https://ansne.du.ac.ir/ Analytical and Numerical Solutions for Nonlinear Equations en daily 1 Fri, 04 Sep 2026 00:00:00 +0330 Fri, 04 Sep 2026 00:00:00 +0330 https://ansne.du.ac.ir/article_2119.html In this research we obtain new results  of approximate solution for nonlinear singular $p$-Laplacian Caputo-Hadamard fractional $\mathtt{q}$-differential equation under infinite-point boundary conditions. The existence of unique iterative positive, error estimation, and convergence rate of approximate solution are obtained based on  the description of Green function with special properties and employing  appropriate substitution and appropriate cone. A few applications are demonstrated the validity of our achievements. https://ansne.du.ac.ir/article_2120.html This study presents a hybrid computational framework integrating analytical methods, artificial neural networks (ANN), and physics-informed neural networks (PINNs) for solving nonlinear fractional biological models governed by local fractional operators. In contrast to conventional approaches, the proposed framework incorporates local fractional calculus to effectively represent fractal and heterogeneous biological structures. An analytical solution is first derived using the Natural transform in conjunction with the Adomian decomposition method, yielding closed-form expressions in terms of Mittag--Leffler functions. This analytical formulation is subsequently utilized as a knowledge-driven prior to train an ANN-based surrogate model, thereby improving convergence efficiency and reducing computational complexity. Furthermore, a physics-informed neural network is constructed by embedding the governing fractional differential equation into the loss function via a Gr"unwald--Letnikov approximation, enabling accurate learning of long-memory effects without requiring explicit analytical solutions. Comparative analysis with classical numerical solutions obtained using the \texttt{bvp4c} solver demonstrates that the ANN surrogate achieves high-accuracy predictions with error magnitudes below $\mathcal{O}(10^{-4})$ while reducing computational cost by approximately 45\%. The PINN framework further exhibits strong capability in capturing intrinsic fractional dynamics under limited data conditions. Overall, the proposed ADM--ANN--PINN hybrid architecture provides a robust, efficient, and physically consistent framework for modeling complex nonlinear fractional biological systems, thereby advancing the state-of-the-art in fractional computational intelligence. https://ansne.du.ac.ir/article_2121.html In this paper, an optimized numerical method for solving time-fractional diffusion equations in three-dimensional space is proposed. Fractional-order differential equations can model complex physical processes with memory effects more accurately than differential equations with integer order derivatives. On the other hand, numerical solutions of this type of problem are usually challenging due to the existence of a singularity near the initial time. In the proposed method, in order to overcome this issue, the $L1$ formula has been used on a graded mesh to maintain the accuracy of calculations near the initial time. Also, a fourth-order compact operator has been used to discretize the Laplace operator in three-dimensional space. Stability and convergence analyses show that this method has a convergence order proportional to the grading parameter in time and a fourth-order convergence order in space. The obtained results from numerical simulations confirm the high accuracy of the proposed method and the efficiency of the optimal selection of the scaling parameter in achieving the suitable convergence order, even in the presence of initial singularities. https://ansne.du.ac.ir/article_2123.html The rapid proliferation of Internet of Things (IoT) devices has created complex network environments increasingly vulnerable to sophisticated cyber attacks. Detecting anomalous communication patterns in such heterogeneous networks requires mathematical models capable of capturing the multi-faceted nature of IoT traffic. This paper develops a multi-layer graph-theoretic framework for detecting anomalous communication patterns in IoT networks. The proposed model represents network traffic as a multi-layer graph where each layer corresponds to a different communication modality including TCP, UDP, ICMP, HTTP, and MQTT. Unlike prior works that assume stationary Poisson processes, we propose a dynamic negative binomial model with an overdispersion parameter to capture burstiness and a time-varying function to model diurnal patterns. The framework integrates three complementary mathematical approaches: spectral analysis using random matrix theory for global structural anomalies, local neighborhood analysis using graph signal processing for node-level behavioral deviations, and inter-layer correlation analysis using tensor decomposition for coordinated multi-vector attacks. To ensure practical robustness under non-stationary conditions, we introduce permutation-based threshold calibration that controls false positive rates even when theoretical assumptions are violated. Comprehensive sensitivity analysis is provided for all hyperparameters including integration weights, time window length, and tensor rank. Fair comparative evaluation is conducted against six state-of-the-art graph-based methods including Graph Convolutional Networks (GCN), Dynamic Graph Neural Networks (DyGNN), Multi-layer Graph Convolutional Networks (M-GCN), GraphSAGE, Graph Attention Networks (GAT), and Ensemble Graph Convolutional Networks (E-GCN). Numerical experiments on real IoT traffic datasets from CICIDS2017 and Bot-IoT demonstrate that the proposed framework achieves a detection rate of 89.2\% with a false positive rate of 3.8\%, outperforming the leading baseline M-GCN by 1.7\% in detection rate. The computational complexity scales linearly with network size, enabling near real-time deployment in large-scale IoT environments. https://ansne.du.ac.ir/article_2126.html The penalized Cox proportional hazard model is a popular analytical approach for survival data with a large set of covariates. Such problems are especially challenging when covariates vary over follow-up time (i.e., the covariates are time-dependent), leading to increased computational complexity and difficulties in efficient variable selection. In this paper, we propose an Adaptive Active-Set Coordinate Descent (AACD) algorithm for LASSO-penalized time-dependent Cox models. The proposed method combines coordinate descent with an adaptive active-set strategy and warm starts along the regularization path, allowing the algorithm to focus computation on relevant variables and better exploit sparsity. Simulation studies demonstrate that AACD consistently outperforms \texttt{glmnet}, achieving uniformly lower mean squared errors with relative efficiency greater than one across all settings, and reducing estimation error by up to 37\% in small samples. The method remains robust as dimensionality increases, improves variable selection by reducing false negatives, and produces more parsimonious models with comparable predictive performance while reducing runtime by approximately 36\% in real data analysis. https://ansne.du.ac.ir/article_2125.html Cardiac disease is a major public health concern in India, and accurate statistical techniques are required for early risk assessment. Traditional regression analysis generally depends on fixed clinical thresholds; however, cardiovascular risk factors such as blood pressure, cholesterol, diabetes, obesity, tobacco use, and physical activity involve uncertainty because they gradually move from the absence of risk to the presence of risk. This study applies fuzzy regression to develop a statistical modelling framework for analysing cardiovascular disease risk factors using the Heart Attack Risk and Prediction dataset from Kaggle. The proposed modelling process includes data preprocessing, descriptive analysis, fuzzy membership construction, fuzzy regression modelling, and comparison with conventional regression models. The selected clinical and lifestyle variables are transformed into linguistic risk levels such as low, moderate, and high risk, thereby allowing the uncertainty associated with individual risk profiles to be represented more meaningfully. The performance of the models is evaluated using statistical measures such as mean absolute error, root mean square error, accuracy, sensitivity, specificity, and receiver operating characteristic area under the curve. The fuzzy statistical modelling approach improves interpretability and provides a flexible risk assessment structure compared with conventional modelling methods. Overall, this study presents a fuzzy regression-based statistical framework to support data-driven cardiovascular risk assessment in the Indian healthcare context. https://ansne.du.ac.ir/article_2127.html Regular expressions (regex) play a fundamental role in modern software development, enabling efficient data validation, text processing, and pattern matching across a wide range of applications. Despite their versatility, the intricate syntax and abstract structure of regex patterns often create significant challenges for developers, particularly in terms of construction, debugging, comprehension, and maintenance. To address these issues, specialized regex development and debugging tools have been introduced to improve usability and reduce development effort. This study presents a qualitative comparative analysis of widely used regex tools, including Regex101, RegExr, Debuggex, and RegexBuddy, together with an assessment of regex support provided by modern integrated development environments (IDEs). The evaluation is conducted with respect to key usability criteria, including learnability, ease of use, debugging support, visualization capabilities, and workflow integration. In addition, a practical case study is presented in which regex-based techniques are employed to parse and extract structural components of nonlinear mathematical equations. The case study demonstrates the effectiveness of these tools in facilitating complex pattern recognition, improving program comprehension, and supporting software analysis tasks. The results indicate that the choice of an appropriate regex tool can significantly enhance developer productivity, reduce debugging effort, and improve the overall understanding of complex regular-expression patterns. These findings provide useful guidance for both practitioners and researchers seeking effective tool support for regex-intensive software development activities. https://ansne.du.ac.ir/article_2128.html This study introduces a hybrid approach based on a finite difference scheme and radial basis functions (RBF) for solving the Caputo time-fractional telegraph equation numerically. To handle the temporal fractional derivatives, a finite difference formulation of the L1/L2 type is adopted, while the spatial derivatives are approximated using an RBF collocation technique. This combination results in a  discretized system characterized by a sparse matrix structure. Through the application of energy methods, the stability and convergence assessment of the temporal discretization is conducted, yielding a theoretical error estimate of $\mathcal{O}(\delta_t^{3-\alpha})$ in the time direction. The reliability and effectiveness of the proposed numerical scheme are verified through a representative test problem, confirming the capability of the proposed method to achieve high accuracy. https://ansne.du.ac.ir/article_2132.html Bootstrap is a resampling method based on high calculations, which can help us a lot for statistical inference in cases where the amount of data that we have is limited. For example, in the design of hydraulic structures such as bridges or dams, etc. there is a need to estimate hydrological events such as, floods or precipitations by statistical inference of quantiles of a probability distribution. In this paper, we aim to estimate precipitation quantiles. For calculating this estimation, the confidence interval for quantiles has been introduced with percentile bootstrap, accelerated bias-corrected bootstrap, t-bootstrap methods; that in this article, we want to compare these methods with the confidence interval made by the highest density method based on bootstrap data and we obtain the average length of the confidence intervals as a criterion to evaluate the methods. To calculate the average length of confidence intervals using different methods, first, the best distribution among commonly used distributions is fitted to the original data, and its parameters are estimated using the maximum likelihood method, and quantiles are obtained from it. Then, we continue until the coverage probability of the real quantile reaches the nominal confidence level of 95$\%$ by repeating the simulated bootstrap samples. The results of the performed simulation show that the bootstrap highest density method has the smallest average length of confidence intervals among all methods. The data used in the article are 24-hour annual maximum precipitation records in five meteorological stations in Mexico, which are compared with the data of five stations in Gilan province of Iran. https://ansne.du.ac.ir/article_2139.html We develop an improved physics-informed neural network (PINN) framework for solving nonlinear inverse partial differential equations. Two separate network architectures are explored: one utilizing Legendre polynomials as activation functions and another employing rational Legendre polynomials. In the latter case, there is a crucial hyperparameter, $L$, that significantly impacts both performance and efficiency. To optimize this parameter, we employ the cuckoo optimization algorithm (COA), which fine-tunes $L$ based on the problem's characteristics. The proposed methods are tested on two benchmark problems, the Fisher equation and the Burgers--Fisher equation. The results highlight the effectiveness of both approaches, demonstrating superior accuracy in reconstructing unknown boundary conditions. All experiments were conducted in a Jupyter Notebook environment using PyTorch, confirming the potential of tailored activation functions and optimization techniques in physics-informed deep learning. https://ansne.du.ac.ir/article_2141.html This paper studies networks of dynamic duopoly systems with the objectives of maximization of firms' discounted profits and of convergence of market prices in all markets. We model each duopoly as a dynamical process of price adjustment, driven by production decisions of firms. The profit maximization problem is reformulated as a quadratic optimal control problem by representing the dynamics of the duopoly in a linear time-invariant (LTI) state-space form. In order to impose the price convergence between the markets, a consensus term is added to the cost function. We use the Hamilton--Jacobi--Bellman (HJB) approach to derive a control law that encourages profit maximization and market price convergence over a finite time horizon. The proposed framework reformulates the dynamic duopoly problem in networked markets within an optimal control framework, allowing simultaneous profit optimization and market price coordination. https://ansne.du.ac.ir/article_2154.html A rigorous fixed-point framework is developed for linear and nonlinear second-order mechanical vibration systems governed by ordinary differential equations with damping, stiffness, external forcing, and nonlinear restoring forces. The governing equation is reformulated as a Volterra operator equation in an appropriate Banach space, yielding a convenient integral formulation for constructing a Picard fixed-point iteration. Global existence, uniqueness, and convergence are established under suitable contraction conditions, with explicit bounds on the contraction constant and the corresponding geometric convergence rate of the iterative sequence. Energy-based stability estimates are derived for both forced and unforced systems, providing bounds on the mechanical energy and clarifying the dissipative behavior of the underlying dynamics. A discrete quadrature-based implementation of the Volterra formulation is developed and analyzed in terms of convergence order and stability properties under appropriate conditions. Numerical experiments for representative linear and nonlinear vibration problems are presented to validate the theoretical findings and to demonstrate the accuracy, convergence, and effectiveness of the proposed fixed-point approach. https://ansne.du.ac.ir/article_2158.html This paper introduces a novel triple hybrid framework for solving ordinary differential equations (ODEs) and differential-algebraic equations (DAEs) by synergistically combining three distinct machine learning and optimization paradigms. The proposed method integrates Neural Ordinary Differential Equations (Neural ODEs) with a Thermodynamic-Inspired Search Algorithm (TSA) and an adaptive machine learning-based step size controller. Neural ODEs provide a continuous-depth representation that naturally captures the differential structure of physical systems, while TSA---inspired by energy minimization, heat exchange, and entropy control---offers robust global exploration capabilities and prevents premature convergence. The third component employs a lightweight neural network that dynamically adjusts step sizes based on local error estimates, enhancing computational efficiency without sacrificing accuracy. This triple hybrid architecture addresses the limitations of existing methods: gradient instability in Physics-Informed Neural Networks, poor exploration in traditional metaheuristics, and the rigidity of fixed step size strategies. The method is evaluated on a diverse suite of benchmark problems including linear and nonlinear boundary value problems, stiff ODEs, systems of ODEs, and index-1 DAEs. Extensive statistical analysis over 50 independent runs demonstrates superior performance, with average improvements of 6.2\% in solution accuracy compared to state-of-the-art hybrid methods, 28\% reduction in standard deviation, and 35\% decrease in computational time through adaptive step sizing. The mesh-free nature of the approach provides differentiable closed-form solutions, making it particularly valuable for applications requiring sensitivity analysis or integration with downstream tasks. https://ansne.du.ac.ir/article_2159.html This paper introduces the general fuzzy machine (GFM) as a principled framework for modeling and analyzing fuzzy automata through the lens of fixed-point theory. The system dynamics are formulated as a fixed-point equation \( \mu = T_\theta(\mu) \) on the compact membership space \( \mathcal{F}(Q) = [0,1]^n \), where \( T_\theta \) is a parameterized fuzzy transition operator constructed from the aggregation functions \( F_1 \) and \( F_2 \), and the transition membership function \( \delta \). We investigate the existence, uniqueness, and parametric sensitivity of the fixed point \( \mu^*(\theta) \) using the Banach contraction principle and the implicit function theorem. The contraction condition guarantees not only a unique steady-state fuzzy vector but also exponential convergence independent of the initial configuration. Analytical results include Lipschitz-type stability bounds that quantify the robustness of the equilibrium state under operator perturbations, and a closed-form expression for the sensitivity Jacobian \( \frac{d\mu^*}{d\theta} \), which provides a precise measure of how the fixed point responds to small changes in the tunable parameters. Numerical examples confirm that actual sensitivities are often significantly smaller than the worst-case bounds, indicating that GFMs can exhibit robust behavior under parameter perturbations. We also identify structural limitations arising from non-contractive transition operators, where multiple fixed points may exist and Banach's theorem becomes inapplicable, underscoring the need for principled operator design in fuzzy systems when robustness and predictability are required. https://ansne.du.ac.ir/article_2160.html Financial models frequently exhibit nonlinear behavior, necessitating robust numerical techniques for precise analysis. This study investigates the numerical implementation of a generalized Newton method within a Local-Stochastic Volatility framework. The nonlinear partial differential equation (PDE) governing the model is linearized via the generalized Newton procedure, employing the Fr\'echet derivative to construct the iterative scheme. By imposing appropriate boundary conditions and utilizing representative financial data, we examine the convergence properties of the resulting sequence. Numerical results demonstrate that the iterative method achieves stable and rapid convergence, thereby validating the existence of a unique solution within this framework and highlighting the efficacy of Newton-type approaches for solving nonlinear PDEs in financial modeling. Furthermore, this approach significantly reduces computational costs compared to traditional methods, offering a practical and efficient tool for real-time pricing in dynamic market environments. The proposed method also ensures high accuracy in handling complex derivative structures, making it a valuable asset for financial institutions seeking to optimize their risk management strategies. https://ansne.du.ac.ir/article_2171.html This study examines the dynamics of inflation and the effectiveness of monetary policy in the Iranian economy. For this purpose, a nonlinear dynamic stochastic general equilibrium model with state‑dependent features and endogenous indexation has been developed. The results show that inflation in Iran is nonlinear and regime-dependent in nature. In periods of high and stable inflation, strengthening indexation and the formation of adaptive expectations leads to self-reinforcing inflation and weakening the effectiveness of monetary policy. Also, forward-looking monetary policy rules with a medium-term horizon perform better in stabilizing expectations and reducing recession costs compared to simultaneous rules. These findings emphasize the necessity of designing state-based monetary policies and carrying out structural reforms to contain inflationary dynamics in Iran.  https://ansne.du.ac.ir/article_2176.html Fuzzy nonlinear regression learns functional relationships from imprecise observations, yet common estimators suffer from two persistent difficulties: shrinkage rules that cannot discard redundant basis functions, and endpoint fits that may violate the defining properties of a fuzzy number. This paper develops a shape-preserving adaptive elastic net framework for fuzzy responses with crisp or fuzzy inputs. The fuzzy response is decomposed at every membership level into a center and a nonnegative radius, and each component is regressed on a hybrid dictionary joining a polynomial tail, Gaussian atoms with adaptively rescaled dispersions, and a stacked local smoothing pilot atom. An adaptive elastic net penalty with data driven weights selects and shrinks the dictionary, all tuning constants are chosen by an internal cross validation that respects the stacking structure, and a non expansive projection restores nestedness of the estimated level sets. We prove existence and uniqueness of the estimator, convergence of its coordinate descent solver, and that the shape repair can never increase the Hausdorff estimation error. Monte Carlo experiments on four simulated benchmarks and a real clinical dataset in ten dimensions compare the proposal with ridge regularized fuzzy regression, local linear smoothing, and kernel smoothing under a nested cross validation protocol. The proposed method improves on the ridge estimator uniformly, attains the best cross validation error and bias in three simulated designs and the best mean cross validation error on the real data, is statistically tied with the best local smoother elsewhere, and alone guarantees valid fuzzy outputs. https://ansne.du.ac.ir/article_2177.html Variable-order (VO) fractional diffusion equations provide a flexible framework for modeling transport phenomena with time-dependent memory effects‎. ‎However‎, ‎the nonlocality of fractional operators‎, ‎combined with the variable-order structure‎, ‎makes the development of stable and accurate numerical schemes considerably more challenging‎. ‎In this paper‎, ‎a fully discrete numerical method is developed for VO time-fractional diffusion equations involving the Caputo fractional derivative‎. ‎The spatial discretization is based on cubic trigonometric B-spline collocation‎, ‎which provides a smooth approximation to the spatial solution‎. ‎For the temporal discretization‎, ‎an adapted VO--L1 scheme is employed to approximate the VO Caputo derivative‎. ‎The variable-order function $\lambda(t)$ incorporates the time-dependent memory effects into the model‎, ‎and the global temporal convergence rate is of order $2-\lambda_{\max}$‎, ‎where $\lambda_{\max}=\max_{0\le t\le T_F}\lambda(t)$‎. ‎By incorporating the Dirichlet boundary conditions into the collocation formulation‎, ‎the fully discrete system reduces at each time level to a tridiagonal linear algebraic system‎, ‎which can be solved efficiently‎. ‎A matrix-based stability analysis is carried out under suitable assumptions on the coefficient matrix‎. ‎Moreover‎, ‎a convergence analysis is provided by combining the consistency of the temporal and spatial discretizations with the stability of the overall scheme‎. ‎Under appropriate regularity assumptions‎, ‎the proposed method satisfies the global maximum-norm error estimate $\|\mathbf{e}^{n}\|_{\infty} \leq C\bigl(h_x^2+h_t^{\,2-\lambda_{\max}}\bigr)$‎, ‎where $C$ is independent of $h_x$ and $h_t$‎. ‎Several numerical experiments are conducted to validate the theoretical findings and assess the accuracy and efficiency of the proposed approach‎. ‎The results support the theoretical analysis and demonstrate the effectiveness of the cubic trigonometric B-spline VO--L1 collocation scheme for solving VO fractional diffusion problems with time-dependent memory effects‎. https://ansne.du.ac.ir/article_2179.html This paper presents a constrained hybrid FEM-LSSVR framework for the numerical solution of nonlinear partial differential equations, with a particular focus on FitzHugh-Nagumo (FHN)-type model variants. In the proposed approach, Least Squares Support Vector Regression (LSSVR) is not employed as a post-processing or data-fitting tool; instead, it is formulated as a local primal optimization problem in which the governing PDE, boundary conditions, and inter-element continuity are explicitly enforced as constraints. The finite element method (FEM) is incorporated to provide structural consistency, embedding boundary and continuity information directly into the learning formulation rather than using FEM solutions as training data. The resulting method establishes a tight interaction between numerical discretization and constrained learning, yielding a flexible yet physically consistent approximation framework. The performance of the proposed approach is investigated through numerical experiments on several FHN-type nonlinear PDE variants, including studies with varying mesh resolutions, different final time horizons, and data-driven settings. Comparisons with standard FEM solutions demonstrate improved robustness and accuracy, particularly in scenarios involving limited or noisy data. These results indicate that the constrained FEM-LSSVR formulation provides a viable alternative to traditional numerical solvers for nonlinear PDEs in data-driven contexts. https://ansne.du.ac.ir/article_2180.html In this paper, a novel numerical method based on hybrid Fourier and block-pulse functions (HFBPFs) is proposed for solving the nonlinear Duffing equation. Although hybrid Fourier-block-pulse bases have been previously applied to linear integral equations and calculus of variations problems, their application to the Duffing equation---a benchmark nonlinear oscillator with cubic stiffness---has not been reported in the literature. The proposed method combines the local behavior of block-pulse functions with the spectral accuracy of Fourier series within each subinterval. The operational matrix of integration and the product operational matrix are derived for the hybrid basis, and the Duffing equation is reduced to a system of nonlinear algebraic equations. Numerical examples demonstrate that the HFBPFs method achieves approximately three orders of magnitude improvement in accuracy compared with pure block-pulse functions, while avoiding the Gibbs phenomenon that affects pure Fourier approximations for non-smooth solutions. The method offers dual tunability through the number of subintervals and the number of Fourier components per subinterval, providing flexibility in balancing accuracy and computational cost.