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Numerical solutions of stochastic delay integro-differential equations by block pulse functions

APPLIED NUMERICAL MATHEMATICS 2025年 208卷 214-230页

作者： Jiang, Guo Chen, Yuanqin Ying, Jiayi Hubei Normal Univ Sch Math & Stat Huangshi Key Lab Metaverse & Virtual Simulat Huangshi 435002 Peoples R China

This paper presents an efficient numerical method for solving nonlinear stochastic delay integrodifferential equations based on block pulse functions. Firstly, the equation is transformed into an algebraic system by the integral delay operator matrixes of block pulse functions. Then, error analysis is conducted on the method. Finally, some numerical examples are provided to validate the method. This work provides numerical solutions for the stochastic delay integro-differential equations by global approximation method. This method has the advantages of simple calculation and higher error accuracy.

关键词： Stochastic delay integro-differential equations block pulse functions Integral delay operator matrixes Numerical analysis

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block pulse functions for solving fractional Poisson type equations with Dirichlet and Neumann boundary conditions

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BOUNDARY VALUE PROBLEMS 2017年第1期2017卷 1页

作者： Xie, Jiaquan Huang, Qingxue Zhao, Fuqiang Gui, Hailian Taiyuan Univ Sci & Technol Coll Mech Engn Taiyuan 030024 Shanxi Peoples R China Taiyuan Univ Technol Coll Mech Engn Taiyuan 030024 Shanxi Peoples R China Shanxi Prov Key Lab Met Device Design Theory & Te Taiyuan Shanxi Peoples R China

In this study, the numerical technique based on two-dimensional block pulse functions (2D-BPFs) has been developed to approximate the solution of fractional Poisson type equations with Dirichlet and Neumann boundary conditions. These functions are orthonormal and have compact support on [0, 1]. The proposed method reduces the original problems to a system of linear algebra equations that can be solved easily by any usual numerical method. The obtained numerical results have been compared with those obtained by the Legendre and CAS wavelet methods. In addition an error analysis of the method is discussed. Illustrative examples are included to demonstrate the validity and robustness of the technique.

关键词： block pulse functions fractional Poisson type equations numerical solution Dirichlet and Neumann boundary conditions error analysis

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RETRACTED: Solution of nonlinear Fredholm integro-differential equations using a hybrid of block pulse functions and normalized Bernstein polynomials (Retracted Article)

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2014年 260卷 258-265页

作者： Behiry, S. H. King Abdulaziz Univ Jeddah Community Coll Gen Required Courses Dept Jeddah 21589 Saudi Arabia

A numerical method for solving nonlinear Fredholm integro-differential equations is proposed. The method is based on hybrid function approximations. The properties of a hybrid of block pulse functions and orthonormal Bernstein polynomials are presented and utilized to reduce the problem to the solution of nonlinear algebraic equations. Numerical examples are introduced to illustrate the effectiveness and simplicity of the present method. (C) 2013 Elsevier B.V. All rights reserved.

关键词： Nonlinear Fredholm integro-differential equations Hybrid functions block pulse functions Bernstein polynomials

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Parameter identification of fractional order systems using block pulse functions

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SIGNAL PROCESSING 2015年 107卷 272-281页

作者： Tang, Yinggan Liu, Haifang Wang, Weiwei Lian, Qiusheng Guan, Xinping Yanshan Univ Sch Informat Sci & Engn Qinhuangdao 066004 Hebei Peoples R China Minist Educ Key Lab Syst Control & Informat Proc Shanghai 200240 Peoples R China Yanshan Univ Inst Elect Engn Qinhuangdao 066004 Hebei Peoples R China

In this paper, a novel method is proposed to identify the parameters of fractional-order systems. The proposed method converts the fractional differential equation to an algebraic one through a generalized operational matrix of block pulse functions. And thus, the output of the fractional system to be identified is represented by a matrix equation. The parameter identification of the fractional order system is converted to a multi-dimensional optimization problem, whose goal is to minimize the error between the output of the actual fractional order system and that of the identified system. The proposed method can simultaneously identify the parameters and the fractional differential orders of the fractional order system and avoid the drawbacks in the literature that the fractional differential orders should be known or commensurate. Furthermore, the proposed method avoids complex calculations of the fractional derivative of input and output signals. Illustrative examples covering both fractional and integer systems are given to demonstrate the validity of the proposed method. (C) 2014 Elsevier B.V. All rights reserved.

关键词： Identification Fractional order system block pulse functions Operational matrix

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Parameter Identification of Fractional Order Systems Using a Hybrid of Bernoulli Polynomials and block pulse functions

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IEEE ACCESS 2021年 9卷 40178-40186页

作者： Zhang, Bo Tang, Yinggan Zhang, Xuguang Zhang, Chunjiang Yanshan Univ Inst Elect Engn Qinhuangdao 066004 Hebei Peoples R China Yanshan Univ LiRen Coll Qinhuangdao 066004 Hebei Peoples R China Hangzhou Dianzi Univ Sch Commun Engn Hangzhou 310018 Peoples R China

block pulse functions (BPFs) are piecewise constant and not sufficiently smooth. Therefore, their accuracy is limited when it comes to identifying the parameters of fractional order systems (FOSs). This in turn means that BPFs are incapable of offering highly accurate parameter identification results. However, using a great number of BPFs would significantly increase the dimension of the operational matrix and thereby adds to the computational complexity and burden. To overcome this problem, we present here a hybrid function method for identifying FOSs. The method utilizes a hybrid of Bernoulli polynomials and block pulse functions (HBPBPFs) as the base functions to approximate input and output signals. The fractional integral operational matrix of HBPBPFs is derived and used to convert an FOS to an algebraic system. The parameters of the FOS are successfully identified by minimizing the mean square error between the output of the true system and that of the algebraic representation of the FOS. The simulation experiment verifies that our proposed HBPBPFs method is effective and can generate more accurate identification results than existing BPFs methods.

关键词： Matrices Frequency-domain analysis Matrix converters Licenses Integral equations Heuristic algorithms Fractional calculus Bernoulli polynomials block pulse functions fractional order system operational matrix parameter identification

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Numerical solution of nonlinear Volterra-Fredholm-Hammerstein integral equations in two-dimensional spaces based on block pulse functions

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2017年 317卷 565-572页

作者： Xie, Jiaquan Huang, Qingxue Zhao, Fuqiang Taiyuan Univ Sci & Technol Coll Mech Engn Taiyuan 030024 Shanxi Peoples R China Taiyuan Univ Technol Coll Mech Engn Taiyuan 030024 Shanxi Peoples R China Shanxi Prov Key Lab Met Device Design Theory & Te Taiyuan Peoples R China

In this paper, the numerical technique based on block pulse functions (BPFs) has been developed to approximate the solutions of nonlinear Volterra-Fredholm-Hammerstein integral equations in two-dimensional spaces. These functions are orthogonal and have compact support on [0, 1]. The proposed method reduces the integral equations to a system of nonlinear algebraic equations that can be easily solved by any numerical method. Also, the convergence of the proposed approach is discussed. Furthermore, in order to show the accuracy and reliability of the above-mentioned algorithm, the new approach is verified through some numerical examples. (C) 2016 Elsevier B.V. All rights reserved.

关键词： block pulse functions Operational matrix Nonlinear integral equation Volterra-Fredholm-Hammerstein

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Active tendons control of structures using block pulse functions

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STRUCTURAL CONTROL & HEALTH MONITORING 2014年第12期21卷 1453-1464页

作者： Ghaffarzadeh, Hosein Younespour, Amir Univ Tabriz Fac Civil Engn Tabriz Iran

Structural active control is one of the most efficient systems available to protect building structures from destructive effects of earthquakes. Active control efficiency is highly dependent on control algorithm. One of the most important and challenging components of structural control is the development of an accurate analytical approach having less computational expenses. block pulse functions have been studied and applied frequently in recent years as a basic set of functions for signal characterizations in systems science and control. The proposed method evolves minimizing computational expenses of analytical approaches by a novel method to suppress the responses of the structure based on block pulse functions. To prove the validity and feasibility of the proposed control method, numerical simulations of earthquake-excited 10-story shear buildings with active tendons are performed, and comparative results are presented. The uncontrolled and controlled responses of structural system are obtained by using the proposed method and compared with linear quadratic regulator method. The results reveal that the proposed method can be beneficial in reducing seismic responses of structures with high accuracy and less computational expenses. Copyright (c) 2014 John Wiley & Sons, Ltd.

关键词： active control active tendon block pulse functions linear quadratic regulator seismic response

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Identification of fractional-order systems with time delays using block pulse functions

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MECHANICAL SYSTEMS AND SIGNAL PROCESSING 2017年 91卷 382-394页

作者： Tang, Yinggan Li, Ning Liu, Minmin Lu, Yao Wang, Weiwei Yanshan Univ Inst Elect Engn Qinhuangdao 066004 Hebei Peoples R China Yanshan Univ Liren Coll Qinhuangdao 066004 Hebei Peoples R China

In this paper, a novel method based on block pulse functions is proposed to identify continuous-time fractional-order systems with time delays. First, the operational matrices of block pulse functions for fractional integral operator and time delay operator are derived. Then, these operational matrices are applied to convert the continuous-time fractional-order systems with time delays to an algebraic equation. Finally, the system's parameters along with the differentiation orders and the time delays are all simultaneously estimated through minimizing a quadric error function. The proposed method reduces the computation complexity of the identification process, and also it does not require the system's differentiation orders to be commensurate. The effectiveness of the proposed method are demonstrated by several numerical examples. (C) 2017 Elsevier Ltd. All rights reserved.

关键词： Identification Fractional order systems Time delays block pulse functions Operational matrix

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OPTIMAL-CONTROL OF DELAY SYSTEMS VIA block pulse functions

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1985年第1期45卷 101-112页

作者： HWANG, C SHIH, YP NATL TAIWAN INST TECHNOL DEPT CHEM ENGN & TECHNOLTAIPEITAIWAN

The concept of coefficient shift matrix is introduced to represent delay variables in block pulse series. The optimal control of a linear delay system with quadratic performance index is then studied via block pulse functions, which convert the problems into the minimization of a quadratic form with linear algebraic equation constraints. The solution of the two-point boundary-value problem with both delay and advanced arguments is circumvented. The control variable obtained is piecewise constant.

关键词： Delay systems optimization block pulse functions linear systems

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A numerical method for solving Fredholm-Volterra integral equations in two-dimensional spaces using block pulse functions and an operational matrix

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2011年第14期235卷 3965-3971页

作者： Babolian, E. Maleknejad, K. Mordad, M. Rahimi, B. Islamic Azad Univ Dept Math Sci & Res Branch Tehran 14778 Iran Iran Univ Sci & Technol Narmak Dept Math Tehran *** Iran

In this paper, the block pulse functions (BPFs) and their operational matrix are used to solve two-dimensional Fredholm-Volterra integral equations (F-VIE). This method converts F-VIE to systems of linear equations whose solutions are the coefficients of block pulse expansions of the solutions of F-VIE. Finally some numerical examples are presented to show the efficiency and accuracy of the method. (C) 2011 Published by Elsevier B.V.

关键词： Two-dimensional Fredholm-Volterra integral equations block pulse functions Operational matrix

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