https://ansne.du.ac.ir/ Analytical and Numerical Solutions for Nonlinear Equations en daily 1 Mon, 23 Feb 2026 00:00:00 +0330 Mon, 23 Feb 2026 00:00:00 +0330 https://ansne.du.ac.ir/article_2069.html This paper introduces a novel approach for solving two-dimensional time-fractional Navier-Stokes and Black-Scholes equations. The method integrates a new integral transform--based on a generalized power function of the form $\frac{t^{\varrho}}{\varrho}$-- with the residual power series method. This combined approach, termed the ``generalized integral transform residual power series method,'' utilizes the Katugampola fractional derivative in the Caputo sense. The convergence of the method is rigorously established, and its efficacy, accuracy, and precision are demonstrated through illustrative examples. The results highlight the method's potential for efficiently solving complex fractional partial differential equations across various scientific and engineering disciplines. https://ansne.du.ac.ir/article_2090.html In this manuscript, we investigate the nonlinear singular $q-$differential equation of Lane-Emden type, by using the general Riemann-Liouville integral and Caputo derivative  of $q-$fractional order operators. First, our approach to prove existence and uniqueness is Banach's  contraction principle. Then, in the next step, to confirm the existence of at least one solution, we take help from fixed point theorem of Schaefer. Moreover,  the stabilities in the sense of Ulam-Hyers and Ulam-Hyers-Rassias are also defined and examined. Finally, we present a comprehensive example to show the applicability of the outcomes. https://ansne.du.ac.ir/article_2084.html This study introduces a novel population-based metaheuristic algorithm, the Pirates of the Caribbean Optimization Algorithm (PCOA), inspired by the adventurous strategies of pirates in cinematic narratives. PCOA models the exploration and exploitation process through the metaphor of pirate crews searching for hidden treasure while navigating unpredictable challenges. The algorithm’s effectiveness is evaluated by extensive comparisons with leading optimization methods across 23 classical functions and the CEC 2019 benchmark suites. Results consistently demonstrate PCOA’s superior solution quality, robustness, and convergence speed. Additionally, PCOA is successfully applied to challenging real-world inverse problems in nonlinear partial differential equations, highlighting its practical potential. The open-source implementation of PCOA further supports reproducibility and future research. https://ansne.du.ac.ir/article_2085.html In this paper, we investigate the numerical solution of fuzzy differential inclusions (FDIs) using characterization results for fuzzy differential equations (FDEs) based on Hukuhara differentiability. By employing the characterization theorem introduced by Bede and the construction of solutions via differential inclusions developed by Kaleva, we establish a rigorous connection among fuzzy differential equations, fuzzy differential inclusions, and systems of ordinary differential equations (ODEs). In particular, under suitable regularity and monotonicity assumptions, it is shown that the solution of a fuzzy differential inclusion coincides with the solution of the corresponding fuzzy differential equation and can be equivalently represented by a system of ODEs defined on the $\alpha$-level sets. This equivalence enables the reduction of fuzzy-valued problems to classical real-valued systems, thereby allowing the direct application of standard numerical methods for ordinary differential equations. Based on this framework, we propose a numerical approach for solving FDIs by first transforming the fuzzy problem into a parametric family of ODEs and then applying the Euler method to approximate the solutions. The validity of the proposed approach is illustrated through a numerical example, in which the approximate fuzzy solution is compared with the exact solution. The results demonstrate that classical numerical schemes can be effectively employed for fuzzy differential inclusions without reformulating them within a fully fuzzy numerical framework. https://ansne.du.ac.ir/article_2110.html In this research, we consider the system of linear Volterra-Fredholm integro-differential equations (SVFIDEs). The main aim of this research is to approximate the integral by Gauss-Kronrod-Legendre quadrature rules and then using quintic B-spline as the bases function. The unknown coefficients in combination determine by collocation method. The arising system of algebraic linear can be solved via iterative method. Error analysis is investigated theoretically. Numerical text problems are considered to justify the applicability and efficient nature of our approach, comparison of the results justify the considerable accuracy and efficiency proposed methods. The extended parameter in valued in the spline can be chosen in such a way to  improve the accuracy also. https://ansne.du.ac.ir/article_2092.html Given the increasing challenges in electricity supply and the necessity of optimizing energy consumption in public institutions, this article designs an efficient and incentive-compatible self-regulating mechanism in the smart electricity grid (based on mechanism design theory) to manage electricity consumption in public governmental institutions. Unlike existing mechanisms that rely on direct supervision or fixed quotas, our approach introduces a composite permitted ceiling that dynamically adapts to each institution's historical performance and operational constraints. The aim of this article is to design a mathematical model and create a mechanism through which agents (public governmental institutions) voluntarily and without coercion choose desirable and optimal consumption behavior. The proposed mechanism encourages agents to reduce consumption and adhere to the permitted ceiling without direct supervision. Our model achieves an improvement in energy balance compared to traditional fixed-ceiling approaches while maintaining voluntary participation.  The proposed mechanism encourages agents to reduce consumption and adhere to the permitted ceiling without direct supervision. The proposed mechanism includes a composite permitted ceiling, mission coefficient, and a system of rewards and penalties that aligns with the interests of each institution and leads to the reform of consumption behavior. https://ansne.du.ac.ir/article_2091.html In this paper, a collocation approach based on reproducing kernels is presented for the numerical solution of the 2D time-fractional diffusion equation. Some finite-dimensional positive definite reproducing kernel spaces are constructed using the bases of the Fibonacci polynomials. The spatial discretization in the proposed method is based on the Fibonacci polynomial reproducing kernel method, which is combined with a finite difference scheme for temporal discretization. Handling the boundary conditions in the numerical solution of partial differential equations is a challenging issue in numerical methods. To deal with the boundary conditions in the proposed method, the reproducing kernels are constructed in a way that exactly satisfy the boundary conditions. Some numerical simulations are conducted to prove the efficiency and ability of the Fibonacci kernel approach combined with the time-stepping scheme. The numerical results show the effectiveness and accuracy of the proposed method. https://ansne.du.ac.ir/article_2086.html We broaden the framework of sandpile monoids and weighted Leavitt path algebras to encompass fuzzy graph structures, establishing key structural relationships within this extended setting. Specifically, we prove that idempotent components in fuzzy sandpile monoids $\text{FSP}(E, \mu, \gamma)$ correspond to fuzzy hereditary saturated subsets $(E, \mu, \gamma)$. Additionally, we demonstrate that these idempotent structures exhibit a lattice organization governed by order ideals within $(\bar{E}, \mu, \gamma)$. Furthermore, this lattice structure aligns with the lattice formed by vertex-generated ideals in the fuzzy weighted Leavitt path algebra $L_1(\bar{E}, \omega, \mu, \gamma)$. We characterize the fuzzy sandpile group through Archimedean equivalence classes and establish that optimal subgroups align exactly with Grothendieck groupoids of these equivalence classes. Our analysis reveals how the lattice of idempotents in $\text{FSP}(E, \mu, \gamma)$ forms a system of graded ideals that preserve invariance under graded automorphisms. https://ansne.du.ac.ir/article_2094.html In this paper, we investigate a nonlinear analytical model for estimating the overlap area between two circular sectors. The overlap problem is formulated in a functional–analytic framework by representing sector regions through indicator functions and interpreting the overlap area as a nonlinear trace expression involving multiplication operators on  $L^{2}(\mathbb{R}^{2})$. This formulation allows the application of classical nonlinear inequalities, including Young’s inequality and related operator bounds, to derive explicit analytical estimates for the overlap area. The resulting bounds depend nonlinearly on the sector parameters, such as angular widths and radii, and avoid direct geometric intersection computations. In addition, the proposed bounds are numerically tractable and can be efficiently evaluated numerically for a wide range of sector configurations. An angular–averaged nonlinear bound is introduced, providing a computable upper estimate that captures the combined angular and radial effects of the sectors. Several illustrative examples demonstrate the effectiveness of the analytical bounds and confirm their numerical consistency. The proposed approach establishes a connection between nonlinear analytical techniques, operator inequalities, and geometric modeling, offering a flexible framework for nonlinear overlap estimation problems. This formulation presents an innovative conceptual reformulation using operator theory for the circular overlap problem, providing a powerful tool for analysis and accurate estimation of the overlap area using classical inequalities. https://ansne.du.ac.ir/article_2109.html In this paper, we present the Residual Integral Solver Network (RISN), a novel neural network architecture designed to solve a wide range of integral and integro-differential equations, including one-dimensional, multi-dimensional, ordinary and partial integro-differential, systems, fractional types, and Helmholtz-type integral equations involving oscillatory kernels. RISN integrates residual connections with high-accuracy numerical methods such as Gaussian quadrature and fractional derivative operational matrices, enabling it to achieve higher accuracy and stability than traditional Physics-Informed Neural Networks (PINN). The residual connections help mitigate vanishing gradient issues, allowing RISN to handle deeper networks and more complex kernels, particularly in multi-dimensional problems. Through extensive experiments, we demonstrate that RISN consistently outperforms not only classical PINNs but also advanced variants such as Auxiliary PINN (A-PINN) and Self-Adaptive PINN (SA-PINN), achieving significantly lower Mean Absolute Errors (MAE) across various types of equations. These results highlight RISN’s robustness and efficiency in solving challenging integral and integro-differential problems, making it a valuable tool for real-world applications where traditional methods often struggle. https://ansne.du.ac.ir/article_2093.html Newton-type methods are essential for solving nonlinear equations and systems, with classical metric-based analysis focusing on quadratic convergence and local error bounds. However, these results overlook the structural stability of iterations under perturbations. This paper introduces a Selje topological framework to analyze the stability of Newton-type methods beyond traditional numerical theory. We associate nonlinear operators with Selje topological structures and study the invariance and stability of iterative sequences via induced operators $\mathcal{T}_{\mathcal{R}}{(\mathbb{X})}$,$\mu_{\mathcal{R}}{(\mathbb{X})}$, $SJ_{\mathcal{R}}{(\mathbb{X})}$ . Sufficient conditions are established for preserving topological stability in Newton-type iterations, interpreting convergence as structural consistency in the Selje space. This framework yields a generalized stability characterization that complements classical convergence theory, advancing the analysis of nonlinear iterative solvers through topology. https://ansne.du.ac.ir/article_2095.html In this paper, a fractional integro-differential model involving the Caputo time-fractional derivative and the Riesz space-fractional operator is proposed and analyzed. The model incorporates both nonlinear reaction terms and nonlocal integral interactions, allowing an accurate description of anomalous diffusion processes with memory and spatial long-range effects. By applying the Fourier transform with respect to the spatial variables and the Laplace transform with respect to time, the governing equation is transformed into an algebraic equation in the transform domain, leading to an explicit representation of the solution in terms of Mittag--Leffler functions. The existence, convergence, and stability of the mild solution are established by means of an iterative scheme combined with fixed-point arguments and a fractional Gronwall inequality. It is shown that the approximate solutions converge uniformly to the unique mild solution and that the solution depends continuously on the initial data. To illustrate the theoretical results, three representative examples are presented, including a pure fractional diffusion model, a reaction--diffusion model, and a multi-mode system with nonzero integral kernels. The obtained exact solutions demonstrate the significant influence of the fractional orders on the temporal decay rate and spatial behavior of the solution. The proposed framework provides a mathematically rigorous and physically meaningful tool for modeling and analyzing fractional-order transport phenomena arising in engineering and industrial applications such as heat conduction in heterogeneous materials, diffusion in porous media, and dynamic processes in complex systems. 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_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_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_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.