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Design of Piecewise Constant Gains for Optimal Control via block-pulse functions

IETE Journal of Research 1992年第6期38卷 307-310页

Using block-pulse functions, a method is presented to determine the piecewise constant feed-back controls for a finite linear regulator problem optimal for time multiplied quadratic performance indices. This method gives a complete solution to the time weighted finite and infinite quadratic regulator problem. This is illustrated with two examples.

关键词： Optimal control block-pulse functions Regulator problem

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Numerical solution of nonlinear partial quadratic integro-differential equations of fractional order via hybrid of block-pulse and parabolic functions

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NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS 2019年第3期35卷 1134-1151页

作者： Mirzaee, Farshid Alipour, Sahar Malayer Univ Fac Math Sci & Stat POB 65719-95863 Malayer Iran

In this paper, an effective numerical approach based on a new two-dimensional hybrid of parabolic and block-pulse functions (2D-PBPFs) is presented for solving nonlinear partial quadratic integro-differential equations of fractional order. Our approach is based on 2D-PBPFs operational matrix method together with the fractional integral operator, described in the Riemann-Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations, which greatly simplifies the problem. By using Newton's iterative method, this system is solved, and the solution of fractional nonlinear partial quadratic integro-differential equations is achieved. Convergence analysis and an error estimate associated with the proposed method is obtained, and it is proved that the numerical convergence order of the suggested numerical method is O(h(3)). The validity and applicability of the method are demonstrated by solving three numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the exact solutions much easier.

关键词： block-pulse functions fractional calculus nonlinear quadratic integral equation operational matrix parabolic functions partial differential equation

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Numerical solution based on hybrid of block-pulse and parabolic functions for solving a system of nonlinear stochastic It(o)over-cap-Volterra integral equations of fractional order

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2019年 349卷 157-171页

作者： Mirzaee, Farshid Alipour, Sahar Samadyar, Nasrin Malayer Univ Fac Math Sci & Stat POB 65719-95863 Malayer Iran

In this study, an effective numerical approach based on the hybrid of block-pulse and parabolic functions (PBPFs) is suggested to obtain an approximate solution of a system of nonlinear stochastic It (o) over cap -Volterra integral equations of fractional order. For this aim, we first introduce these functions and express some of their properties and then calculate fractional and stochastic operational matrices of integration based on these functions. Using the properties of PBPFs and obtained operational matrices, the system of nonlinear stochastic It (o) over cap -Volterra integral equations of fractional order converts to a nonlinear system of algebraic equations which can be easily solved by using Newton's method. Moreover, in order to show the rate of convergence of the suggested approach, we present several theorems on convergence analysis and error estimation which demonstrate the rate of convergence of the proposed method for solving this nonlinear system is O(h(3)). Finally, two examples are included to illustrate the validity, applicability and efficiency of the proposed technique. (C) 2018 Elsevier B.V. All rights reserved.

关键词： Parabolic functions block-pulse functions Stochastic integral equations Brownian motion process Fractional calculus Operational matrix

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Theoretical error analysis and validation in numerical solution of two-dimensional linear stochastic Volterra-Fredholm integral equation by applying the block-pulse functions

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COGENT MATHEMATICS 2017年第1期4卷 0-0页

作者： Fallahpour, M. Khodabin, M. Maleknejad, K. Islamic Azad Univ Karaj Branch Dept Math Karaj Iran

In this paper, we introduce an efficient method based on two-dimensional block-pulse functions (2D-BPFs) to approximate the solution of the 2D-linear stochastic Volterra-Fredholm integral equation. Also, we present convergence analysis of the proposed method. Illustrative examples are included to demonstrate the validity and applicability of the proposed method.

关键词： block-pulse functions 2D-integral equation stochastic integral equation Volterra-Fredholm integral equation operational matrix Brownian motion process Ito integral

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A comparative study of numerical integration based on block-pulse and sinc functions and Chebyshev wavelet

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MATHEMATICAL SCIENCES 2012年第1期6卷 8-8页

作者： Rostami, Majid Hashemizadeh, Elham Heidari, Mohammd Islamic Azad Univ Dept Math Naragh Branch Naragh Iran Islamic Azad Univ Karaj Branch Young Res Club 2 Karaj Iran

In this paper, numerical integration rules based on block-pulse functions and Chebyshev wavelet are proposed to find approximate values of definite integrals. Errors of these numerical integrations are given. These numerical integrations are compared by sinc functions numerical integration method. Some numerical examples are provided to illustrate the accuracy of proposed rules and comparison between them. The main advantage of proposed numerical integration methods are their efficiency and simple applicability.

关键词： Numerical integration block-pulse functions Sinc functions Chebyshev wavelet

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Numerical solution of linear time-varying differential equations using the hybrid of block-pulse and rationalized Haar functions

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JOURNAL OF VIBRATION AND CONTROL 2006年第10期12卷 1081-1092页

作者： Arabzadeh, B. Razzaghi, M. Ordokhani, Y. Amirkabir Univ Technol Dept Appl Math Tehran Iran Mississippi State Univ Dept Math & Stat Mississippi State MS 39762 USA Alzahra Univ Dept Math Tehran Iran

A numerical method for finding the solution of linear time-varying differential equations is proposed. The properties of the hybrid functions, which consist of block-pulse functions plus rationalized Haar functions, are presented. The properties of the hybrid functions together with the operational matrices of integration and product are then utilized to reduce the solution of differential equations to the solution of algebraic equations. Examples are included to demonstrate the validity and applicability of the technique.

关键词： hybrid rationalized Haar functions block-pulse functions differential equations

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A reliable numerical algorithm mixed with hypergeometric function for analyzing fractional variational problems

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NUMERICAL ALGORITHMS 2025年第4期98卷 2081-2112页

作者： Zarvan, Z. Sayevand, K. Ganji, R. M. Jafari, H. Malayer Univ Fac Math Sci & Stat Malayer Iran Postdoctoral Researcher Univ Mazandaran Babolsar Iran Univ South Africa Dept Math Sci UNISA0003 Pretoria South Africa

The present study aims to introduce a numerical approach based on the hybrid of block-pulse functions (BPFs), Bernoulli polynomials (BPs), and hypergeometric function for analyzing a class of fractional variational problems (FVPs). The FVPs are made by the Caputo derivative sense. To analyze this problem, first, we create an approximate for the Riemann-Liouville fractional integral operator for BPFs and BPs of the fractional order. In this framework and using the Gauss-Legendre points, the main problem is converted into a system of algebraic equations. In the follow-up, an accurate upper bound is obtained and some theorems are established on the convergence analysis. Moreover, the computational order of convergence and solvability of the proposed approach are displayed and approximated theoretically and numerically. Meanwhile, the thrust of the proposed scheme is compared with other sophisticated examples in the literature, demonstrating that the process is accurate and efficient.

关键词： Bernoulli polynomial block-pulse functions Fractional variational problems Gauss-Legendre points Hypergeometric function

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A hybrid analysis direct method in the calculus of variations

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2000年第3期75卷 259-269页

作者： Razzaghi, M Marzban, HR Mississippi State Univ Dept Math & Stat Mississippi State MS 39762 USA Amirkabir Univ Dept Math Tehran Iran

This paper presents a direct method for finding the solution of variational problems using a hybrid function. The properties of hybrid functions which consists of block-pulse functions plus Legendre polynomials are presented. An operational matrix of integration and the cross product of two hybrid function vectors are used to reduce a variational problem to the solution of algebraic equation. Illustrative examples are included to demonstrate the validity and applicability of the technique.

关键词： calculus of variations direct method hybrid functions Legendre polynomials block-pulse functions

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Numerical expansion-iterative method for analysis of integral equation models arising in one- and two-dimensional electromagnetic scattering

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2012年第3期36卷 416-422页

作者： Hatamzadeh-Varmazyar, Saeed Masouri, Zahra Islamic Azad Univ Islamshahr Branch Dept Elect Engn Tehran Iran Islamic Azad Univ Khorramabad Branch Dept Math Khorramabad Iran

Most integral equations of the first kind are ill-posed, and obtaining their numerical solution needs often to solve a linear system of algebraic equations of large condition number. So, solving this system may be difficult or impossible. Since many problems in one- and two-dimensional scattering from perfectly conducting bodies can be modeled by Fredholm integral equations of the first kind, this paper presents an effective numerical expansion-iterative method for solving them. This method is based on vector forms of block-pulse functions. By using this approach, solving the first kind integral equation reduces to solve a recurrence relation. The approximate solution is most easily produced iteratively via the recurrence relation. Therefore, computing the numerical solution does not need to directly solve any linear system of algebraic equations and to use any matrix inversion. Also, the method practically transforms solving of the first kind Fredholm integral equation which is inherently ill-posed into solving second kind Fredholm integral equation. Another advantage is low cost of setting up the equations without applying any projection method such as collocation, Galerkin, etc. To show convergence and stability of the method, some computable error bounds are obtained. Test problems are provided to illustrate its accuracy and computational efficiency, and some practical one- and two-dimensional scatterers are analyzed by it. (C) 2011 Elsevier Ltd. All rights reserved.

关键词： One- and two-dimensional electromagnetic scattering Perfectly conducting bodies First kind Fredholm integral equation Expansion method Iterative method block-pulse functions Vector forms

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Optimal Control of Singular Systems via Orthogonal functions

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INTERNATIONAL JOURNAL OF CONTROL AUTOMATION AND SYSTEMS 2011年第1期9卷 145-150页

作者： Mohan, Bosukonda M. Kar, Sanjeeb Kumar Indian Inst Technol Dept Elect Engn Kharagpur 721302 W Bengal India SOA Univ Inst Tech Educ & Res Dept Elect Engn Bhubaneswar 751030 Orissa India

In this paper two new recursive algorithms are presented for computing optimal control law of linear time-invariant singular systems with quadratic performance index by using the elegant properties of block-pulse functions (BPFs) and shifted Legendre polynomials (SLPs). Also a unified approach is given to solve the optimal control problem of singular systems via BPFs or SLPs. Two numerical examples are included to demonstrate the validity of the proposed algorithms and approach.

关键词： block-pulse functions optimal control orthogonal functions shifted legendre polynomials singular systems

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