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An expansion-iterative method for numerically solving Volterra integral equation of the first kind

COMPUTERS & MATHEMATICS WITH APPLICATIONS 2010年第4期59卷 1491-1499页

作者： Masouri, Z. Babolian, E. Hatamadeh-Varmazyar, S. IAU Dept Math Sci & Res Branch Tehran Iran IAU Dept Elect Engn Sci & Res Branch Tehran Iran

Most integral equations of the first kind are ill-posed, and obtaining their numerical solution often leads to solving a linear system of algebraic equations of a large condition number. So, solving this system is difficult or impossible. For numerically solving Volterra integral equation of the first kind an efficient expansion-iterative method based on the block-pulse functions is proposed. Using this method, solving the first kind integral equation reduces to solving a recurrence relation. The approximate solution is most easily produced iteratively via the recurrence relation. Therefore, computing the numerical solution does not need to solve any linear system of algebraic equations. To show the convergence and stability of the method, some computable error bounds are obtained. Numerical examples are provided to illustrate that the method is practical and has good accuracy. (C) 2009 Elsevier Ltd. All rights reserved.

关键词： First kind Volterra integral equation Expansion method Iterative method block-pulse functions Operational matrix

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A composite collocation method for the nonlinear mixed Volterra-Fredholm-Hammerstein integral equations

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COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION 2011年第3期16卷 1186-1194页

作者： Marzban, H. R. Tabrizidooz, H. R. Razzaghi, M. Mississippi State Univ Dept Math & Stat Mississippi State MS 39762 USA Isfahan Univ Technol Dept Math Esfahan Iran Univ Kashan Fac Sci Dept Math Kashan Iran Amirkabir Univ Technol Dept Appl Math Tehran Iran

This paper presents a computational technique for the solution of the nonlinear mixed Volterra-Fredholm-Hammerstein integral equations. The method is based on the composite collocation method. The properties of hybrid of block-pulse functions and Lagrange polynomials are discussed and utilized to define the composite interpolation operator. The estimates for the errors are given. The composite interpolation operator together with the Gaussian integration formula are then used to transform the nonlinear mixed Volterra-Fredholm-Hammerstein integral equations into a system of nonlinear equations. The efficiency and accuracy of the proposed method is illustrated by four numerical examples. (C) 2010 Elsevier B.V. All rights reserved.

关键词： Mixed Volterra-Fredholm-Hammerstein integral equations Composite collocation method Hybrid functions block-pulse functions Lagrange interpolating polynomials Nonlinear integral equations

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Numerical solution of delay systems containing inverse time by hybrid functions

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APPLIED MATHEMATICS AND COMPUTATION 2006年第1期173卷 535-546页

作者： Wang, XT Harbin Inst Technol Dept Math Harbin 150001 Peoples R China

The hybrid functions consisting of general block-pulse functions and Legendre polynomials are presented to solve delay systems containing inverse time. The direct algorithm for a product of a matrix function and a vector function is given. The general operational matrix is introduced. The delay function and inverse time function are expanded by hybrid functions. The approximate solution of delay systems containing inverse time is derived. Numerical examples illustrate that the algorithms are applicable. (c) 2005 Elsevier Inc. All rights reserved.

关键词： delay systems containing inverse time block-pulse functions Legendre polynomials

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Fractional-order hybrid functions combining simulated annealing algorithm for solving fractional pantograph differential equations

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JOURNAL OF COMPUTATIONAL SCIENCE 2023年 74卷

作者： Zhou, Fengying Xu, Xiaoyong Jiangxi Sci & Technol Normal Univ Sch Math & Comp Sci Nanchang 330036 Jiangxi Peoples R China East China Univ Technol Sch Sci Nanchang 330013 Jiangxi Peoples R China

A new and novel numerical method has been developed based on the fractional-order hybrid functions combining simulated annealing (SA) algorithm for the solutions of fractional pantograph delay differential equations. First of all, a fractional-order hybrid of block-pulse functions and Chebyshev polynomials (FOHBPCs) is defined. With the aid of regularized beta function, the exact formulas of FOHBPCs are derived under the definition of Riemann-Liouville fractional integral. And then, by the properties of FOHBPCs and the exact formulas together with the collocation method, the problem under consideration is simplified into algebraic equations, which are solved by Gaussian elimination method and Picard iteration method for linear and nonlinear cases, respectively. The error analysis of the proposed method is investigated. Due to the importance of the parameter alpha in the fractional-order hybrid functions method, SA algorithm is considered to find the optimal parameter alpha. Finally, the effectiveness and applicability of the suggested method are verified through some numerical examples.

关键词： block-pulse functions Chebyshev polynomials Fractional pantograph differential equation Picard iteration Simulated annealing

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Operational matrices based on hybrid functions for solving general nonlinear two-dimensional fractional integro-differential equations

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COMPUTATIONAL & APPLIED MATHEMATICS 2020年第2期39卷 1-34页

作者： Maleknejad, Khosrow Rashidinia, Jalil Eftekhari, Tahereh Iran Univ Sci & Technol Sch Math Tehran *** Iran

In the present paper, an attempt was made to develop a numerical method for solving a general form of two-dimensional nonlinear fractional integro-differential equations using operational matrices. Our approach is based on the hybrid of two-dimensional block-pulse functions and two-variable shifted Legendre polynomials. Error bound and convergence analysis of the proposed method are discussed. We prove that the order of convergence of our method is O(122M-1NMM!). The presented method is tested by seven test problems to demonstrate the accuracy and computational efficiency of the proposed method and to compare our results with other well-known methods. The comparison highlighted that the proposed method exhibits superior performance than the existing methods, even using a few numbers of bases.

关键词： Two-dimensional nonlinear fractional integro-differential equations Two-variable hybrid functions block-pulse functions Shifted Legendre polynomials Operational matrices Convergence analysis

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A class of nonlinear optimal control problems governed by Fredholm integro-differential equations with delay

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INTERNATIONAL JOURNAL OF CONTROL 2020年第9期93卷 2199-2211页

作者： Marzban, Hamid Reza Rostami Ashani, Mehrdad Isfahan Univ Technol Dept Math Sci Esfahan *** Iran

This paper introduces an efficient numerical scheme for solving a significant class of nonlinear delay optimal control problem governed by Fredholm integro-differential equations. A direct approach based on a combination of block-pulse functions and orthonormal Taylor polynomials is utilised to transcribe the problem under study into a nonlinear programming one. The resulting optimisation problem is then solved by employing the method of Lagrange multipliers. An upper error bound for the hybrid functions with respect to the maximum norm is obtained. In addition, the convergence of the hybrid functions is established. Numerous numerical experiments are carried out to evaluate the performance and effectiveness of the suggested framework.

关键词： block-pulse functions orthonormal Taylor polynomials Fredholm integro-differential equation nonlinear optimal control time-delay system

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Optimal control of linear multi-delay systems based on a multi-interval decomposition scheme

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OPTIMAL CONTROL APPLICATIONS & METHODS 2016年第1期37卷 190-211页

作者： Marzban, Hamid Reza Isfahan Univ Technol Dept Math Sci Esfahan Iran

This paper presents a novel approximation scheme to the numerical treatment of linear time-varying multi-delay systems with a quadratic performance index. A direct approach based on a hybrid of block-pulse functions and Chebyshev polynomials is successfully developed. The operational matrix of delay associated to multi-delay systems is constructed by an efficient manner. The excellent properties of hybrid functions together with the operational matrices of integration, delay, and product are then used to transform the optimal control problem into a mathematical optimization problem whose solution is much more easier than the original one. The procedure described in the current paper can be regarded as a multi-interval decomposition scheme. The convergence of the proposed method is verified numerically. A wide variety of multi-delay systems are investigated to demonstrate the effectiveness and computational efficiency of the proposed numerical scheme. The method has a simple structure, is easy to implement, and provides very accurate solutions. Copyright (C) 2015 John Wiley & Sons, Ltd.

关键词： optimal control multi-delay block-pulse functions Chebyshev polynomials hybrid functions operational matrix

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A hybrid approach to parameter identification of linear delay differential equations involving multiple delays

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INTERNATIONAL JOURNAL OF CONTROL 2018年第5期91卷 1034-1052页

作者： Marzban, Hamid Reza Isfahan Univ Technol Dept Math Sci Esfahan Iran

In this paper, we are concerned with the parameter identification of linear time-invariant systems containing multiple delays. The approach is based upon a hybrid of block-pulse functions and Legendre's polynomials. The convergence of the proposed procedure is established and an upper error bound with respect to the L-2-norm associated with the hybrid functions is derived. The problem under consideration is first transformed into a system of algebraic equations. The least squares technique is then employed for identification of the desired parameters. Several multi-delay systems of varying complexity are investigated to evaluate the performance and capability of the proposed approximation method. It is shown that the proposed approach is also applicable to a class of nonlinear multi-delay systems. It is demonstrated that the suggested procedure provides accurate results for the desired parameters.

关键词： Parameter identification block-pulse functions Legendre's polynomials hybrid functions multi-delay systems least squares technique

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Optimal control of linear multi-delay systems with piecewise constant delays

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IMA JOURNAL OF MATHEMATICAL CONTROL AND INFORMATION 2018年第1期35卷 183-212页

作者： Marzban, Hamid Reza Hoseini, Sayyed Mohammad Isfahan Univ Technol Dept Math Sci POB *** Esfahan Iran

This article is concerned with the numerical solution of optimal control of linear multi-delay systems involving piecewise constant time delays. The proposed approach is based on direct method using a hybrid of block-pulse functions and Legendre polynomials. Combining block-pulse functions and Legendre polynomials allows one to simultaneously make use of the properties of the two mentioned bases. The operational matrices of delay and derivative corresponding to the proposed hybrid functions are constructed. Two error bounds related to hybrid functions are established. In addition, the necessary conditions of optimality are derived. These conditions are expressed in the form of a Pontryagin type maximum principle. The excellent properties of the hybrid functions together with the associated operational matrices are used to transform the delayed optimal control problem into a parameter optimization problem whose solution is much easier than the original one. An illustrative example is investigated to demonstrate the effectiveness of the suggested numerical scheme.

关键词： block-pulse functions Legendre polynomials hybrid functions piecewise constant delay optimal control multi-delay systems necessary optimality conditions

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A composite Chebyshev finite difference method for nonlinear optimal control problems

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COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION 2013年第6期18卷 1347-1361页

作者： Marzban, H. R. Hoseini, S. M. Isfahan Univ Technol Dept Math Sci Esfahan Iran

In this paper, a composite Chebyshev finite difference method is introduced and is successfully employed for solving nonlinear optimal control problems. The proposed method is an extension of the Chebyshev finite difference scheme. This method can be regarded as a non-uniform finite difference scheme and is based on a hybrid of block-pulse functions and Chebyshev polynomials using the well-known Chebyshev-Gauss-Lobatto points. The convergence of the method is established. The nice properties of hybrid functions are then used to convert the nonlinear optimal control problem into a nonlinear mathematical programming one that can be solved efficiently by a globally convergent algorithm. The validity and applicability of the proposed method are demonstrated through some numerical examples. The method is simple, easy to implement and yields very accurate results. (C) 2012 Elsevier B. V. All rights reserved.

关键词： Nonlinear optimal control Composite Chebyshev finite difference block-pulse functions Hybrid functions Chebyshev-Gauss-Lobatto points

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