Damghan University PressAnalytical and Numerical Solutions for Nonlinear Equations3060-785X9120250420Solving the Fokker-Planck Equation with Neural Networks: A Performance Improvement Approach11147610.22128/ansne.2025.967.1130ENHassan Dana MazraehDepartment of Computer and Data Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, IranPegah MotaharinezhadDepartment of Computer and Data Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, IranNafiseh DaneshianDepartment of Computer and Data Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, IranKourosh ParandDepartment of Computer and Data Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran; Department of Cognitive Modeling, Institute for Cognitive and Brain Sciences, Shahid Beheshti University, Tehran, IranJournal Article20250227The Fokker-Planck equation models the evolution of probability densities in fields such as physics, biology, and finance. Traditional numerical methods for solving this equation can be computationally expensive and struggle with complex, high-dimensional problems. In this work, we propose a Physics-Informed Neural Networks (PINNs) approach to efficiently approximate solutions of the Fokker-Planck equation. Our method employs a fully connected feedforward neural network using two activation functions--Tanh and SiLU--with fixed learning rates (0.001 for Tanh and 0.01 for SiLU) and varied spatial discretization. The loss function is designed to enforce the governing differential equation as well as the initial and boundary conditions. Experimental results, evaluated using standard error metrics (RMS, Relative $L_2$-Norm Error, and MAE), demonstrate that our PINN approach achieves competitive accuracy with improved convergence and lower computational costs compared to traditional methods. This study underscores the potential of neural network-based solvers for complex differential equations and sets the stage for future optimization.https://ansne.du.ac.ir/article_476_9c8e5e69a0af2b426668d0c187d4ba8d.pdfDamghan University PressAnalytical and Numerical Solutions for Nonlinear Equations3060-785X9120250420Solving the Transportation Problem using Meta-Heuristic Algorithms121947710.22128/ansne.2025.956.1129ENAhmad Aliyari BoroujeniDepartment of Mathematics and Computer Science, Faculty of Sciences, University of Zanjan, Zanjan, Iran0000-0002-9318-3920Mohammad Reza GhaemiDepartment of Mathematics and Computer Science, Faculty of Sciences, University of Zanjan, Zanjan, Iran0009-0003-1490-0317Reza PourgholiSchool of Mathematics and Computer Sciences, Damghan University, P.O. Box 36715-364, Damghan, Iran0000-0003-4111-5130Journal Article20250215In this paper, the transportation problem is thoroughly analyzed and solved using three different meta-heuristic algorithms. The transportation problem, a fundamental optimization issue in operations research, involves determining the most efficient way to distribute goods from multiple supply sources to multiple destinations while minimizing overall transportation costs. Traditional exact methods may struggle to provide solutions in a reasonable time frame, especially as the size and complexity of the problem grow. In contrast, meta-heuristic algorithms offer the potential to find near-optimal solutions more efficiently, making them a valuable approach for large-scale problems. This study focuses on three algorithms: Genetic Algorithm (GA), Teaching-Learning-Based Optimization (TLBO), and an improved variant of TLBO known as ITLBO. Each of these algorithms was applied to the transportation problem, and their performance was evaluated in terms of solution quality, convergence speed, and computational efficiency. The results demonstrate that while all three algorithms can solve the transportation problem, ITLBO consistently outperforms GA and TLBO {in terms of accuracy}. Specifically, ITLBO shows a faster convergence to the optimal solution and a significant reduction in execution time, particularly for large problem instances. The improved efficiency of ITLBO makes it a more practical and scalable option for solving complex transportation problems. https://ansne.du.ac.ir/article_477_4080100aa5a4fe71c0561f60e37756ce.pdfDamghan University PressAnalytical and Numerical Solutions for Nonlinear Equations3060-785X9120250420A Pseudo-Spectral Approach to Solving the Fractional Cable Equation Using Lagrange Polynomials203347810.22128/ansne.2025.946.1126ENSoheila MirzaeiDepartment of Mathematics, Faculty of Sciences, University of Zanjan, Zanjan, IranAli ShokriDepartment of Mathematics, Faculty of Sciences, University of Zanjan, Zanjan, Iran0000-0002-5942-5260Journal Article20250123This paper focuses on addressing the fractional cable equation through the pseudo-spectral method, providing an innovative approach for solving problems in fractional calculus. The proposed method utilizes Lagrange polynomials at Chebyshev points to approximate spatial derivatives, ensuring high computational precision. One of the noteworthy features of this study is the time discretization scheme introduced, which is unconditionally stable and boasts a convergence order of $\mathcal{O}(\tau^2)$. Such stability is crucial for solving time-dependent fractional differential equations, especially when long-term simulations are required. Pseudo-spectral methods are known for their exponential accuracy, and the approach detailed in this paper exemplifies this capability. To demonstrate the robustness and reliability of the proposed technique, several numerical examples are presented. These examples highlight the method's efficiency in achieving highly accurate solutions with minimal computational effort. By leveraging spectral techniques, this research offers insights into solving fractional differential equations with greater precision, paving the way for future advancements in this field. The outcomes underscore the proposed method's effectiveness and emphasize its potential applicability to a broad range of scientific and engineering problems.https://ansne.du.ac.ir/article_478_da3a107eec886bd35a06d233e7560b1c.pdfDamghan University PressAnalytical and Numerical Solutions for Nonlinear Equations3060-785X9120250517Enhancing Kernel Ridge Regression Models with Compact Support Wendland Functions344748210.22128/ansne.2025.916.1123ENZahra BehdaniDepartment of Mathematics and statistics, Behbahan Khatam Alanbia University of Technology, Khouzestan, IranMajid DarehmirakiDepartment of Mathematics and statistics, Behbahan Khatam Alanbia University of Technology, Khouzestan, IranJournal Article20241130Radial Basis Functions (RBFs) have gained significant attention in various machine learning applications, including regression modeling, due to their ability to approximate complex, nonlinear relationships. RBFs offer a flexible approach to capturing intricate dependencies between input features and the target variable, making them particularly useful in high-dimensional and nonparametric settings. This paper investigates the use of a specific class of compactly supported RBFs, known as Wendland functions, within the framework of kernel ridge regression (KRR). We discuss their theoretical advantages—such as sparsity enforcement and computational efficiency as well as practical challenges, including parameter selection and scalability. A comprehensive overview of RBFs is provided, along with their mathematical formulation and a comparison of different RBF kernels in terms of smoothness and locality. We detail the integration of Wendland functions into KRR models, emphasizing their suitability for problems requiring robustness and interpretability. Through extensive simulation studies, the performance of the proposed approach is evaluated against conventional RBF kernels and other widely used regression techniques. Our results demonstrate that Wendland-based KRR achieves competitive accuracy while offering improved stability in the presence of noise and outliers. Furthermore, real-world case studies illustrate the effectiveness of Wendland functions in handling datasets with high collinearity, where traditional kernels often struggle. The practical implications of our findings are discussed, along with guidelines for implementation and potential extensions to large-scale or sparse data scenarios. This work contributes to the growing body of research on interpretable and efficient kernel methods, providing insights for both theoretical and applied machine learning practitioners.https://ansne.du.ac.ir/article_482_df258dc8ae0b92e60a209b02c18af79b.pdfDamghan University PressAnalytical and Numerical Solutions for Nonlinear Equations3060-785X9120250525Approximation Solution to the Heat Conduction Equation in a Rectangular Channel Influenced by Airflow using the Chebyshev Pseudo-Spectral Method485948410.22128/ansne.2025.959.1128ENEsmaeil YousefiDepartment of Mathematics and Computer Science, SR.C., Islamic Azad University, Tehran, Iran0000-0002-0977-7030Mahdi AzhiniDepartment of Mathematics and Computer Science, SR.C., Islamic Azad University, Tehran, IranAmir Mohammad IranbodiDepartment of Mathematics and Computer Science, SR.C., Islamic Azad University, Tehran, IranJournal Article20250212In this paper, the approximation of the solution to the one-dimensional convection-diffusion equation is studied as a model for heat transfer in a rectangular channel influenced by airflow. First, the convection-diffusion equation is derived using basic principles of conservation, including convection and diffusion. Then, the existence and uniqueness of the solution to this equation are briefly discussed, taking into account the properties of the convective and diffusive coefficients, boundary conditions, and initial conditions. Numerical solution methods for the problem have been explored by researchers, leading to various approaches. As examples, \cite{Kho,Baz1,Ism,Smit1,Mor1} used the Chebyshev pseudo-spectral method for spatial discretization and the fourth-order Runge-Kutta (RK4) method for temporal discretization to solve this equation. The numerical characteristics of this method, including accuracy, stability, and convergence rate, are analyzed using eigenvalue analysis of the system and stability regions. The results obtained include temperature distribution, absolute error, and a three-dimensional analysis of the temperature distribution in space and time. Additionally, the impact of the time step on the stability of the numerical method has been investigated, and it is shown that the proposed method can achieve desirable accuracy and stability with proper parameter adjustments. This study confirms the effectiveness of the Chebyshev pseudo-spectral method in solving dynamic problems such as heat transfer and related applications providing a foundation for using this method in more complex problems.https://ansne.du.ac.ir/article_484_dafcd3f04224c0a1cb6dc4e06bb1bf16.pdfDamghan University PressAnalytical and Numerical Solutions for Nonlinear Equations3060-785X9120250531Hub Number of Incidence and Power Graph606448510.22128/ansne.2025.978.1132ENAbolfazl BahmaniDepartment of Mathematics, University of Zanjan, Zanjan, IranOzra NaserianDepartment of Mathematics, Zanjan Branch, Islamic Azad university, Zanjan, Iran0000-0003-4888-2965Mohammad Reza GhaemiDepartment of Mathematics, University of Zanjan, Zanjan, Iran0009-0003-1490-0317Mohammad Reza BonabifardDepartment of Mathematics, University of Zanjan, Zanjan, Iran0009-0009-1205-2661Journal Article20250312In graph theory, a set $H \subseteq V (G)$ is defined as a hub set if every pair of non-adjacent vertices outside $H$ can be interconnected by a path that exclusively traverses through the internal vertices contained in $H$. The hub number of a graph $G$ refers to the minimal cardinality of such a hub set, providing crucial insights into the structural connectivity of the graph. This paper delves into the exploration of the hub number across various graph structures, specifically focusing on incidence graphs and square graphs, both of which possess unique characteristics impacting their connectivity properties. We establish theoretical bounds for the hub numbers of these graphs, facilitating a clearer understanding of their structural complexities. Furthermore, we derive explicit values for the hub numbers of several special types of graphs, including path graphs, star graphs and complete graphs. Through rigorous analysis and evaluation, this study contributes to the broader field of connectivity in graphs by not only identifying the hub numbers for specific examples but also by proposing methodologies for their computation. These findings have important implications for applications in network design and graph optimization, enhancing the utility of hub sets in practical scenarios.https://ansne.du.ac.ir/article_485_cb9f526f12da836b2ca9c67b28c1627f.pdfDamghan University PressAnalytical and Numerical Solutions for Nonlinear Equations3060-785X9120250529Solving the Fractional HIV Model using Bell Polynomials and the Tau Method657348610.22128/ansne.2025.1001.1136ENReza BoroghaniDepartment of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences,
Semnan University, 35195-363, Semnan, IranKazem NouriDepartment of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University,
35195-363, Semnan, Iran0000-0002-7922-5848Journal Article20250128This paper introduces a Tau-based numerical approach utilizing Bell polynomials to solve a fractional HIV model involving $CD4^+ T$ cells. The model comprises a system of three fractional nonlinear ordinary differential equations. The method begins with deriving the operational matrix for the fractional derivative of Bell polynomials. Next, any function within the space \(L^2[0,1]\) is approximated using Bell polynomials, and these approximations are substituted into the model equations. The resulting expressions are used to compute Chebyshev collocation points within the interval \([0,1]\). By applying the Tau method along with the boundary conditions, the problem is converted into a system of algebraic equations that can be solved using standard numerical techniques such as Newton's method. An example is provided to illustrate the accuracy and effectiveness of the proposed method.https://ansne.du.ac.ir/article_486_3f1802e50a8e2a582a3c90a4af351319.pdfDamghan University PressAnalytical and Numerical Solutions for Nonlinear Equations3060-785X9120250602A Estimation of Ridge-Based in a Type-2 Fuzzy Non-Parametric Regression748848710.22128/ansne.2025.982.1133ENJavad GhasemianSchool of Mathematics and Computer Science, Damghan University, Damghan, IranMahmoud MoallemSchool of Mathematics and Computer Science, Damghan University, Damghan, IranFatemeh HamidiradSchool of Mathematics and Computer Science, Damghan University, Damghan, IranZahra KarimiSchool of Mathematics and Computer Science, Damghan University, Damghan, IranJournal Article20250325This paper focuses on estimating ridge in a type-2 fuzzy non-parametric regression model that utilizes non-fuzzy inputs, type-2 fuzzy output data, and type-2 fuzzy coefficients within a dual Lagrange space. It begins with definitions of type-2 fuzzy sets (T2FSs) and presents a closed parametric form for complete triangular T2FSs. The proposed framework underpins a local linear smoothing method that incorporates a cross-validation procedure for optimizing ridge parameters and smoothing values. The research advances statistical modeling with type-2 fuzzy systems, offering innovative techniques for regression analysis in complex data situations. The combination of ridge estimation, local linear smoothing, and cross-validation is highlighted for its potential to yield precise results. Our work is able to model complex and nonlinear relationships between variables, which more effectively deals with uncertainties and ambiguities in the data, prevents overfitting, and ultimately improves the accuracy and reliability of predictions. Numerical simulations are included to validate the theoretical findings and demonstrate the method's effectiveness.https://ansne.du.ac.ir/article_487_d6cba6f23307852927af9536972d9708.pdfDamghan University PressAnalytical and Numerical Solutions for Nonlinear Equations3060-785X9120250710System of Volterra Fredholm Integro-Fractional Differential Equations: Application of Fibonacci Polynomials89101185910.22128/ansne.2025.993.1135ENZahra GilaniDepartment of Mathematics, Faculty of Basic Science, Babol Noshirvani University of Technology, Babol, IranMohsen AlipourDepartment of Mathematics, Faculty of Basic Science, Babol Noshirvani University of Technology, Babol, IranSanaz RivazDepartment of Mathematics, Faculty of Basic Science, Babol Noshirvani University of Technology, Babol, IranJournal Article20250418In this paper, we introduce the Fibonacci polynomials (FPs) and approximate functions using them. Furthermore, several lemmas and corollaries present the properties of FPs. Also, we derive the Fibonacci polynomials operational matrix for the fractional derivative in the Caputo sense, which has not been undertaken before. As applications of the Fibonacci polynomials operational matrix, we solve the system of Volterra Fredholm integro-fractional differential equations. In this scheme, we approximate one and two variable functions based on Fibonacci basis. Then by applying Fibonacci polynomials operational matrix, the system of Volterra Fredholm integro-fractional differential equations is reduced to a system of algebraic equations that is easily solvable with the help of a software (version 13 of the Mathematica software). The obtained results are in good agreement with the exact solutions and with those in literature. As anticipated, the solutions converge to classical solutions as the fractional derivative order approaches integer values.https://ansne.du.ac.ir/article_1859_207062d32665a511549c0cd2470fafb6.pdfDamghan University PressAnalytical and Numerical Solutions for Nonlinear Equations3060-785X9120250713Double Barrier Option Pricing Formulas of an Uncertain Stock Model10211346810.22128/ansne.2025.958.1127ENBehzad AbbasiDepartment of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Iran0009-0008-1247-7291Farahnaz OmidiDepartment of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, IranKazem NouriDepartment of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Iran0000-0002-7922-5848Leila TorkzadehDepartment of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Iran0000-0002-2504-4048Journal Article20250528The valuation of options is an essential topic in the financial markets, and barrier options represent a widely utilized category of options that may gain or lose value once the price of the underlying asset hits a specified threshold. A double barrier option includes two barriers, one above and one below the current stock price. It is classified as path dependent due to the fact that the holder's return is influenced by the stock price's breach of these barriers. The double barrier option contract defines three specific payoffs, which are contingent upon whether the upper barrier or lower barrier is breached, or if there is no breach of either barrier throughout the option's duration. In this paper, pricing of the double barrier options when the underlying asset price follows the uncertain stock model is investigated, and also pricing formulas for different types of double barrier options (knock-in and knock-out) are derived by $ \alpha $-paths of uncertain differential equations in the uncertain environment.https://ansne.du.ac.ir/article_468_eee009f52682a7e97bffab243abdf75d.pdfDamghan University PressAnalytical and Numerical Solutions for Nonlinear Equations3060-785X9120250807$C_{0}$-Groups and $C$-Groups on Non-Archimedean Quasi-Banach Spaces114132186210.22128/ansne.2025.1021.1137ENJawad EttaybC. High school of Hauman El fetouaki, Had Soualem, MoroccoJournal Article20250602In this paper, we introduce and study $C_{0}$-groups and $C$-groups of bounded linear operators on non-Archimedean quasi-Banach spaces over $\mathbb{K}.$ In particular, we show some results related to them. In contrast with the classical framework, the parameter of $C_{0}$-groups and $C$-groups families of bounded linear operators belongs to a open ball $\Omega_{r}$ of a non-Archimedean field $\mathbb{K}.$ As an illustration, we shall discuss the solvability of some homogeneous $p$-adic differential equations for $C_{0}$-groups and $C$-groups. Also, we provide some examples to illustrate our study.https://ansne.du.ac.ir/article_1862_8d9dc940ac50dcbb6002d5a14e2c9a58.pdfDamghan University PressAnalytical and Numerical Solutions for Nonlinear Equations3060-785X9120250808Fiscal and Monetary Equilibrium: A Differential Game Between the Government and the Central Bank133149186310.22128/ansne.2025.968.1131ENBehzad ShahbaieDepartment of Mathematics, Semnan University, Semnan 35195-363, IranAlireza BahiraieDepartment of Mathematics, Semnan University, Semnan 35195-363, Iran;
Department of Mathematics, Washington University, St.Louis, MO, USAJournal Article20250304This study employs a differential game-theoretic framework to analyze the strategic interaction between the government and the central bank in the context of public debt management. The core objective is to develop an analytical model that stabilizes government debt by optimizing three critical policy instruments: tax revenue, government spending, and the money supply. A major innovation of this research lies in disaggregating the fiscal deficit into two distinct control variables—taxation and spending, which allows for the derivation of optimal equilibrium trajectories for each. Numerical simulations, based on a 20-year dataset from the United States, reveal key differences in outcomes depending on the degree of independence or interdependence between fiscal instruments. When tax revenue and government spending are treated as independent from the monetary authority, the model converges to equilibrium values of 0.8992 for public debt, 0.0982 for the fiscal deficit, and 0.1288 for the monetary base. Conversely, when tax and spending decisions are jointly determined as components of the fiscal deficit, the corresponding equilibrium values shift slightly to 0.8975, 0.0852, and 0.1157, respectively. These findings suggest that enhanced coordination between the fiscal authority’s instruments and the central bank’s control over money creation can improve debt sustainability and overall macroeconomic stability. While the empirical focus is on the U.S. economy, the proposed framework offers a flexible foundation for evaluating fiscal-monetary dynamics in other institutional settings.https://ansne.du.ac.ir/article_1863_29f7cbf341fb9a355e2058e114d128bb.pdf