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A Novel Method of Accurate Calculation of the Magnetic Dipole Field in Both the Near and Far Fields 近场和远场磁偶极子场精确计算的新方法
IF 1.6 4区 工程技术 Q3 ENGINEERING, ELECTRICAL & ELECTRONIC Pub Date : 2025-06-11 DOI: 10.1002/jnm.70068
Jiaqi Liu, Guoqiang Wang, Chaobo Liu, Qiancheng Zhang, Lifei Meng, Zhong Yi, Qi Xiao, Tielong Zhang

The magnetic dipole serves as a fundamental concept in understanding electromagnetic phenomena. It has extensive applications across various fields such as geophysics and indoor navigation, which require accurate determination of its magnetic field. Although the magnetic dipole approximation yields satisfactory results in the far field, its computational accuracy is poor in the near-field region. Here, we propose a method of accurately calculating the magnetic dipole field in both the near and far fields. This method encompasses three steps: first, calculating the magnetic field strength BT at the position r; second, determining the direction of the magnetic field at r; and third, calculating three components of the magnetic field. Numerical tests show that the calculation error of BT is < 1% at r > 1.2 R, and is < 0.1% at r > 10 R, where R is the radius of the magnetic dipole. Additionally, the magnetic field direction can be precisely modeled via multi-parameter fitting, yielding angular errors < 0.1° in most regions at r > 1.2 R. Integration of the direction and BT enables us to accurately calculate three components of the magnetic field with an error of < 1% at r > 1.8 R. These results indicate that our method is able to achieve high accurate calculation of the magnetic dipole field in both the near and far fields. This method can provide an effective computational algorithm for the applications relying on magnetic dipoles.

磁偶极子是理解电磁现象的一个基本概念。它在地球物理和室内导航等需要精确测定其磁场的各个领域都有广泛的应用。磁偶极子近似在远场得到满意的结果,但在近场计算精度较差。本文提出了一种精确计算近场和远场磁偶极子场的方法。该方法分为三个步骤:首先,计算r位置的磁场强度BT;第二,确定r处磁场的方向;第三,计算磁场的三个分量。数值试验表明,在r >; 1.2 r处,BT的计算误差为<; 1%,在r >; 10 r处,BT的计算误差为<; 0.1%,其中r为磁偶极子半径。此外,通过多参数拟合可以精确地模拟磁场方向,在r >; 1.2 r下,大多数区域的角误差为<; 0.1°。方向和BT的积分使我们能够准确地计算出磁场的三个分量,在r >; 1.8 r时误差为<; 1%。这些结果表明,我们的方法能够在近场和远场实现高精度的磁偶极子场计算。该方法可为依赖磁偶极子的应用提供一种有效的计算算法。
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引用次数: 0
Enhanced Pixel Antenna Design and Optimization Through Dynamic Updating of Initial Structure 基于初始结构动态更新的改进像素天线设计与优化
IF 1.6 4区 工程技术 Q3 ENGINEERING, ELECTRICAL & ELECTRONIC Pub Date : 2025-06-09 DOI: 10.1002/jnm.70067
Fan Jiang, Huiyao Tan, Lulu Chen, Liang Hua Ye, Jian-Feng Li, Duo-Long Wu

Dynamic updating technique for initial structure in pixel antenna design and optimization is proposed. The conventional approach to pixel antenna design employs a fixed initial pixel structure set at the start of the entire process, while rarely studying the setting of the initial structure; therefore, the performance potential is not fully exploited. The proposed approach adaptively updates the initial structure to enhance the performance of the pixel antenna design, aiming to find the optimal initial pixel structure that achieves miniaturization and broadband capabilities. In general, the design procedure starts with an initial structure with relatively big element size and small overall size, then gradually reduces the element size and expands the overall size of the pixel area. A two-port pixel antenna is used as a design example to validate the proposed updating technique. The goal was to design a dual-port pixel antenna operating in the band of 2.4–3.2 GHz, using a miniaturized size. After two rounds of updates, the obtained −10 dB impedance bandwidths increased from 0.44 GHz (2.47–2.91GHz) to 0.75 GHz (2.45–3.20 GHz) and to 1.07 GHz (2.35–3.42 GHz), while having isolation better than −15 dB. The statistical results of 10 optimization runs for 3 initial structures also showed the performance enhancement of each updated initial structure. The proposed updated technology can be applied to other types of pixel antenna designs, with different design specifications.

提出了像素天线初始结构的动态更新技术。传统的像元天线设计方法在整个设计过程开始时设置固定的初始像元结构,很少对初始结构的设置进行研究;因此,性能潜力没有得到充分利用。该方法自适应更新初始结构,提高像素天线设计的性能,旨在找到实现小型化和宽带性能的最优初始像素结构。一般来说,设计过程是从一个元素尺寸比较大,整体尺寸较小的初始结构开始,然后逐渐减小元素尺寸,扩大像素区域的整体尺寸。以双端口像素天线为设计实例,验证了该方法的有效性。目标是设计一种工作在2.4-3.2 GHz频段的双端口像素天线,使用小型化的尺寸。经过两轮更新,获得的−10 dB阻抗带宽从0.44 GHz (2.47-2.91GHz)增加到0.75 GHz (2.45-3.20 GHz)和1.07 GHz (2.35-3.42 GHz),同时隔离性优于−15 dB。对3个初始结构进行10次优化的统计结果也显示了每次更新初始结构的性能提升。提出的更新技术可以应用于其他类型的像素天线设计,具有不同的设计规范。
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引用次数: 0
A Hyperaccurate Semi–Analytical Method With Error Bound Analysis for Treating Fractional Integral Equations With Functional Kernels and Variable Delays 含变时滞泛函核分数阶积分方程的高精度半解析误差界分析方法
IF 1.6 4区 工程技术 Q3 ENGINEERING, ELECTRICAL & ELECTRONIC Pub Date : 2025-06-04 DOI: 10.1002/jnm.70061
Ömür Kıvanç Kürkçü

This study is concerned with treating the fractional integral equations with functional kernels and variable delays, introducing a hyperaccurate semi–analytical method based on the Stieltjes–Wigert polynomials, matrix expansions, and the Laplace transform. After analytically converting the terms in the governing equation into the matrix expansions of the Stieltjes–Wigert polynomials type at the collocation points, the method gathers these matrices into a unique matrix equation and then readily solves it by an elimination technique. The residual improvement technique is also introduced to correct the obtained solutions. The residual error bound analysis is theoretically proved via algebraical properties and the mean value theorem for fractional integral calculus, respectively. Six model equations are treated via the method, which runs on a devised computer program. Based on the outcomes, the method is straightforward to treat model equations and to encode its mainframe on a mathematical software.

本文研究了具有泛函核和变时滞的分数阶积分方程,引入了一种基于stieltje - wigert多项式、矩阵展开式和拉普拉斯变换的高精度半解析方法。该方法将控制方程中的项解析转化为stieltje - wigert多项式型的矩阵展展开,将这些矩阵集合成一个唯一的矩阵方程,然后利用消元技术进行求解。并介绍了残差改进技术对得到的解进行修正。分别利用分数阶积分的代数性质和中值定理从理论上证明了残差界分析。用该方法处理了6个模型方程,并在设计的计算机程序上运行。根据结果,该方法可以直接处理模型方程并在数学软件上对其主机进行编码。
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引用次数: 0
An Accurate Dynamic Model Identification Method for Industrial Robots Based on Improved Excitation Trajectory 基于改进激励轨迹的工业机器人动态模型精确辨识方法
IF 1.6 4区 工程技术 Q3 ENGINEERING, ELECTRICAL & ELECTRONIC Pub Date : 2025-06-04 DOI: 10.1002/jnm.70062
Xiao Lin, Junyang Li, Yankui Song, Yogendra Arya, Yu Xia

This article focuses on dynamic parameter identification for industrial robots and proposes a parameter identification method based on an improved excitation trajectory. First, a complex nonlinear friction model is adopted and modified according to joint friction characteristics, with a genetic algorithm utilized to determine its six parameters. Second, a weighted optimal excitation trajectory is designed to address nonlinear friction requirements and smooth operation constraints. Then, a global parameter optimization algorithm based on the least squares method and the modified sparrow search algorithm is proposed. Finally, the proposed method is validated on a self-developed six-axis industrial robot. Experimental results demonstrate that the proposed method achieves higher identification accuracy compared with two representative identification approaches.

针对工业机器人的动态参数辨识问题,提出了一种基于改进激励轨迹的参数辨识方法。首先,采用复杂非线性摩擦模型,并根据关节摩擦特性对其进行修正,利用遗传算法确定其6个参数;其次,设计了加权最优激励轨迹,以满足非线性摩擦要求和平滑运行约束。然后,提出了一种基于最小二乘法和改进的麻雀搜索算法的全局参数优化算法。最后,在自行研制的六轴工业机器人上对该方法进行了验证。实验结果表明,与两种具有代表性的识别方法相比,该方法具有较高的识别精度。
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引用次数: 0
Special Issue: The 13th International Symposium on Electric and Magnetic Fields (EMF 2023) 特刊:第十三届国际电磁场研讨会(EMF 2023)
IF 1.6 4区 工程技术 Q3 ENGINEERING, ELECTRICAL & ELECTRONIC Pub Date : 2025-06-04 DOI: 10.1002/jnm.70066
Christophe Geuzaine, André Nicolet

On behalf of the Editorial Board, we are pleased to present a selection of papers related to the 13th International Symposium on Electric and Magnetic Fields (EMF 2023), held in Marseille, France, on August 29–31, 2023.

The EMF symposium series, whose first edition was held in Liège, Belgium, in 1992, aims at building a bridge between recent research advances in mathematical and numerical modeling of electromagnetic fields and the growing number of industrial problems requiring such techniques. The 13th edition was organized at Aix-Marseille Université and attracted 75 participants from 16 countries.

Among the 65 presentations in the symposium program, 13 papers were selected for publication in this special issue of the International Journal of Numerical Modelling: Electronic Networks, Devices and Fields. The high scientific and technical quality of the symposium is well reflected in the quality of the manuscripts contained in this special issue.

Three broad topics were covered during the 13th edition of the symposium. The first major topic considers the mathematical modeling of electromagnetic problems in view of their eventual numerical solution on computers, with contributions on the calculation of forces [1], homogenization [2] and material models [3, 4]. The second major topic treats general methodological advances in numerical methods, from quasi-static [5, 6] to high-frequency problems [7, 8]. The third major topic is the application of modeling to the study, design, and optimization of a wide range of technological devices, spanning low- to high-frequency electromagnetic regimes [9-13]. This underlines the spirit of the conference, which encompasses both theory and practical applications.

We express our gratitude to the members of the EMF scientific committee—Florian Bentivegna, Oszkár Bró, Markus Clemens, Stéphane Clénet, Willie Cronje, Luc Dupré, Johan Gyselinck, Kay Hameyer, Lauri Kettunen, Vincent Mazauric, Gérard Meunier, Axel Modave, Ronan Perrussel, Martin Petrun, Adel Razek, Maurizio Repetto, Ruth Sabariego, Sebastian Schöps, Jan Sykulski—as well as to all the reviewers who provided the necessary volunteer time and expertise to conduct a fair and detailed review, ensuring high publication standards for the selected manuscripts. We also thank the staff from the Association des Ingénieurs de Montefiore (AIM), and especially Céline Dizier and Louisa Kara, for their help organizing the EMF symposium series.

We hope that you will enjoy reading this selection of articles.

谨代表编辑委员会,我们很高兴地介绍与第13届国际电磁场研讨会(EMF 2023)有关的论文选集,该研讨会将于2023年8月29日至31日在法国马赛举行。EMF专题讨论会系列的第一版于1992年在比利时li日举行,其目的是在电磁场数学和数值模拟方面的最新研究进展与需要这种技术的日益增多的工业问题之间建立一座桥梁。第13届会议在艾克斯-马赛大学举办,吸引了来自16个国家的75名与会者。在研讨会的65篇报告中,有13篇论文被选中发表在本期《国际数值模拟杂志:电子网络,设备和领域》的特刊上。这次专题讨论会的高科学和技术质量很好地反映在本期特刊所载手稿的质量上。第十三届研讨会讨论了三个广泛的议题。第一个主要课题考虑了电磁问题的数学建模,考虑了它们最终在计算机上的数值解,并对力[1]、均匀化[2]和材料模型的计算做出了贡献[3,4]。第二个主要主题涉及数值方法的一般方法学进展,从准静态问题[5,6]到高频问题[7,8]。第三个主要主题是将建模应用于研究、设计和优化广泛的技术设备,涵盖低至高频电磁状态[9-13]。这突出了会议的精神,它包括理论和实际应用。我们感谢EMF科学委员会的成员——florian Bentivegna, Oszkár Bró, Markus Clemens, stsamphane cl, Willie Cronje, Luc dupraud, Johan Gyselinck, Kay Hameyer, Lauri Kettunen, Vincent Mazauric, gsamrard Meunier, Axel Modave, Ronan Perrussel, Martin Petrun, Adel Razek, Maurizio Repetto, Ruth Sabariego, Sebastian Schöps, Jan sykulski——以及所有提供必要的志愿时间和专业知识来进行公平和详细审查的审审员。确保所选稿件的高出版标准。我们还要感谢Montefiore ingsamnieurs协会(AIM)的工作人员,特别是csamline Dizier和Louisa Kara,感谢他们帮助组织了EMF系列研讨会。我们希望你会喜欢阅读这些精选的文章。
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引用次数: 0
Long Short-Term Memory (LSTM)-Based Modeling of Negative Bias Temperature Instability (NBTI) in 40 nm MOSFETs 基于LSTM的40 nm mosfet负偏置温度不稳定性(NBTI)建模
IF 1.6 4区 工程技术 Q3 ENGINEERING, ELECTRICAL & ELECTRONIC Pub Date : 2025-06-02 DOI: 10.1002/jnm.70059
Fikret Başar Gencer, Xhesila Xhafa, Ali Doğuş Güngördü, Mustafa Berke Yelten

Bias temperature instability (BTI) is a time-based degradation mechanism that causes serious damage to the performance of analog and digital integrated circuits. The increasingly probabilistic nature of this phenomenon renders machine learning-based modeling approaches more advantageous, as they can deliver more accurate results in that context compared to analytical methods. In this paper, the Long Short-Term Memory (LSTM) method, a time-series approach, has been adopted to model BTI in 40 nm CMOS p-type metal-oxide-semiconductor field-effect transistors (MOSFETs). The aging model has been established by training the experimental data collected from a dedicated test chip. A bi-directional LSTM structure has been employed in model generation. Mean-square error (MSE) results indicate that the model can be effectively utilized in interpolation exercises where the test data falls within the same interval as the training data, with great accuracy. Moreover, the model has yielded promising outcomes in extrapolation exercises where the test data lies outside the defined training range. This property potentially qualifies the proposed approach for time-to-market and cost-reduction efforts.

偏置温度不稳定性(BTI)是一种基于时间的退化机制,对模拟和数字集成电路的性能造成严重损害。这种现象的概率性越来越高,这使得基于机器学习的建模方法更具优势,因为与分析方法相比,它们可以在这种情况下提供更准确的结果。本文采用一种时间序列方法——长短期记忆(LSTM)方法对40 nm CMOS p型金属氧化物半导体场效应晶体管(mosfet)中的BTI进行了建模。通过对专用测试芯片采集的实验数据进行训练,建立了老化模型。模型生成采用双向LSTM结构。均方误差(MSE)结果表明,该模型可以有效地用于测试数据与训练数据在同一区间内的插值练习,具有较高的精度。此外,该模型在测试数据位于定义的训练范围之外的外推练习中产生了有希望的结果。这一特性可能使所提议的方法符合上市时间和降低成本的要求。
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引用次数: 0
Design of Compact and Quad Band Gap Coupled Ring-Shape Microstrip Patch Antenna for WLAN/ISM/WiMAX/5G Applications 用于WLAN/ISM/WiMAX/5G应用的紧凑型四带隙耦合环形微带贴片天线设计
IF 1.6 4区 工程技术 Q3 ENGINEERING, ELECTRICAL & ELECTRONIC Pub Date : 2025-06-02 DOI: 10.1002/jnm.70060
Surjeet Raikwar, Akanksha Gupta, Karunesh Srivastava, Maninder Singh, Nishant Anand, Ramesh Kumar Verma

In this paper, a gap coupled ring-shape multi-band antenna of quad-band characteristics has been designed. The geometry of the proposed gap coupled ring-shape antenna is obtained by loading two horizontal and six vertical strips of the same width and the same gap. The quad band characteristic of the proposed antenna is obtained at frequencies 2.48, 3.85, 5.14, and 5.61 GHz. The quad band lies within frequency ranges 2.34–2.59 GHz (first band), 3.56–4.06 GHz (second band), 5.09–5.21 GHz (third band) and 5.47–5.96 GHz (fourth band). The second and fourth bands of the antenna arise due to outer vertical strips, while the first and third bands of the antenna are due to middle and inner vertical strips, respectively. The return loss of −17.31, −29.18, −18.40, and − 19.19 dB is obtained at frequencies 2.48, 3.85, 5.14, and 5.61 GHz, respectively. Additionally, a parametric analysis is also conducted to enhance the performance of the proposed gap coupled antenna. To validate the geometry of the proposed antenna, experimental measurement is performed with the prototype antenna. The first band (2.34–2.59 GHz) of the antenna covers WLAN, ISM band, and WiMAX; the second band (3.56–4.06 GHz) of the antenna covers 5G; the third band (5.09–5.21 GHz) of the antenna covers WLAN; and the fourth band (5.47–5.96 GHz) of the antenna covers WiMAX and WLAN applications. The proposed quad band antenna is a cost-effective and efficient solution for multi-band communication devices. It eliminates the need for multiple antennas and lowers hardware costs.

本文设计了一种具有四波段特性的间隙耦合环形多波段天线。通过加载相同宽度和相同间距的两条水平带和六条垂直带,得到了该缝隙耦合环形天线的几何形状。在频率为2.48、3.85、5.14和5.61 GHz时获得了天线的四频特性。四频段包括2.34-2.59 GHz(第一频段)、3.56-4.06 GHz(第二频段)、5.09-5.21 GHz(第三频段)和5.47-5.96 GHz(第四频段)。天线的第二和第四波段是由外部垂直带产生的,而天线的第一和第三波段分别是由中间和内部垂直带产生的。在频率为2.48、3.85、5.14和5.61 GHz时,回波损耗分别为- 17.31、- 29.18、- 18.40和- 19.19 dB。此外,还进行了参数分析,以提高所提出的间隙耦合天线的性能。为了验证所提出的天线的几何形状,用原型天线进行了实验测量。天线第一频段(2.34-2.59 GHz)覆盖WLAN、ISM频段和WiMAX;天线的第二频段(3.56-4.06 GHz)覆盖5G;天线的第三频段(5.09-5.21 GHz)覆盖WLAN;天线的第四个频段(5.47-5.96 GHz)覆盖WiMAX和WLAN应用。提出的四频带天线是一种经济高效的多频带通信器件解决方案。它消除了对多个天线的需求,并降低了硬件成本。
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引用次数: 0
Wearable Textile Antenna With Low SAR and High Fidelity for BAN and Medical Applications 用于BAN和医疗应用的低SAR高保真可穿戴纺织天线
IF 1.6 4区 工程技术 Q3 ENGINEERING, ELECTRICAL & ELECTRONIC Pub Date : 2025-05-29 DOI: 10.1002/jnm.70058
Raghav Dwivedi, D. K. Srivastava, Vinod Kumar Singh

This paper presents a novel wearable antenna fabricated using denim material, designed for flexible electronics and medical monitoring applications. The proposed antenna leverages common jean fabric as the substrate material, offering a cost-effective and readily available solution while combining esthetic appeal with practical functionality through its unique configuration. Operating across 2.269–19.42 GHz with a maximum gain of 6.75 dB, the antenna achieves an enhanced bandwidth of 158.15%. Notably, the design measured as 0.488λo× 0.488λo × 0.008λo exhibits a low specific absorption rate (SAR) compared to FCC standards that is 1.6 W/kg averaged over 1 g of tissue, making it particularly suitable for medical monitoring applications. We obtained a maximum SAR value for the antenna as 1.61, 1.01 W/kg for 1 and 10 g at 2 mm from the body phantom, 0.488 and 0.769 W/kg for 1 and 10 g when placed at 5 mm from the human phantom, and 1.02, 0.73 W/kg for 1 and 10 g at on-body placement of the antenna. Experimental results demonstrate the antenna's effectiveness for vital signs surveillance while maintaining wearer safety and comfort. The use of denim as the substrate material not only ensures flexibility and durability but also provides an eco-friendly approach by utilizing common textile materials. The high-fidelity factor and wideband characteristics ensure reliable data transmission, making this design a promising solution for next-generation wearable healthcare devices.

本文提出了一种新型的可穿戴天线,采用牛仔布材料制成,设计用于柔性电子和医疗监测应用。该天线利用普通牛仔织物作为基板材料,通过其独特的配置将美学吸引力与实用功能结合起来,提供了一种具有成本效益且易于获得的解决方案。该天线工作在2.269-19.42 GHz之间,最大增益为6.75 dB,实现了158.15%的带宽增强。值得注意的是,该设计的测量值为0.488λo × 0.488λo × 0.008λo,与FCC标准的1.6 W/kg平均超过1 g组织相比,显示出较低的比吸收率(SAR),使其特别适合医疗监测应用。我们获得的天线最大SAR值分别为:距离人体幻影2毫米处1和10克时的1.61、1.01 W/kg,距离人体幻影5毫米处1和10克时的0.488和0.769 W/kg,距离人体1和10克时的1.02、0.73 W/kg。实验结果表明,该天线在保持佩戴者安全和舒适的同时,对生命体征监测是有效的。使用牛仔布作为衬底材料不仅确保了灵活性和耐用性,而且通过使用常见的纺织材料提供了一种环保的方法。高保真度因素和宽带特性确保可靠的数据传输,使该设计成为下一代可穿戴医疗保健设备的有前途的解决方案。
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引用次数: 0
Bernoulli Collocation Method for Solving Time-Fractional Diffusion Equation Arising in Physics 求解物理中时间分数扩散方程的伯努利配点法
IF 1.6 4区 工程技术 Q3 ENGINEERING, ELECTRICAL & ELECTRONIC Pub Date : 2025-05-26 DOI: 10.1002/jnm.70052
Jalil Rashidinia, Arefeh Momeni

This research presents an effective spectral collocation scheme based on orthogonalized Bernoulli polynomials for solving the time-fractional diffusion equation (TFDE). To provide a numerical method, we consider the Bernoulli polynomials and estimate the derivatives as well as the Caputo fractional derivative by operational matrices. By collocating the discretized equations, we obtain a system of algebraic equations. By solving this system, we obtain the approximate solution. The advantages of the suggested method are its low computational cost and exponential convergence. Also, the convergence analysis of the presented method is discussed. Finally, we present several test problems to demonstrate the capability of the proposed method. The obtained results are compared with the existing methods in the literature.

提出了一种有效的基于正交伯努利多项式的谱配置方案,用于求解时间分数阶扩散方程。为了提供一种数值方法,我们考虑伯努利多项式,并利用运算矩阵估计其导数和卡普托分数阶导数。通过对离散方程进行配置,得到了一个代数方程组。通过求解这个方程组,我们得到了近似解。该方法的优点是计算成本低,具有指数收敛性。并对该方法的收敛性进行了分析。最后,我们提出了几个测试问题来证明所提出方法的能力。所得结果与文献中已有的方法进行了比较。
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引用次数: 0
New Technique Based on Vieta–Lucas Polynomials for Solving Nonlinear Stochastic Itô-Volterra Integral Equation 基于Vieta-Lucas多项式求解非线性随机Itô-Volterra积分方程的新技术
IF 1.6 4区 工程技术 Q3 ENGINEERING, ELECTRICAL & ELECTRONIC Pub Date : 2025-05-26 DOI: 10.1002/jnm.70044
Narges Barzegar, Farshid Mirzaee, Erfan solhi

In the present study, we introduce an iterative technique grounded in shifted Vieta–Lucas polynomials for the numerical solution of nonlinear stochastic Volterra integral equations. Notably, our iterative approach is fast and provides solutions without solving algebraic equations. This method addresses nonlinear problems with high accuracy, making it very useful. We present an error estimation for the suggested approach, theoretically confirming its accuracy. Several numerical examples illustrate the practicality and efficacy of our technique. Furthermore, we compare the numerical outcomes of our method with those reported in existing literature and, whenever available, with exact solutions. This comparative analysis affirms the practicality and high precision of the suggested approach.

在本研究中,我们引入了一种基于移位的Vieta-Lucas多项式的迭代技术来求解非线性随机Volterra积分方程的数值解。值得注意的是,我们的迭代方法是快速的,并提供解决方案,而不需要求解代数方程。该方法求解非线性问题精度高,非常实用。我们给出了该方法的误差估计,从理论上证实了其准确性。数个算例说明了该方法的实用性和有效性。此外,我们将我们的方法的数值结果与现有文献中报道的结果进行比较,并在可用的情况下与精确解进行比较。通过对比分析,证实了该方法的实用性和较高的精度。
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引用次数: 0
期刊
International Journal of Numerical Modelling-Electronic Networks Devices and Fields
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