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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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A new method for solving variable coefficients fractional differential equations based on a hybrid of Bernoulli polynomials and block pulse functions

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2023年第7期46卷 8054-8073页

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

block pulse functions (BPFs) are constant functions on each subinterval and not so smooth. This property makes BPFs incapable of approximating function accurately. Therefore, the existing BPFs method is insufficiently accurate to numerically solve variable coefficients fractional differential equations (VCFDEs). To obtain highly accurate solutions with fewer computational burden, we propose an efficient numerical method to find the approximate solutions of VCFDEs. The proposed method uses a hybrid of Bernoulli polynomials and BPFs (HBPBPFs) to overcome the disadvantage that BPFs are piecewise constant and smooth enough. For this aim, a fractional integral operational matrix of HBPBPFs is derived and used to convert VCFDEs into a system of algebraic equations. The solutions of VCFDEs are obtained by solving the algebraic equations. Some simulation examples are presented to verify the effectiveness of our proposed method. Numerical results from our proposed method are compared with those from the existing BPFs numerical method. It is demonstrated that our method can obtain more accurate approximate solutions than the BPFs method.

关键词： Bernoulli polynomials block pulse functions fractional differential equation operational matrix variable coefficient

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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 fractional differential equations using hybrid Bernoulli polynomials and block pulse functions

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MATHEMATICAL SCIENCES 2021年第3期15卷 293-304页

作者： Zhang, Bo Tang, Yinggan Zhang, Xuguang 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 Zhejiang Peoples R China

block pulse functions are piecewise constant and not smooth enough. Therefore, they offer limited accuracy when used to approximate functions and unable to find highly accurate numerical solutions of fractional differential equations (FDEs). To overcome this problem, we present in this paper a new efficient numerical method for solving FDEs. A hybrid Bernoulli polynomials and block pulse functions operational matrix of fractional order integrals is derived and used to convert the underlying FDEs into a system of algebraic equations. The solutions of the FDEs are obtained by solving the algebraic equations. Simulation examples are given to verify the effectiveness of our proposed method, and the results show that the method is much more efficient and accurate than other known methods.

关键词： Fractional differential equations Numerical solutions Bernoulli polynomials block pulse functions Operational matrix

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Identification of fractional-order systems with both nonzero initial conditions and unknown time delays based on block pulse functions

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MECHANICAL SYSTEMS AND SIGNAL PROCESSING 2022年 169卷 108646-108646页

作者： Sin, Myong-Hyok Sin, Cholmin Ji, Song Kim, Su-Yon Kang, Yun-Hui Kim II Sung Univ Fac Math Pyongyang North Korea Acad Sci Inst Math Pyongyang North Korea

In this paper, we present an effective method for the identification of fractional order system (FOS) with both nonzero initial conditions and unknown time delays based on a construction of suitable separable nonlinear least squares problem including both. The separable nonlinear least squares problem is first projected into the least squares problem only with respect to nonlinear parameters, time delays. Next, a bias compensated recursive least squares identification method is used to estimate the remaining linear parameters involving initial conditions of FOS. The effectiveness of the proposed method is illustrated by several simulations.

关键词： Identification Fractional order system block pulse functions Nonzero initial conditions Time delay Least square regression

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Sliding Mode Control of Nonlinear Systems Under Nonstationary Random Vibrations via an Equivalent Linearization Method Using block pulse functions 1

Sliding Mode Control of Nonlinear Systems Under Nonstationar...

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1st International Nonlinear Dynamics Conference (NODYCON)

作者： Younespour, Amir Cheng, Shaohong Univ Windsor Dept Civil & Environm Engn Windsor ON Canada

ISBN: (纸本)9783030347475

This chapter proposes a sliding mode control (SMC) method for nonlinear systems subjected to stationary and nonstationary excitations. The equivalent linearization (EL) method is commonly used as an effective tool in seeking approximate solutions to nonlinear systems, and SMC has numerous successful applications in the field of nonlinear system control due to its robustness in handling uncertainties in dynamic systems. In the current study, the nonlinear system is first linearized by a proposed block pulse (BP) function-based equivalent linearization method, then a continuous form of SMC method is developed to control the linearized nonlinear system. Numerical simulations reveal that compared to other existing methods, the proposed BP function-based EL method can predict the nonlinear system responses more accurately at lower computational cost, and the chattering-free SMC approach can effectively improve the performance of a nonlinear system when subjected to various types of random excitations including seismic load.

关键词： Equivalent linearization Sliding mode control block pulse functions

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Solving fractional differential equation using block-pulse functions and Bernstein polynomials

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2021年第7期44卷 5501-5519页

作者： Zhang, Jie Tang, Yinggan Liu, Fucai Jin, Zhaopeng Lu, Yao Yanshan Univ Liren Coll Qinhuangdao Hebei Peoples R China Yanshan Univ Inst Elect Engn Qinhuangdao Hebei Peoples R China Yanshan Univ Key Lab Parallel Robot & Mechatron Syst Qinhuangdao Hebei Peoples R China

The method based on block pulse functions (BPFs) has been proposed to solve different kinds of fractional differential equations (FDEs). However, high accuracy requires considerable BPFs because they are piecewise constant and not so smooth. As a result, it increases the dimension of operational matrix and computational burden. To overcome this deficiency, a novel numerical method is developed to solve fractional differential equations. The method is based upon hybrid of BPFs and Bernstein polynomials (HBBPs), which are piecewise smooth. The HBBPs operational matrix of fractional-order integral is derived to reduce the FDEs to a system of algebraic equations. Then the numerical solution of the FDEs is obtained through solving the system of algebraic equations. The convergence analysis is conducted for the suggested scheme, and the upper bound of error of the solution is given. Finally, illustrative examples are presented to demonstrate the validity, applicability, and efficiency of the proposed technique in contrast with other approaches.

关键词： Bernstein polynomials block pulse functions fractional differential equations numerical solution operational matrix

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Parameter identification of fractional order systems with nonzero initial conditions based on block pulse functions

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MEASUREMENT 2020年 158卷 107684-000页

作者： Lu, Yao Tang, Yinggan Zhang, Xuguang Wang, Shuen 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 Zhejiang Peoples R China Hulunbuir Univ Coll Mech & Elect Engn Hailar District 021008 Inner Mongolia Peoples R China

Ignoring the initial conditions in the identification of fractional order systems (FOS) usually results in incorrect parameter estimations. In this paper, we present a method for the online identification of FOS with nonzero initial conditions. The initial conditions are treated as extra parameters and identified together with the system parameters. To reduce the complexity involved with calculating the fractional order derivatives of input and output signals, generalized operational matrices of block pulse functions are adopted. A recursive least squares identification method and a bias compensated recursive least squares identification method are designed to estimate the parameters of FOS without and with noise, respectively. Identification examples are given and the results demonstrate that the identification accuracy is significantly improved if the nonzero initial conditions are considered. (C) 2020 Elsevier Ltd. All rights reserved.

关键词： Identification Fractional order system block pulse functions Recursive least squares Nonzero initial conditions

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Numerical Solution of Two-Dimensional Nonlinear Stochastic Ito-Volterra Integral Equations by Applying block pulse functions

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Advances in Pure Mathematics 2019年第2期9卷 53-66页

作者： Guo Jiang Xiaoyan Sang Jieheng Wu Biwen Li School of Mathematics and Statistics Hubei Normal University Huangshi China

This paper investigates the numerical solution of two-dimensional nonlinear stochastic Itô-Volterra integral equations based on block pulse functions. The nonlinear stochastic integral equation is transformed into a set of algebraic equations by operational matrix of block pulse functions. Then, we give error analysis and prove that the rate of convergence of this method is efficient. Lastly, a numerical example is given to confirm the method.

关键词： block pulse functions Integration Operational Matrix Stochastic It?-Volterra Integral Equations

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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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