Damghan University PressAnalytical and Numerical Solutions for Nonlinear Equations3060-785X11120260223Solving Fractional Black-Scholes and Navier-Stokes Equations via a New $\frac{t^{\varrho}}{\varrho}$-Integral Transform and Residual Power Series131206910.22128/ansne.2026.3084.1159ENAbbas PoyaDepartment of Mathematics, Daykondi University, Nili, AfghanistanMohammad Ali ZirakDepartment of Mathematics, Daykondi University, Nili, AfghanistanMohammad Hossein AkramiDepartment of Mathematical Sciences, Yazd University, Yazd, IranJournal Article20251005This 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_2069_413a6fda792abd1e1ad14818da4ff2f2.pdfDamghan University PressAnalytical and Numerical Solutions for Nonlinear Equations3060-785X11120260607Application of Fractional Quantum Calculus on Lane-Emden Type Problem Involving Two Fractional q_Derivatives3250209010.22128/ansne.2026.3155.1180ENMohamed HouasLaboratory, FIMA, UDBKM, Khemis Miliana University, Algeria0000-0001-6256-0511Mohammad Esmael SameiDepartment of Mathematics, Faculty of Science, Bu-Ali Sina University, Hamedan, Iran0000-0002-5450-3127Mohammed K. A. KaabarGofa Camp, Near Gofa Industrial College and German Adebabay, Nifas Silk-Lafto, 26649 Addis Ababa, Ethiopia;
Institute of Mathematical Sciences, Faculty of Science, University of Malaya, Kuala Lumpur 50603, Malaysia0000-0002-5450-3127Manochehr KazemiDepartment of Mathematics, Ashtian Branch, Islamic Azad University, Ashtian, Iran0000-0001-8392-6690Journal Article20251117In 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_2090_a2e8588b0c3d56c7a45b7e2e7a036536.pdfDamghan University PressAnalytical and Numerical Solutions for Nonlinear Equations3060-785X11120260607Pirates of the Caribbean Metaheuristic: A Novel Optimization Algorithm Inspired by Cinematic Metaphors for Solving Complex Optimization Problems5183208410.22128/ansne.2026.3221.1187ENYasin AlipourSchool of Mathematics and Computer Science, Damghan University, Damghan, Iran0009-0001-2863-4924Abdolali BasiriSchool of Mathematics and Computer Science, Damghan University, Damghan, Iran0000-0003-4454-4379Reza PourgholiSchool of Mathematics and Computer Science, Damghan University, Damghan, Iran0000-0003-4111-5130Mahmoud MoallemSchool of Mathematics and Computer Science, Damghan University, Damghan, IranHossein MollajafarySchool of Mathematics and Computer Science, Damghan University, Damghan, Iran0009-0003-8540-7300Journal Article20260119This 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_2084_0fc8741761a6e1c3f6faa3ba2c495726.pdfDamghan University PressAnalytical and Numerical Solutions for Nonlinear Equations3060-785X11120260607Characterization Theorem for the Numerical Solution of Fuzzy Differential Inclusions (FDIs)8492208510.22128/ansne.2026.3222.1188ENSabba KarbalaeiPhd Student, Department of Mathematics and Computer Science, SR.C, Islamic Azad University, Tehran, IranEsmaeil YousefiDepartment of Mathematics and Computer Science, SR.C, Islamic Azad University, Tehran, Iran0000-0002-0977-7030Samira SiamansouriUniversity of Applied Science and Technology, Central of Tehran Shahdad Milk Industries, Iran0000-0003-1750-6393Samanh NeyshabouriDepartment of Mathematics Qazvin Branch, Islamic Azad University, Qazvin, IranJournal Article20260124In 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_2085_632b872c1be2461b7444789d1c9b8ff9.pdfDamghan University PressAnalytical and Numerical Solutions for Nonlinear Equations3060-785X11120260607The Solution for Systems of High-Order Linear Volterra-Fredholm Integro-Differential Equations93100211010.22128/ansne.2026.3231.1190ENZahra MahmoodiDepartment of Mathematics, WT.C., Islamic Azad University, Tehran, Iran0000-0002-0855-6017Journal Article20260128In 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_2110_6d099679fda2571dc36c00c92f12780d.pdfDamghan University PressAnalytical and Numerical Solutions for Nonlinear Equations3060-785X11120260607Designing an Efficient and Incentive-Compatible Mechanism of Self-Adjusting in the Smart Power Grid101115209210.22128/ansne.2026.3234.1191ENFazeleh AkbarianNon Government Higher Education Institute of Fazilat, Semnan, IranEsmaiel AbounooriDepartment of Economics, Semnan University, Semnan, IranMahdi FarzinfarSchool of Engineering, Damghan University, P.O. Box 36716-41167, Damghan, IranNader AsghariDepartment of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, IranJournal Article20260129Given 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_2092_3e301937b2dd3dd6fe75c185906dea99.pdfDamghan University PressAnalytical and Numerical Solutions for Nonlinear Equations3060-785X11120260607Fibonacci Polynomial Reproducing Kernel Collocation Method for 2D Time-Fractional Diffusion Equations116129209110.22128/ansne.2026.3235.1192ENMahdi EmamjomehBasic Sciences Group, Golpayegan College of Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran0000-0002-5557-463XJournal Article20260129In 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_2091_25c943776b7a94c4731c6adb8825dfdd.pdfDamghan University PressAnalytical and Numerical Solutions for Nonlinear Equations3060-785X11120260607Fuzzy Graph Extensions of Sandpile Monoids and Their Connections to Leavitt Path Algebras130141208610.22128/ansne.2026.3237.1193ENShanookha AliDepartment of General Science, Birla Institute of Technology & Science, Pilani, Dubai Campus, Dubai 345055, United Arab Emirates0000-0001-8720-3764Farshid MofidnakhaeiDepartment of Physics, Sari Branch, Islamic Azad University, Sari, Iran0000-0002-2903-0428Nitha Niralda PCDepartment of Mathematics and Statistics, Providence Women's College, Calicut, Kerala, India0000-0002-5573-3944Shafeequdheen PalengaraDepartment of Mathematics, SRM University, Andhra Pradesh, IndiaJournal Article20260131We 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_2086_20b3d4cb3234ee380492f9d4875d7cd7.pdfDamghan University PressAnalytical and Numerical Solutions for Nonlinear Equations3060-785X11120260607Analytical and Numerical Bounds for a Nonlinear Overlap Model of Circular Sectors142152209410.22128/ansne.2026.3249.1195ENFereshteh TahmasiDepartment of Mathematics, Lorestan University P. O. Box 465, Khoramabad, IranMahmood ShakooriDepartment of Mathematics, Lorestan University
P. O. Box 465, Khoramabad, IranLeila NasiriDepartment of Mathematics, Lorestan University P. O. Box 465, Khoramabad, IranJournal Article20260216In 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_2094_1b46800df46dfa379b5a46f71c50a335.pdfDamghan University PressAnalytical and Numerical Solutions for Nonlinear Equations3060-785X11120260607Advanced Physics-Informed Neural Network with Residuals for Solving Complex Integral Equations153173210910.22128/ansne.2026.3261.1200ENMahdi Movahedian MoghaddamDepartment of Computer and Data Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran0000-0002-6513-5251Kourosh 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 UniversitySaeed Reza KheradpishehDepartment of Computer and Data Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, IranJournal Article20260218In 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_2109_d49a926a5977e1492d7556602484fdb9.pdfDamghan University PressAnalytical and Numerical Solutions for Nonlinear Equations3060-785X11120260607From Quadratic Convergence to Structural Invariance: A Selje Topological Framework for Newton-Type Methods174185209310.22128/ansne.2026.3276.1203ENSelva Nandhini NAssistant Professor Faculty in Mathematics, Dr. Mahalingam College of Engineering and Technology, Pollachi, India0009-0002-1862-6784Maryam AkhoundiClinical Research Development Unit of Rouhani Hospital, Babol University of Medical Sciences, Babol, Iran0000-0002-2419-1732Senbaga Priya KAssistant Professor Faculty in Mathematics, Dr. Mahalingam College of Engineering and Technology, Pollachi, IndiaMostafa NouriJouybariPayame Noor University (PNU) Tehran, Iran0000-0003-2295-1504Farshid MofidnakhaeiDepartment of Physics, Sar.C., Islamic Azad University, Sari, Iran0000-0002-2903-0428Journal Article20260220Newton-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_2093_4247dd41848ce59c482dc54eb75ba048.pdfDamghan University PressAnalytical and Numerical Solutions for Nonlinear Equations3060-785X11120260607Exact Solutions, Convergence, and Stability Analysis of Fractional Diffusion Models with Nonlocal Interactions186203209510.22128/ansne.2026.3284.1206ENAmeer Jabbar MunshidDepartment of Mathematics, Babol Noshirvani University of Technology, Shariati Ave., Babol, IranAzizollah BabakhaniDepartment of Mathematics, Babol Noshirvani University of Technology, Shariati Ave., Babol, IranJournal Article20260226In 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_2095_560a48e1dc1f3239b0a78d588b70d6d7.pdf