AppliedMath, Volume 6, Issue 8 (August 2026) – 22 articles

A Lyapunov-based framework for stability analysis and synthesis of adaptive neural-network (NN) controllers for a class of uncertain second-order nonlinear systems (SNS) with bounded external perturbations and unmodelled dynamics is presented. Online learning is employed for the reconstruction of the plant nonlinearity with the use of a radial-basis-function (RBF) network whose weights are adapted using a direct adaptation law deduced from a single composite Lyapunov function. The proposed controller couples the weight update to a persistent robustifying action, while the closed-loop stability is guaranteed throughout the learning transient, in contrast to schemes that guarantee stability after learning has converged. Using a composite Lyapunov function in the filtered tracking error and the weight-estimation error, we prove that all closed-loop signals are uniformly ultimately bounded (UUB) and that the tracking error converges to an explicitly characterized residual set whose radius is governed by the network reconstruction accuracy, the disturbance bound and the design gains. A $\sigma$-modification ensures parameter boundedness without persistency of excitation, and a robustness theorem shows that bounded parametric perturbations of the plant preserve stability and enlarge the ultimate bound only gradually (a graceful degradation, rather than a loss of the guarantee). The open-loop plant (a forced double-well Duffing oscillator) is characterized by means of equilibrium and Jacobian analyses. A bifurcation diagram and the largest Lyapunov exponent are presented, which show a chaotic regime (with ${\lambda }_1\approx 0.17$). Numerical experiments indicate that the proposed controller is able to suppress the chaotic motion with a small value of the ultimate bound, and maintain a smooth reference motion with a small and constant RMS error of order ${10}^{-3}$, which is approximately 26 times less than the RMS error obtained with a tuned fixed-gain baseline, and the theoretical dependence of the ultimate bound on the disturbance and the design gains is confirmed by sensitivity sweeps. Full article

Meningitis is a significant public health problem despite the availability of effective vaccination programs, especially in children and young people. The memory-dependent features of disease transmission, immunity and vaccination dynamics are not often represented in classical integer-order epidemic models. This study proposes a fractional-order model of meningitis transmission with memory using the Atangana–Baleanu–Caputo fractional derivative. The model features susceptible, vaccinated, exposed, infectious, treated, and recovered populations to assess the impact of vaccination coverage, vaccine effectiveness, loss of vaccine immunity, and treatment on the spread of meningitis. The basic mathematical characteristics of the model, such as positivity, existence, uniqueness, and stability of solution are proven. The Laplace–Adomian Decomposition Method (LADM) is used to obtain the approximate analytical solutions, and a numerical simulation is used to analyze the influence of the fractional-order memory and epidemiological parameters on the epidemic process. The most important parameters that influence the basic reproduction number are found in sensitivity analysis to be the transmission rate and the vaccination-related parameters. The results show that simply increasing the vaccination coverage and vaccine effectiveness can substantially decrease the number of disease transmissions, and vaccine coverage can produce memory effects to change the timing and duration of outbreaks. The suggested fractional-order computational framework is a framework that is vital for studying the dynamics of meningitis and can be used for the design of long-term vaccination and disease-control strategies. Full article

SPARQ is an advanced global optimization algorithm, the direct evolution of an earlier method called ARQ2. It searches efficiently for the best solution to problems where the space of options is too vast to check exhaustively, such as tuning antenna arrays or planning fuel-efficient spacecraft trajectories. Like its predecessor, it works with a population of candidate solutions that improve generation after generation, alternating between two complementary search strategies. What sets SPARQ apart is that it makes nearly every part of this process adaptive. Its population shrinks intelligently as the search matures. Its internal settings draw from a memory of many past successful configurations, not a single average. Its escape-from-stagnation mechanisms come in graduated strength, from a gentle nudge to a deeper partial restart. It also adds capabilities its predecessor never had, including a dedicated phase that locally polishes the current best solution using its own memory of productive directions. Every addition is kept only where it showed an overall benefit during development, though a subsequent component-wise analysis shows this benefit varies markedly in size and statistical significance across mechanisms. The result is more reliable than its predecessor on the large majority of tested problems, while staying grounded in the same battle-tested core. Full article

This paper presents a new algorithm based on splitting methods for the numerical solution of the Navier–Stokes equations coupled with the heat equation. This work is set in a context where viscosity and thermal conductivity depend explicitly on temperature We propose an extension of the Coupled Prediction Scheme based on a rotational projection formulation, noted as the Rotational Coupled Prediction Scheme (RCPS). The accuracy and effectiveness of the proposed numerical scheme are demonstrated through a series of numerical tests, and the results obtained are compared with those already existing in the literature. In particular, the Rayleigh–Bénard convection (RBC) problem was used as a reference benchmark. Full article

Electron–positron pair production is one of the fundamental quantum electrodynamical (QED) processes and provides an important framework for investigating photon–matter interactions. In this work, the influence of linear photon polarization on electron–positron pair production in the electromagnetic field of the ${Be}_4^9$ nucleus is investigated using both the conventional Bethe–Heitler formalism and its polarization-dependent extension. The differential cross sections are evaluated deterministically in Wolfram Mathematica 13.3 over the intermediate photon-energy range of 400–600 MeV and emission angles between 30° and 90°. The numerical results show that the largest differential cross sections are obtained at an incident photon energy of 400 MeV and an emission angle of 30°. A detailed examination of the angular distributions reveals a local change in the vicinity of 60°, which is interpreted because of the kinematic structure of the Bethe–Heitler formalism rather than a distinct production mechanism. A direct comparison with the conventional Bethe–Heitler model demonstrates that linear photon polarization consistently enhances the differential cross section throughout the investigated kinematic range. These leading-order theoretical results provide a systematic analysis of the combined energy and angular dependence of polarized pair production in the ${Be}_4^9$ nucleus and contribute to a deeper understanding of polarization effects in intermediate-energy QED processes. Full article

We study a delayed semilinear evolution equation with a Caputo time derivative and the spectral fractional Dirichlet Laplacian on a bounded connected domain. The model separates two forms of memory: the Caputo operator retains the distributed Volterra history, whereas the nonlinear production samples the single past state $u(t-\tau )$. Working in the strongly continuous phase space $C_0(\Omega )$, we prove local well-posedness, positivity, a sup-norm continuation criterion, and a compatible weak formulation. In the delayed-source case with $\mu =0$, the solution exists globally and remains bounded on every finite time interval, while the first Dirichlet mode admits an explicit recursive sequence of positive lower bounds across successive delay windows. In the dissipative case $\mu >0$, $p>q>1$, histories satisfying the explicit smallness conditions remain in an invariant order interval and the $L^2$-energy decays at a Mittag–Leffler rate. The scalar computations are presented only as heuristic first-mode surrogate experiments. In addition, an independent spatially resolved sine spectral-Galerkin/L1 computation of the PDE, with temporal and spectral refinement studies, is included as a numerical illustration. Full article

Missing data remain a major challenge in breast cancer research because they can introduce bias, reduce statistical efficiency, and compromise the performance of predictive models. Although numerous imputation techniques have been proposed, their comparative performance under different missing-data mechanisms and their impact on downstream classification remain inadequately understood. This study systematically compared statistical and machine learning-based imputation methods using two publicly available breast cancer datasets representing complementary clinical settings. The methods were evaluated under simulated Missing Completely at Random (MCAR), Missing at Random (MAR), and Missing Not at Random (MNAR) mechanisms using both reconstruction accuracy and downstream classification performance. The results showed that no single imputation method consistently achieved the best performance across both datasets. Regularized regression and machine learning-based methods generally outperformed conventional statistical approaches, although the optimal method depended on the characteristics of the dataset. Furthermore, the best-performing imputation methods preserved downstream classification performance despite the introduction of missing data, demonstrating that reconstruction accuracy alone is insufficient for selecting imputation strategies intended for predictive modelling. Overall, the findings highlight the importance of considering dataset characteristics, missing-data mechanisms, and the intended analytical objective when selecting imputation methods. The proposed evaluation framework provides a robust approach for assessing missing-data handling strategies in breast cancer prediction studies and other biomedical machine learning applications. Full article

Suppose that T is a normal spanning tree (depth-first search tree) of a graph G. If $e=xy$ and $e^{\prime }=uv$ are edges of G, satisfying $x{\prec }_{T}u{\prec }_{T}y{\prec }_{T}v$, then they are called secant edges of G with respect to T. Suppose that G has no secant edges with respect to T. If T is a path, Ghazal and Al-Mniny proved that the chromatic number is at most 3. We conjecture that there is a positive constant $\gamma$ such that, for any graph G that has no secant edges with respect to a normal spanning tree T, then $\chi \left(G\right)\le \gamma$. We pose the problem of whether $\gamma =4$ suffices. We establish a positive answer in the case where T has at most one node. Full article

Financial contagion—the propagation of shocks across markets, sectors, and time—remains one of the central questions in empirical finance, and yet most computational approaches model it at the wrong granularity. Existing work captures contagion at the asset level by measuring realised return or volatility comovements between specific securities, which conflates a structural pattern with its surface manifestation. We argue that the genuine causal regularity in contagion is between event types (rate decisions, default announcements, regulatory actions) rather than between individual assets. This paper introduces TECG, a framework that learns time-stamped causal-temporal association edges—predictive dependencies rather than identified causal effects—between abstract financial event types from news text and uses them to forecast multi-step contagion cascades. Methodologically, TECG can be read as a neural impulse response function (IRF) for event sequences: where the canonical structural-VAR IRF traces the dynamic response of one continuous variable to a shock in another, TECG traces the conditional intensity with which one event type fires after a shock to another event type, at a learned and possibly multi-modal lag. The framework couples LLM-based event extraction with a multivariate neural Hawkes process whose intensity functions are parameterised by a temporal graph neural network. The resulting graph has interpretable, time-stamped edges of the form “event-type A triggers event-type B with lag distribution L and conditional intensity κ.” We evaluated TECG on an aggregated corpus of approximately 482,000 financial news items spanning 2007–2023 and report three findings. First, the learned edges recover associations consistent with known causal relationships in finance—central-bank announcements preceding sector-level earnings revisions, default events at one institution preceding due-diligence events at competitors—without supervision on these relationships. Second, edges learned from data up to 2018 transfer to held-out cascades from 2020 (COVID equity crash) and 2023 (US regional banking stress) with substantially better fidelity than baselines that ignore temporal structure or operate at the asset level. Third, the framework supports generalised impulse response analysis: given a hypothetical seed event, it produces a distribution over downstream event chains that an analyst can interrogate, time-stamp, and entity-resolve. We are explicit that the empirical validation is observational and that the causal interpretation rests on assumptions we discuss at length. All headline comparisons are supported by paired-bootstrap significance tests; the extraction front end is independently evaluated on a manually annotated benchmark, with an explicit error-propagation analysis; and additional comparisons against recent temporal point-process and temporal-graph baselines are reported. The framework’s principal limitation is that it does not separately identify endogenous fire-sale or leverage-spiral dynamics; we point to where these gaps could be closed. Full article

Task-evoked EEG can reveal individual differences in cognitive processing, but it also creates a machine-learning challenge: many epochs are recorded from relatively few participants, and evaluation can be misleading if within-subject dependence is ignored. This study compared handcrafted, ROCKET, and EEGNet representations under subject-wise validation for classifying questionnaire-derived Dark Triad profiles from task-onset EEG. Dark Triad traits were assessed with the Dirty Dozen and clustered into four exploratory multivariate profiles using k-means. EEG was recorded during a visual speeded decision task, and epochs were extracted from −200 to 1000 ms around task onset. The final dataset included 1780 retained task-onset epochs from 30 participants. Three representations were evaluated under identical five-fold subject-wise cross-validation: XGBoost with handcrafted EEG features, XGBoost with ROCKET-derived time-series features, and compact EEGNet. All performance estimates were based only on predictions from held-out participants. The handcrafted model achieved balanced accuracy of 60.9%, whereas ROCKET and EEGNet improved performance to 74.3% and 76.2%, respectively, with only a modest difference between the waveform-based representations. A participant-level label-shuffling analysis of the out-of-fold predictions indicated that prediction–label alignment exceeded chance for all models. Across models, the mean probability assigned to the true cluster increased with psychometric cluster centrality, suggesting that borderline profiles were harder to classify. Signed channel-wise ablation of EEGNet suggested distributed model sensitivity, with the largest positive effects over posterior/parietal electrodes. The findings highlight the importance of participant-level validation, EEG signal representation, and psychometric label structure, while emphasizing the need for external validation in larger independent cohorts. Full article

This study obtained the energy levels and examined the partition function (Z) of a quantum system described by an Eckart-like potential model. By adopting the Greene–Aldrich approximation scheme for the centrifugal term, the radial Schrödinger equation (SE) is solved and the analytic expression of the energy eigenvalues is obtained. The ro-vibrational Z is computed by explicitly incorporating the rotational quantum number, a feature often neglected or misapplied in many studies. This result is used to evaluate the key thermodynamic properties (TP), including the Gibbs free energy (G), entropy (S), and enthalpy (H). Numerical analysis reveals that the Z increases monotonically with temperature, while the G decreases in accordance with statistical thermodynamics. The S exhibits saturation-like behaviour at higher temperatures, while the H displays convex growth with increasing thermal energy. Parametric studies demonstrate that the Eckart-like potential allows for the controlled tuning of TP, with variations in the potential parameters, including the screening parameter, having distinct effects. The results generalise existing models, reproduce the Hulthén potential under specific conditions, show the effect of the rotational quantum number of TP, and provide new insights into the ro-vibrational statistical mechanics of exponential-type potentials. Full article

We introduce finite-observation conditional guarded language operators on the complete length-based ultrametric space of formal languages over a finite alphabet. A finite observation map selects a guarded branch according to the membership of prescribed test words. Although each branch is Banach-contractive, branch switching may destroy global Banach contractivity. We derive depth–residual conditions yielding a max-type Kannan inequality with an explicit constant $\alpha <1/2$. Consequently, the operator has a unique fixed language, and the Picard iteration converges from every initial language. We also construct multi-branch operators that are Kannan-contractive but not Banach contractive and obtain residual error estimates, explicit convergence bounds, finite-depth certification, and eventual stabilization of the selected branch. Full article

The Rayleigh–Logarithmic distribution provides a useful framework for modelling lifetime data by combining continuous lifetime variability with a logarithmic compounding mechanism. This study introduces a three-parameter Alpha Power Rayleigh–Logarithmic (APRL) distribution by applying the Alpha Power transformation to the classical Rayleigh–Logarithmic model. The additional transformation parameter allows the distributional shape, skewness, tail behaviour, and rate of increase in the hazard function to be adjusted while retaining the underlying structure of the baseline model. Several mathematical and reliability properties of the APRL distribution are derived, including the probability density and cumulative distribution functions, survival and hazard rate functions, quantile function, moments, order statistics, and mean residual life function. Model parameters are estimated by maximum likelihood using a multiple-start numerical optimization procedure, and the finite-sample performance of the estimators is investigated through Monte Carlo simulations under different parameter configurations and sample sizes. The simulation results show that estimation accuracy generally improves with increasing sample size, as reflected by decreasing bias, MSE, and RMSE, although estimation of the transformation parameter may exhibit greater variability for more extreme parameter settings. The practical performance of the APRL distribution is examined using the Aircraft Windshield Failure Times and Breaking Stress of Carbon Fibres datasets. Model comparisons based on information criteria, bootstrap-based goodness-of-fit assessment, and graphical diagnostics show that the APRL distribution provides competitive fits relative to several established lifetime distributions. In addition, mean time to failure and mean residual life analyses illustrate the practical interpretation of the reliability measures derived for the proposed model. Overall, the results support the APRL distribution as a useful alternative for the statistical analysis of lifetime and reliability data. Full article

Fractional Brownian motion (fBm) provides a powerful stochastic framework for modeling long-range temporal dependence that cannot be represented by classical Brownian motion. This study presents a numerical and theoretical investigation of constrained fractional Brownian motion with applications to stochastic financial systems. An efficient simulation framework combining Fast Fourier Transform (FFT)-based circulant embedding and Markov Chain Monte Carlo (MCMC) sampling is developed to generate long correlated trajectories under absorbing boundary conditions. The proposed algorithm enables simulations with trajectory lengths up to L = 10⁷ while reducing the computational complexity from O (L³) for direct covariance decomposition to approximately O(L log L). Numerical results accurately reproduce the theoretical autocorrelation function of fBm and confirm the expected persistence behavior governed by the Hurst exponent. Super-diffusive regimes (H > 0.5) exhibit persistent long-range correlations and enhanced survival probabilities, whereas sub-diffusive regimes (H < 0.5) display anti-persistent dynamics and increased boundary absorption. The fractional stochastic volatility formulation captures important characteristics associated with long-memory financial systems, including persistent volatility dynamics and implied-volatility structures. The proposed biophysical-based FFT–MCMC methodology provides an accurate, scalable, and computationally efficient framework for studying constrained fractional stochastic processes and offers a foundation for future investigations of fractional volatility models and related financial applications. A conceptual Adaptive Hurst Momentum framework is briefly discussed as a possible direction for future research. Full article

We study a Camassa–Holm-type equation with a prescribed, time-independent bottom profile, $m_t+(u+h\left(x\right))m_x+2u_{x}m+{\displaystyle \frac{1}{2}}{h}_x\left(x\right)m=0,m=(1-{\partial }_x^2)u.$ The model is considered here as a mathematically motivated bottom-modified Camassa–Holm equation. The bottom modifies the transport velocity and the lower-order term is chosen so that the basic momentum balance keeps the same cancellation structure as in the flat-bottom case. We clarify the meaning of a gentle bottom in terms of bounded multiplier norms of the prescribed profile and do not claim a complete asymptotic derivation from the Euler equations. Under suitable regularity assumptions on h, we prove local well-posedness in Sobolev spaces by verifying the hypotheses of Kato’s quasilinear semigroup theorem. We also derive an $L^2$ momentum identity and a continuation criterion based on the integrability of $\parallel u_x{\parallel }_{L^{\infty }}$. The proof of the continuation criterion is strengthened by combining the momentum bound with a high-order $H^s$ energy estimate. Full article

Complex interval-valued fuzzy sets are powerful tools for representing uncertainty and periodicity that occur in many real-life problems. They not only take both periodicity semantics and uncertainty into account but also describe the information using interval-valued membership grades, which gives experts more freedom to solve complex real-life problems effectively. Complex interval-valued fuzzy sets have been successfully employed many times in medical diagnosis and pattern recognition problems. In this article, two novel methods for computing the correlation coefficient and weighted correlation coefficient of complex interval-valued fuzzy sets are proposed. The methods employed fuzzy statistical parameters such as mean, variance, and covariance of complex interval-valued fuzzy sets. Several important mathematical properties are rigorously established. In order to establish the efficacy and implementation of the methods, a real-life application related to pattern recognition for mineral identification is discussed in detail. Furthermore, based on the proposed correlation and weighted correlation, a classification algorithm is developed. The time and space complexities of the proposed algorithm are analyzed. The proposed algorithm is validated through experiments conducted on two real benchmark datasets. The results demonstrate that the proposed approaches are more reliable and accurate than several existing approaches by achieving classification accuracies exceeding 98%. Full article

Acetaldehyde is routinely monitored in neutral alcohol, but pooled precision summaries can obscure changes in performance across the working range. This study evaluated 446 valid GC–FID verification results obtained by three anonymized analysts on two gas chromatographic systems (GC 8890 and GC 7890B) at 0.6, 2.5, and 9.0 mg L⁻¹. The analysis included measured concentration, recovery, relative bias, relative standard deviation (RSD), confidence intervals, and fixed-effect factorial analysis of variance (ANOVA). Mean recoveries for the GC 8890 and GC 7890B systems were 48.7% and 157.2% at 0.6 mg L⁻¹, 92.1% and 124.8% at 2.5 mg L⁻¹, and 92.0% and 103.1% at 9.0 mg L⁻¹, respectively. Recovery RSD decreased from 47.8% and 18.4% at the lowest level to 4.2% and 3.0% at the highest level. Type III ANOVA identified significant verification-level, analyst, GC-system, and interaction effects. A contextual comparison with concentration-matched AOAC Appendix F targets classified both systems as outside the recovery and RSD benchmarks at the lowest level, showed system-specific partial conformity at the middle level, and placed both systems within the screening benchmarks at the highest level. The contribution of the study is an integrated decision framework in which predefined recovery and RSD targets are interpreted together with factorial interactions to determine when results may be pooled and when level- or system-specific control is required. Full article

This paper establishes new Fejér and Hermite–Hadamard-type inequalities for functions of two variables whose mixed second-order partial derivatives satisfy coordinated GA-convexity or coordinated GA-quasi-convexity on a rectangle in the positive quadrant. Our main results are formulated for non-negative continuous weight functions that are not necessarily symmetric with respect to the geometric means of the interval endpoints, thereby extending the classical framework to genuinely asymmetric weights. However, to obtain explicit and sharp integral bounds in certain cases, we also employ a technical lemma that assumes a special symmetric setting where the weight function is symmetric on each coordinate with respect to $\sqrt{h_1{h}_2}$ and $\sqrt{k_1{k}_2}$. We clearly distinguish which theorems hold for general asymmetric weights and which depend on this symmetry condition. Our findings unify and extend numerous previously known results for both symmetric and non-symmetric weight functions. Full article

In this study, we propose a mathematical model for the dynamics of atmospheric carbon dioxide (CO₂) with human population growth and an age-structured vehicle population. The positivity and boundedness of the model’s solutions are verified, and the global stability of an interior equilibrium point is proved. We calibrate the model parameters using the least-squares method. Simulations of the model show good alignment with the reported atmospheric CO₂ data. Moreover, the sensitivity analysis shows that human-related emission factors and emissions from old vehicles are the main contributors to CO₂ accumulation. The results of this study recommend that reducing anthropogenic emissions, accelerating the retirement of old vehicles, and strengthening carbon sequestration initiatives be considered to mitigate future atmospheric CO₂ accumulation. Full article

This study investigates the efficacy of early multimodal data fusion for identifying academic performance profiles and predicting academic risk in specialized postgraduate education under small-sample constraints ($n=51$ graduate students distributed across three independent cohorts). Student numerical grades and multi-source qualitative faculty evaluations (self-, peer-, and hetero-evaluation) were integrated at the feature level prior to modeling. To overcome the limitations of traditional small-sample validation, a multi-layered framework was deployed, combining non-linear Kernel Principal Component Analysis (KPCA), multi-method clustering consensus, bootstrap resampling (1000 iterations), stratified nested cross-validation, and an external cohort validation strategy. The analysis revealed a bimodal distribution of academic risk (low-risk: 71.4%, high-risk: 28.6%) with high consensus across unsupervised algorithms (Adjusted Rand Index values spanning 0.922–1.000). Supervised predictive models yielded an unbiased nested cross-validation accuracy of $0.980\pm 0.040$, while external cohort validation demonstrated generalizability with a mean cross-cohort accuracy of $0.970\pm 0.052$ and a mean AUC of $0.997\pm 0.001$ (95% CI: $0.996–0.998$). SHAP feature importance analysis identified Machine Learning (28.68%), Artificial Intelligence (17.93%), and Big Data (16.66%) as the primary drivers of risk differentiation. This study contributes a methodologically rigorous, moderately stable, and reproducible workflow for educational data mining in sample-constrained environments, providing actionable insights for evidence-based curriculum design and early pedagogical interventions. However, given the limited sample size, results should be interpreted as exploratory evidence rather than definitive population-level conclusions. Full article

External sulfate attack constitutes a major durability challenge for structural concrete, yet the influence of mechanical loads on sulfate transport remains incompletely understood. This study develops a coupled sulfate–water transport model that embeds the effects of axial static loads and fatigue loads into the effective sulfate diffusion coefficient through physically motivated correction factors, without explicitly resolving chemical reactions or physical crystallization. Numerical simulations were performed for coupled scenarios of static loading, fatigue loading, and drying–wetting cycles. Three principal findings emerge. First, drying–wetting cycles produce a stable convection enrichment peak at 1 to 3 mm from the exposed surface, rather than a monotonic decay profile. Second, axial static loads exhibit pronounced tension–compression asymmetry: tensile stress enhances sulfate transport, while compressive stress exerts a comparatively weak inhibition effect. Third, fatigue loading introduces time-dependent accumulation and nonlinear acceleration driven by the dynamic evolution of microcrack connectivity. The tensile zone consistently exhibits markedly higher sulfate enrichment and deeper penetration than the compressive zone, identifying it as the dominant region for coupled deterioration. These findings provide a quantitative foundation for durability assessment of concrete structures under complex loading environments. Full article

This paper studies symmetrized neural network (SNN) operators generated by an adjustable half-hyperbolic tangent activation function. The construction is based on the paired density kernels ${\aleph }_t$ and ${\aleph }_{1/t}$, whose average defines the symmetric kernel $\mathcal{F}.$ This kernel is positive, even, normalized, and preserves the partition of unity. Using $\mathcal{F}$, we define finite-interval and whole-line SNN operators in the Banach space-valued setting. Pointwise and uniform convergence estimates are obtained through the first modulus of continuity. Higher-order and fractional approximation estimates are also derived, the latter using Caputo–Bochner fractional derivatives. The numerical part compares the nonsymmetrized operator ${\mathcal{L}}_n$ and the symmetrized operator ${\mathcal{L}}_n^s$. For $n=80$, the uniform error decreases from $0.010463$ to $0.003479$, the root mean square error (RMSE) decreases from $0.006530$ to $0.000826$, and the coefficient of determination ($R^2$) improves from $0.999675$ to $0.999995$. This improvement is accompanied by an increase in central processing unit (CPU) time from $0.014161$ s to $0.027088$ s. The parameter tests further show that the performance depends on the joint choice of n, t, and $\xi$. Overall, the results indicate that symmetrization improves approximation accuracy, while parameter tuning remains necessary. Full article