algebraic product of a matrix A and its conjugate (negative transpose) matrix $A^{*}$ always yields a pseudo skew-symmetric matrix. We address the inverse problem: given a matrix $\tilde{A}$ of this type, can one determine a matrix A such that $A\otimes A^{*}=\tilde{A}$? A particular case involving $3\times 3$ matrices is examined, and one procedure is explained in detail. The method illustrates how such a matrix A can be constructed column by column while also indicating directions for further generalization and computational exploration. Full article

The operationalization of machine learning (ML) introduces distinct engineering and lifecycle management challenges—such as extreme data dependence, silent model degradation (concept drift), and inherent non-determinism—which traditional software engineering workflows fail to adequately address. This systematic literature review provides a rigorous, comprehensive mapping of the ML lifecycle domain between 2015 and 2025 using the PRISMA protocol. Out of an initial pool of 12,450 articles, a highly specialized cohort of 22 primary studies was extracted, classified, and synthesized to map out contemporary Machine Learning Operations (MLOps) patterns, technical debt structures, governance models, and security vulnerabilities. To address the documented “production gap,” this paper formalizes the findings into a synthesized operational mapping and introduces a preliminary conceptual layout for an Adaptive Lifecycle Framework (ALF), juxtaposing it with legacy paradigms like CRISP-DM. Furthermore, we expand the scope to investigate domain-specific lifecycle complexities in healthcare systems and Large Language Model (LLM) pipelines, providing an essential evolutionary baseline for sustainable MLOps. Full article

This study presents a comparative numerical investigation of second-order and fourth-order Runge–Kutta methods for solving chaotic dynamical systems. The Lorenz, Genesio–Tesi, and Rössler systems are considered because of their nonlinear behavior and high sensitivity to initial conditions. The numerical schemes investigated include the Midpoint, Improved Euler, Ralston, and fourth-order Runge–Kutta (RK4) methods. The performance of the methods is evaluated in terms of convergence behavior, numerical accuracy, stability characteristics, and computational cost. A stability analysis of each chaotic system is carried out through equilibrium point determination and Jacobian eigenvalue analysis. Numerical simulations are implemented in MATLAB 2023 version, and comparisons are performed using different step sizes. The results indicate that all numerical methods converge as the step size decreases; however, the RK4 method consistently provides significantly smaller errors and improved stability properties compared with the second-order schemes. The findings further demonstrate that higher-order numerical integration methods provide superior performance for highly sensitive chaotic systems where accuracy and reliability are essential. Full article

This paper solves the open problem mentioned by Lin that was published in 2026 of Algorithms to prove the uniqueness property for the solution procedure proposed by Lin. The developed mathematical verification fulfills the research gap left by Lin. Moreover, for the local maximum point near the starting point (denoted as $3\times {10}^{-7}$) of the amelioration inventory model, this study presents an alternative explanation to help researchers execute their numerical methods with confidence such that, without the analytical procedure developed by Lin, only a rough numerical method can help researchers locate two local maximum points. Full article

To address jagged boundaries in conventional fixed-mesh topology optimization and the discontinuous transfer of topology information after remeshing, this study proposes an improved Movable Morphable Smooth-Boundary (MMSB) method for displacement-constrained continuum topology optimization. A topology optimization model is established using the Independent Continuous Mapping (ICM) method, with structural weight minimization as the objective and displacement as the constraint. In the proposed framework, a triangular mesh is adopted as the current analysis mesh, threshold boundary points are identified using a holographic scanning strategy, and a fixed background mesh is introduced as an intermediate carrier for topology-variable transfer before and after remeshing. Three numerical examples are used to validate the proposed method and to compare it with the original MMSB method. The results show that the proposed method produces clearer boundary representations and more distinct load-transfer paths. In Examples 1–3, the number of result analyses is reduced from 60 to 24, from 144 to 36, and from 112 to 24, respectively. The number of boundary movements is also reduced from 5 to 4, from 8 to 6, and from 7 to 4, respectively. Meanwhile, the final structural weights are 22.0 kg, 101.4 kg, and 1.64 kg, which are close to or slightly lower than those obtained by the original MMSB method. These results indicate that the proposed method improves topology-information continuity and boundary representation while maintaining structural performance. Full article

Complex-valued knowledge graph embedding (KGE) models like ComplEx effectively capture asymmetric relations but are fundamentally constrained by integer-order transformations. This restriction limits their ability to model multi-scale interactions, hierarchical correlations, and non-local semantic dependencies inherent in heterogeneous graphs. To address these limitations, this paper introduces FracComplEx, a novel fractional-order extension that embeds fractional calculus into the complex latent space. By leveraging fractional operators, the framework introduces non-local dynamics and memory-aware mechanisms to continuously generalize standard linear transformations. The core architecture employs a fractional-order parameter $\alpha$ as a controllable scaling mechanism that balances local relational details with global topology, optimizing representation smoothness and flexibility. We provide rigorous theoretical findings demonstrating that fractional transformations enhance the embedding’s expressive capacity, spectral characteristics, and perturbation robustness beyond conventional integer-order benchmarks. Extensive experiments on FB15k-237, WN18RR, and CoDEx-M establish the empirical superiority of FracComplEx, yielding significant improvements in Mean Reciprocal Rank (MRR) and Hits@K metrics over classical baselines, particularly under severe structural data sparsity. Full article

This study constructs a novel fractional-order eco-epidemiological predator–prey model, in which disease spreads among predators through environmental transmission, and both cooperative hunting behavior and disease latency delay are incorporated simultaneously. Different from classical integer-order predator–prey models, fractional derivative is adopted to describe the memory-dependent mechanism of ecological populations, and the infection can alter the hunting strategy of diseased predators. The existence, non-negativity, and boundedness of system solutions are proved theoretically. The local stability of all equilibrium points is analyzed, and the conditions for the occurrence of Hopf bifurcation induced by latency delay are derived. Numerical simulations further verify the theoretical results, and quantitatively reveal the separate and combined effects of the fractional order, cooperative hunting coefficient, and latency delay on the dynamical evolution of the population system. Full article

This study presents an ideal-cycle thermodynamic analysis of an advanced compressed air energy storage (A-CAES) system with single thermal energy storage (TES) and an external heat boost. The additional heat is represented by a solar heat source, although the analysis is equally applicable to other forms of externally supplied thermal energy. Following the classical thermodynamic approach used for ideal cycles such as the Brayton, Otto and Diesel cycles, the objective is to establish analytical relationships and performance bounds for the integrated system rather than to model a specific engineering configuration. Three principal performance measures are examined: the electrical round-trip coefficient of performance (CoP), the marginal thermal coefficient of performance associated with external heat addition, and the overall second-law efficiency. Closed-form analytical expressions are derived for these quantities under idealised but still practically relevant assumptions. The analysis identifies distinct operating regimes governed by the level of external heat input and establishes analytical transition conditions between them. It is shown that external heat addition can substantially increase the round-trip coefficient of performance and lead to high marginal heat-utilisation effectiveness. A rigorous upper bound on the second-law efficiency is also obtained from a complete-cycle exergy analysis, demonstrating consistency with the laws of thermodynamics. The results provide analytical insight into the fundamental thermodynamic structure of solar-assisted A-CAES systems and establish performance bounds that are independent of any particular engineering implementation. Full article

The rapid growth of high-dimensional predictive models in science and industry has intensified the need for statistically rigorous interpretability tools. Although model-agnostic feature importance methods are widely used to explain black-box models, they lack formal uncertainty quantification, leading to unreliable conclusions in high-dimensional settings where spurious correlations are common. We propose a Bootstrap-of-Bootstrap (BoB) inference framework that enables valid uncertainty quantification and hypothesis testing for any model-agnostic feature importance measure. To overcome the high computational cost of nested resampling, we develop an efficient analytical approximation based on influence function theory. The proposed approach provides calibrated confidence intervals and a stability score for each feature, strengthening the statistical foundations of explainable AI. Simulation studies and real-world applications in cancer genomics and credit risk modeling demonstrate its effectiveness, providing reliable, auditable explanations for high-stakes decision-making. Full article

Viscous effects play an important role in the mechanical characterization of soft biological tissues under high-strain-rate loading. Accurate modeling of these behaviors is important for impact biomechanics, injury prediction, and crash safety analysis, in which biological tissues may experience high-strain-rate deformation. To describe the dynamic mechanical responses of soft tissues, a reliable constitutive framework is therefore needed to represent the dynamic response of soft tissues under high-strain-rate loading. The objective of this study is to develop and evaluate a viscohyperelastic constitutive framework for describing the dynamic compressive responses of multiple soft tissues. The proposed formulation is constructed within a continuum mechanics framework, in which the viscous contribution is expressed using objective invariant functions, namely $J_2$, $J_6$, and $J_7$. The developed analytical formulations are calibrated against high-strain-rate experimental data from different soft biological tissues, namely porcine meniscus, bovine liver, and ovine brain tissues. To find the material model parameters, genetic algorithm optimization is used to identify the material parameters and assess the robustness of the fitting procedure. In order to assess the robustness of the proposed constitutive framework across different loading rates, a multi-objective optimization strategy is used to calibrate the model parameters by fitting multiple strain-rate-dependent responses at the same time. This approach enables the model predictive capability to be evaluated over a range of high-strain-rate conditions. These results show that the proposed framework can reasonably describe the nonlinear and rate-dependent mechanical responses of different soft tissues under dynamic compression. Full article

This study proposes a transformed two-part bootstrap confidence interval (TTB-CI) for zero-inflated count data. The method combines a standard zero-inflated mixture formulation, parametric bootstrap, and monotone transformations to improve inference for practically meaningful estimands, including the marginal mean, zero probability, and positive-part mean. Simulation studies under zero-inflated Poisson (ZIP) and zero-inflated negative binomial (ZINB) data-generating processes show that the proposed method maintains nominal or near-nominal coverage while reducing interval width, particularly for the positive-part mean. Compared with conventional Poisson- and negative binomial-based confidence intervals, the proposed TTB-CI provides a more favorable coverage and width tradeoff and yields more informative intervals for positive count inference. These results indicate that the proposed method offers a practical and efficient confidence interval framework for zero-inflated count data. Full article

Bipolar fuzzy sets provide an effective framework for representing both positive and negative aspects of information. The necessity of a mathematically rigorous and valid distance measure in bipolar fuzzy environments motivates us to introduce a new real-valued function on the set of bipolar fuzzy sets defined over both discrete and continuous universes of discourse. The proposed function is shown to define a valid metric on the set of bipolar fuzzy sets, as it satisfies all the metric axioms. The metric induced by the real-valued function is inspired by the Canberra distance, and it can effectively quantify the dissimilarity between bipolar fuzzy sets in a normalized and interpretable manner. The practical utility of the proposed metric is demonstrated in a pattern recognition problem, where it successfully recognizes an unknown pattern using known bipolar fuzzy patterns. Using the proposed metric, a bipolar fuzzy C-means clustering algorithm is developed for sentiment analysis. The time complexity of the aforementioned algorithm is also analysed. Experiments conducted on the IMDb Movie Review Dataset demonstrate that the proposed algorithm outperforms k-means, fuzzy C-means, and intuitionistic fuzzy C-means algorithms. The proposed bipolar fuzzy C-means algorithm achieves an accuracy of 90.04%, a precision of 90.51%, a recall of 89.01%, an F1-score of 89.75%, a Root mean square error of 0.1191, and a Silhouette score of 0.75. The findings establish that the proposed metric and the associated bipolar fuzzy clustering approach provide a robust and effective framework of handling sentiment data associated with simultaneous positive and negative opinions. Full article

This technical note aims to present a theoretical analysis for cybertronics engineering (CE) for a class of dynamic supply chains for artificial intelligence (AI)-based products, services or hybrid solutions. The cybertronics-based analysis encompasses three classes of supply chains: (1) energy-based dynamic supply chains (DSC); (2) semiconductor manufacturing and quantum supply chains; and (3) retailing of the AI-based DSC solutions generated. Considering the nonlinear nature of DSC, to provide solutions for products and services based on AI-chained supply chains, novel robust control is addressed via sliding mode control (SMC) with proper stability analysis for the DSC. Full article