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Original Localized meshless methods based on polynomial basis functions for solving axisymmetric equations

MATHEMATICS AND COMPUTERS IN SIMULATION 2020年 177卷 487-499页

作者： Chang, Wanru Chen, C. S. Liu, Xiao-Yan Li, J. Zhejiang Univ Sch Math Sci Hangzhou Zhejiang Peoples R China Univ Southern Mississippi Dept Math Hattiesburg MS 39406 USA Taiyuan Univ Technol Coll Data Sci Taiyuan Peoples R China Changsha Univ Sci & Technol Sch Math & Stat Changsha Peoples R China

In this paper, two localized meshless methods based on polynomial basis functions are utilized to solve axisymmetric problems. In the first approach, we applied the localized method of particular solutions (LMPS) and the closed-form particular solution to simplify the two-stage approach using Chebyshev polynomial as the basis functions for solving axisymmetric problems. We also propose the modified local Pascal polynomial method (MLPM) to compare the results with LMPS. Since only the low order polynomial basis functions are used, no preconditioning treatment is required and the solution is quite stable. Four numerical examples are given to demonstrate the effectiveness of the proposed methods.

关键词： Particular solutions Axisymmetric problems polynomial basis functions Localized method of particular solutions

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An efficient method of approximate particular solutions using polynomial basis functions

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2020年 111卷 1-8页

作者： Deng, Cheng Zheng, Hui Fu, Mingfu Xiong, Jingang Chen, C. S. Nanchang Univ Sch Civil Engn & Architecture Nanchang 330031 Jiangxi Peoples R China Commun Design & Res Inst Co Ltd Jiangxi Prov Nanchang Jiangxi Peoples R China Nanchang Inst Technol Nanchang 330029 Jiangxi Peoples R China Gongqing Inst Sci & Technol Jiujiang 332020 Peoples R China Univ Southern Mississippi Dept Math Hattiesburg MS 39406 USA

The most challenging task of the method of approximate particular solutions (MAPS) is the generation of the closed-form particular solutions with respect to the given differential operator using various basis functions. These particular solutions have to be generated prior to the solution process of the partial differential equations. In this paper, we propose a different approach without the tedious and inefficient solution procedure using symbolic computation to produce the closed-form particular solutions. The proposed approach is introduced and extended to solve a large class of elliptic partial differential equations (PDEs) based on the method of approximate particular solutions (MAPS). Numerical results show the proposed approach is simple, efficient, accurate, and stable. Five different numerical examples are presented to demonstrate the effectiveness of the proposed method.

关键词： Method of approximate particular solutions Helmholtz-type equation polynomial basis functions Multiple scale technique

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Localized method of particular solutions using polynomial basis functions for solving two-dimensional nonlinear partial differential equations

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Partial Differential Equations in Applied Mathematics 2021年 4卷

作者： Dangal, T. Khatri Ghimire, B. Lamichhane, A.R. Department of Mathematics and Computer Science Alcorn State University Lorman 39096 MS United States Department of Mathematics and Computer Science Alabama State University Montgomery 36104 AL United States School of Science Technology and Mathematics Ohio Northern University Ada 45810 OH United States

The localized method is one of the popular approaches in solving large-scale problems in science and engineering. In this paper, we implement the localized method of particular solutions using polynomial basis functions for solving various nonlinear problems. To validate our proposed numerical method, we present four numerical examples in regular and irregular domains which are solved by using localized method of particular solution with polynomial basis functions. We compared our numerical method with localized method of particular solutions using multiquadric radial basis function and numerical results clearly show that our numerical method is highly accurate, efficient, and outperformed the method using multiquadric radial basis function. © 2021 The Authors

关键词： Localized method of particular solutions Multiquadric Nonlinear problems Particular solutions polynomial basis functions Radial basis functions

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Adaptive fractional physics-informed neural networks for solving forward and inverse problems of anomalous heat conduction in functionally graded materials

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INTERNATIONAL JOURNAL OF HEAT AND MASS TRANSFER 2025年 236卷

作者： Ma, Xingdan Qiu, Lin Zhang, Benrong Wu, Guozheng Wang, Fajie Qingdao Univ Coll Mech & Elect Engn Natl Engn Res Ctr Intelligent Elect Vehicle Power Qingdao 266071 Peoples R China Qingdao Univ Sch Math & Stat Qingdao 266071 Peoples R China Navy Submarine Acad Qingdao 266000 Peoples R China

In this paper, adaptive fractional physics-informed neural networks (PINNs) are employed to solve the forward and inverse problems of anomalous heat conduction in three-dimensional functionally graded materials. In adaptive fractional PINNs, the finite difference L1 scheme is employed to discretize the time fractional derivatives in anomalous heat conduction problems. By combining the finite difference L1 scheme with automatic differentiation technique, adaptive fractional PINNs minimize a loss function constructed based on the governing equation, initial and boundary conditions to obtain solutions to heat conduction problems. To avoid competition among various loss terms during the training process, an adaptive loss balancing algorithm is adopted to balance the interactions among different terms of the loss functions. In addition, polynomial basis functions are used to expand unknown functions characterizing material parameters or heat sources, aiming to enhance the performance of adaptive fractional PINNs in resolving inverse problems. Five numerical examples involving forward, inverse, and nonlinear problems related to anomalous heat conduction are carried out to demonstrate the feasibility and effectiveness of adaptive fractional PINNs.

关键词： Fractional physics-informed neural networks Anomalous heat conduction polynomial basis functions Adaptive loss balancing algorithm Functionally graded materials

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An investigation of radial basis functions for fractional derivatives and their applications

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COMPUTATIONAL MECHANICS 2020年第2期65卷 475-486页

作者： Liu, Qingxia Zhuang, Pinghui Liu, Fawang Lai, Junjiang Anh, Vo Chen, Shanzhen Xiamen Univ Sch Math Sci Xiamen 361005 Fujian Peoples R China Xiamen Univ Fujian Prov Key Lab Math Modeling & High Performa Xiamen 361005 Fujian Peoples R China Queensland Univ Technol Sch Math Sci Brisbane Qld Australia Minjiang Univ Coll Math & Data Sci Fuzhou 350108 Fujian Peoples R China Swinburne Univ Technol Fac Sci Engn & Technol POB 218 Hawthorn Vic 3122 Australia Southwestern Univ Finance & Econ Sch Econ Math Swufe Chengdu 611130 Sichuan Peoples R China

In the present study, the radial basis functions (RBF) are combined with polynomial basis functions to approximate the fractional derivatives specifically. We explore two new types of local support fields for the first time in the literature. In addition, we apply the RBF, combined with an appropriate number of polynomial basis functions featuring a new type of local support field to solve the single-term, multi-term ordinary fractional equations, time-space and two-side space fractional partial differential equations. Finally, the accuracy of the presented method is demonstrated via numerical experiments.

关键词： Fractional derivatives Mixed RBF Local support fields polynomial basis functions Jacobi-Gauss-Lobatto quadratures

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Higher Order Hybrid FEM-MoM Technique for Analysis of Antennas and Scatterers

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION 2009年第5期57卷 1452-1460页

作者： Ilic, Milan M. Djordjevic, Miroslav Ilic, Andjelija Z. Notaros, Branislav M. Univ Belgrade Sch Elect Engn Belgrade 11120 Serbia Colorado State Univ Dept Elect & Comp Engn Ft Collins CO 80523 USA ICT Coll Belgrade 11000 Serbia Vinca Inst Nucl Sci Lab Phys 010 Belgrade 11001 Serbia

A novel higher order large-domain hybrid computational electromagnetic technique based on the finite element method (FEM) and method of moments (MoM) is proposed for three-dimensional analysis of antennas and scatterers in the frequency domain. The geometry of the structure is modeled using generalized curved parametric hexahedral and quadrilateral elements of arbitrary geometrical orders. The fields and currents on elements are modeled using curl- and divergence-conforming hierarchical polynomial vector basis functions of arbitrary approximation orders, and the Galerkin method is used for testing. The elements can be as large as about two wavelengths in each dimension. As multiple MoM objects are possible in a global exterior region, the MoM part provides much greater modeling versatility and potential for applications, especially in antenna problems, than just as a boundary-integral closure to the FEM part. The examples demonstrate excellent accuracy, convergence, efficiency, and versatility of the new FEM-MoM technique, and very effective large-domain meshes that consist of a very small number of large flat and curved FEM and MoM elements, with p-refined field and current distributions of high approximation orders. The reduction in the number of unknowns is by two orders of magnitude when compared to available data for low-order FEM-MoM modeling.

关键词： Antennas curved parametric elements electromagnetic analysis finite element method (FEM) higher order modeling hybrid methods method of moments (MoM) numerical techniques polynomial basis functions scattering

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Higher order frequency-domain computational electromagnetics

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION 2008年第8期56卷 2251-2276页

作者： Notaros, Branislav M. Colorado State Univ Dept Elect & Comp Engn Ft Collins CO 80523 USA

A review of the higher order computational electromagnetics (CEM) for antenna, wireless, and microwave engineering applications is presented. Higher order CEM techniques use current/field basis functions of higher orders defined on large (e.g., on the order of a wavelength in each dimension) curvilinear geometrical elements, which greatly reduces the number of unknowns for a given problem. The paper reviews all major surface/volume integral- and differential-equation electromagnetic formulations within a higher order computational framework, focusing on frequency-domain solutions. With a systematic and unified review of generalized curved parametric quadrilateral, triangular, hexahedral, and tetrahedral elements and various types of higher order hierarchical and interpolatory vector basis functions, in both divergence- and curl-conforming arrangements, a large number of actual higher order techniques, representing various combinations (of formulations, elements, bases, and solution procedures, are identified and discussed. The examples demonstrate the accuracy, efficiency, and versatility of higher order techniques, and their advantages over low-order discretizations, the most important one being a much faster (higher order) convergence of the solution. It is demonstrated that both components of the higher order modeling, namely, higher order geometrical modeling and higher order current/field modeling are essential for accurate and efficient CEM analysis of general antenna, scattering, and microwave structures.

关键词： antennas curved parametric elements differential-equation techniques electromagnetic analysis fast methods finite element method (FEM) higher order modeling hybrid methods integral-equation techniques method of moments (MoM) numerical techniques polynomial basis functions scattering

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A numerical technique for solving a class of fractional variational problems

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2013年第1期237卷 633-643页

作者： Lotfi, A. Yousefi, S. A. Shahid Beheshti Univ Dept Math GC Tehran Iran

This paper presents a numerical method for solving a class of fractional variational problems (FVPs) with multiple dependent variables, multi order fractional derivatives and a group of boundary conditions. The fractional derivative in the problem is in the Caputo sense. In the presented method, the given optimization problem reduces to a system of algebraic equations using polynomial basis functions. An approximate solution for the RIP is achieved by solving the system. The choice of polynomial basis functions provides the method with such a flexibility that initial and boundary conditions can be easily imposed. We extensively discuss the convergence of the method and finally present illustrative examples to demonstrate validity and applicability of the new technique. (C) 2012 Elsevier B.V. All rights reserved.

关键词： Fractional variational problem Caputo fractional derivative polynomial basis functions

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Chaotic dynamics and convergence analysis of temporal difference algorithms with bang-bang control

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OPTIMAL CONTROL APPLICATIONS & METHODS 2016年第1期37卷 108-126页

作者： Tutsoy, Onder Brown, Martin Univ Manchester Control Syst Ctr Sch Elect & Elect Engn Manchester M13 9PL Lancs England Adana Sci & Technol Univ Elect & Elect Engn Dept Seyhan Adana Turkey

Reinforcement learning is a powerful tool used to obtain optimal control solutions for complex and difficult sequential decision making problems where only a minimal amount of a priori knowledge exists about the system dynamics. As such, it has also been used as a model of cognitive learning in humans and applied to systems, such as humanoid robots, to study embodied cognition. In this paper, a different approach is taken where a simple test problem is used to investigate issues associated with the value function's representation and parametric convergence. In particular, the terminal convergence problem is analyzed with a known optimal control policy where the aim is to accurately learn the value function. For certain initial conditions, the value function is explicitly calculated and it is shown to have a polynomial form. It is parameterized by terms that are functions of the unknown plant's parameters and the value function's discount factor, and their convergence properties are analyzed. It is shown that the temporal difference error introduces a null space associated with the finite horizon basis function during the experiment. The learning problem is only non-singular when the experiment termination is handled correctly and a number of (equivalent) solutions are described. Finally, it is demonstrated that, in general, the test problem's dynamics are chaotic for random initial states and this causes digital offset in the value function learning. The offset is calculated, and a dead zone is defined to switch off learning in the chaotic region. Copyright (C) 2015 John Wiley & Sons, Ltd.

关键词： badly conditioned learning polynomial basis functions rate of convergence temporal difference learning value function approximation

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A numerical approach for fractional partial differential equations by using Ritz approximation

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APPLIED MATHEMATICS AND COMPUTATION 2018年 338卷 711-721页

作者： Firoozjaee, M. A. Yousefi, S. A. Univ Sci & Technol Mazandaran Dept Math Behshahr *** Iran Shahid Beheshti Univ Dept Math GC Tehran Iran

In this article, Ritz approximation have been employed to obtain numerical solutions of fractional partial differential equations (FPDEs) based on the Caputo fractional derivative. Transforming fractional partial differential equations into optimization problem and using polynomial basis functions, we obtain the system of algebraic equation. Then, we solve the system of nonlinear algebraic equation using Mathematica7 and we have the coefficients of polynomial expansion. We extensively discuss the convergence of the method. Some numerical examples are presented which illustrate the theoretical results and the performance of the method. (C) 2018 Elsevier Inc. All rights reserved.

关键词： Fractional partial differential equations Caputo fractional derivative The initial and boundary value fractional problem polynomial basis functions

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