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triangular functions for numerical solution of the nonlinear Volterra integral equations

JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2022年第3期68卷 1979-2002页

作者： Kazemi, Manochehr Islamic Azad Univ Dept Math Ashtian Branch Ashtian Iran

In this paper, a new numerical iterative method based on the successive approximations method for solving nonlinear Hammerstein Volterra integral equations of the second kind is proposed. The main approximation tool is based on triangular functions. Also, the convergence analysis and numerical stability of the proposed method are proved. Finally, some numerical examples verify the theoretical results and show the accuracy of the method.

关键词： Iterative method triangular functions Uniform modulus of continuity Successive approximations

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triangular functions based method for the solution of system of linear Fredholm integral equations via an efficient finite iterative algorithm

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JOURNAL OF INTELLIGENT & FUZZY SYSTEMS 2020年第3期38卷 2847-2858页

作者： Ramadan, Mohamed Oshaba, Heba Kharabsheh, Radwan Menoufia Univ Fac Sci Dept Math & Comp Sci Menoufia Egypt Appl Sci Univ Coll Adm Sci Dept Business Adm Eker Bahrain

Integral equation has been one of the essential tools for various areas of applied mathematics. The aim of this article is to propose a simple, accurate and efficient iterative method for obtaining solutions for Fredholm integral equation systems of one dimension of the second kind. The proposed numerical method is based on orthogonal triangular functions. The orthogonal triangular functions (TFs) based method is first applied to transform the Fredhoim system of integral equations to four coupled system of matrix algebraic equations. A finite iterative algorithm is then applied to solve this system to obtain the coefficients used to get the form of an approximate numerical solution of the unknown solution functions of the integral problems. Some examples are given to clarify the efficiency and accuracy of the method. The obtained numerical results are compared with other numerical methods and the exact solutions.

关键词： Fredhoim integral equation system triangular functions finite iterative algorithm efficiency accuracy

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Solving state feedback control of fractional linear quadratic regulator systems using triangular functions

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COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION 2019年第Jul.期73卷 319-337页

作者： Baghani, Omid Hakim Sabzevari Univ Dept Appl Math Fac Math & Comp Sci POB 397 Sabzevar Iran

The present paper is useful for two reasons. First, as far as we know, the right operational matrix of the Riemann-Liouville fractional integral for triangular functions is introduced for the first time. So, hereafter, it can be applied in approximating any fractional optimal control problem in Caputo sense, over and over. Second, the rate of convergence of the triangular functions under a new-blown norm parallel *** to(p,alpha) in its corresponding space, denoted by L-p,L-alpha, is investigated. So far, the error analysis in L-p,L-alpha space has not been studied. In this article, both left and right operational matrices of the triangular functions for arbitrary fractional order integral alpha>0 in Caputo sense are applied to approximate solutions of fractional linear optimal control systems which have a quadratic performance index. The necessary and sufficient optimality conditions are stated in a fractional two-point boundary value problem. This problem is converted to a set of coupled equations involving left and right Riemann-Liouville integral operators. Using these operational matrices of the triangular functions, an extension of the functions of control, state, co-state and other engaged functions is considered. This technique is a successful approach because of reducing such systems to a system of coupled Sylvester equations. Applying the definition of Kroneker product, a linear algebraic system containing the corresponding matrices of the original problem and the operational matrices of fractional order integral can be constructed. Accordingly, fractional linear quadratic optimal control can be solved indirectly. The advantage of this methodology, in addition to simplicity, is its low computational cost and its flexible precision. The simulation results confirm the reliability and validity of this method. (C) 2019 Elsevier B.V. All rights reserved.

关键词： Caputo fractional derivative Optimal control problem Fractional linear quadratic triangular functions Operational matrix of fractional order

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Three-dimensional triangular functions and their applications for solving nonlinear mixed Volterra-Fredholm integral equations

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ALEXANDRIA ENGINEERING JOURNAL 2016年第3期55卷 2943-2952页

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

In this paper, we used the three-dimensional triangular functions (3D-TFs) for the numerical solution of three-dimensional nonlinear mixed Volterra-Fredholm integral equations. First, 3D-TFs and their properties are described. Then the properties of 3D-TFs together with their operational matrix are used to reduce the problem to a nonlinear system of algebraic equations. Furthermore, existence and uniqueness of the solution of three-dimensional nonlinear mixed Volterra-Fredholm integral equations are proved. Illustrative examples have been discussed to demonstrate the validity and applicability of the technique. Also, some interesting comparisons between proposed method, block-pulse functions (BPFs) method and modified block-pulse functions (MBPFs) method are presented. (C) 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://***/licenses/by-nc-nd/4.0/).

关键词： Nonlinear mixed Volterra-Fredholm integral equations triangular functions Operational matrices Orthogonal functions Numerical solution

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Error estimation and numerical solution of nonlinear fuzzy Fredholm integral equations of the second kind using triangular functions

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JOURNAL OF INTELLIGENT & FUZZY SYSTEMS 2016年第2期30卷 639-649页

作者： Baghmisheh, Mahdi Ezzati, Reza Islamic Azad Univ Sci & Res Branch Dept Math Tehran Iran Islamic Azad Univ Karaj Branch Dept Math Karaj Iran

In this paper we present an efficient iterative procedure based on the triangular functions (TFs) to obtain the numerical solution of the specific nonlinear fuzzy Fredholm integral equations of the second kind. The error estimation of the proposed method is given in terms of uniform modulus of continuity. Also, we prove the stability of the method with respect to the choice of the first iteration. Finally, in order to demonstrate the accuracy and the convergence of the method, some numerical examples are included.

关键词： Nonlinear fuzzy Fredholm integral equation iterative procedure triangular functions numerical stability

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The construction of operational matrix of fractional integration using triangular functions

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APPLIED MATHEMATICAL MODELLING 2015年第3-4期39卷 1341-1351页

作者： Maleknejad, K. Asgari, M. Islamic Azad Univ Karaj Branch Dept Math Karaj Iran Islamic Azad Univ Abhar Branch Dept Engn Abhar Iran

In this paper the operational matrix of triangular functions for fractional order integration in the Caputo sense is derived. Also, the operational matrix of fractional integration is applied for solving multi-order fractional differential equations, Abel's integral equations and nonlinear integro-differential equations. This technique is a successful method because of reducing such problems to solve a system of algebraic equations;so, the problem can be solved directly. The advantage of this method is low cost of setting up the equations. Illustrative examples demonstrate accuracy and efficiency of the method. (C) 2014 Elsevier Inc. All rights reserved.

关键词： Abel's integral equations Fractional calculus Integro-differential equation Multi-order fractional differential triangular functions

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Modeling and analysis of singular systems via orthogonal triangular functions

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控制理论与应用（英文版） 2013年第2期11卷 141-148页

作者： Suchismita GHOSH Anish DEB Srimanti Roy CHOUDHURY Gautam SARKAR Department of Electrical Engineering MCKV Institute of Engineering Department of Applied Physics University of Calcutta92 APC RoadKolkata 700009 India Department of Electrical Engineering Budge Budge Institute of Technology Department of Applied Physics University of Calcutta2 APC RoadKolkata 700009 India

This paper presents a new approach to singular system analysis by modeling the system in terms of orthog- onal triangular functions （TFs）. The proposed method is more accurate compared to block pulse function-based analysis with respect to mean integral square error （MISE）. A numerical example involving four states of a singular system is treated and solutions obtained thereof. Four tables and relevant curves are presented to compare the respective coefficients in block pulse function （BPF） domain as well as in TF domain. The percentage error of the samples determined via TF domain are compared with the exact samples of the states. Furthermore, MISE for both BPF and TF analysis are computed and compared to reveal the efficiency of TF-based analysis.

关键词： Singular systems triangular functions Block pulse functions System analysis Modeling

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Fuzzy Representation of Environmental Flow in Multi-Objective Risk Analysis of Reservoir Operation

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WATER RESOURCES MANAGEMENT 2021年第9期35卷 2845-2861页

作者： Li, Jiqing Huang, Jing Liang, Pengteng Lund, Jay R. North China Elect Power Univ Sch Water Resources & Hydropower Engn Beijing 102206 Peoples R China Xiamen Municipal Engn Design Inst Co Ltd Xiamen 361004 Peoples R China Univ Calif Davis Civil & Environm Engn Davis CA 95616 USA

In the context of worldwide water shortage, environmental flow is the key to alleviating the negative environmental impact of reservoir operation. In reality, there exists a range for the stream flow suitable for the survival and reproduction of aquatic organisms. However, most of current studies set it to a fixed value, which leads to unreasonable resource allocations. In this study, we proposed a fuzzy representation of environmental flow by using the fuzzy theory and the ecological hydraulic radius. Furthermore, we used the Three Gorges-Gezhouba cascade reservoirs as a study case and Four Major Chinese Carps as indicator species. In addition, a multi-objective operation optimization model was established, which was solved by the Evolver Palisade software. Finally, a multi-objective risk analysis method was proposed based on the design reliability and risk rate of various benefit operations. The results show that: (1) Based on the environmental flow membership function, flow ranges suitable for the aquatic organism survival and reproduction at specific locations can be determined to guide reservoir discharge. (2) Taking environmental flow membership as an optimization objective rather than a constraint is conducive to formulating environmentally friendly reservoir operation schemes and making more rational use of water resources. (3) The multi-objective risk analysis can avoid the one-sidedness of single-objective risk analysis and provide more basis for reservoir management. Ecological demands have long been a factor considered when formulating reservoir operation schemes. Therefore, following the environmentally friendly operation scheme is helpful to protect the environment and maximize the overall benefits of reservoirs.

关键词： Environmental flow membership function Multi-objective risk analysis triangular functions Ecological hydraulic radius Four Major Chinese Carps (FMCC)

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Accelerations Asi(MM) and Ati(MM) to find double integrals numerically

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JOURNAL OF INTERDISCIPLINARY MATHEMATICS 2021年第3期24卷 705-712页

作者： Mohammed, Ali Hassan Hasoon, Athraa Mohammed Dheyaa Fadhil, Roaa Aziz Univ Kufa Fac Educ Girls Dept Math Kufa Iraq Univ Alkafeel Coll Pharm Dept Pharmaceut & Pharmaceut Chem Kufa 61001 Iraq Gen Directorate Educ Qadisiyah Al Diwaniyah Iraq

This research investigates two methods which use accelerating of Al Tememe's inverse triangular acceleration of the first kind [1], by using "inverse sine triangular acceleration rule for Al-Tememe of the first kind" in the first method, and "inverse tangent triangular acceleration rule for Al-Tememe of the first kind" in the second method;both of them have a major error of the first kind and these are beneficial to find double integrations with the continuous integrands specified numerically by the midpoint rule on both internal and external dimensions x and t respectively by above methods. We named the first method Asi (MM) and the second method Ati (MM), It was also proposed that the number of subintervals on the interior dimension x was the same as the number of subintervals on the exterior dimension t, obtained from the two methods with highly accurate results with relatively few subintervals and speed with using the program Matlab 2017.

关键词： Al-tememe acceleration Definite integrals Midpoint rule triangular functions

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Direct methods for solving time-varying delay systems

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MATHEMATICAL SCIENCES 2020年第2期14卷 159-166页

作者： Zeynal, Elham Babolian, Esmail Damercheli, Tayebeh Islamic Azad Univ Yadegar E Emam Khomeyni RAH Shahr E Rey Branch Coll Sci Dept Math Tehran Iran Kharazmi Univ Fac Math Sci & Comp Tehran Iran

Direct methods are applied to solve linear time-varying delay systems based on vector forms of block-pulse functions (BPFs) and triangular functions (TFs). Operational matrices of integration of BPFs and TFs are applied to transform linear time-varying delay systems to a linear set of algebraic equations. Further, some numerical examples are provided to indicate reliability and the accuracy of these methods. Convergence analysis of the proposed method has been discussed.

关键词： Linear time-varying delay systems Block-pulse functions triangular functions Direct method

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