41,034 research outputs found

    Numerical Treatment of Non-Linear Fuzzy Integral Equations by Homotopy Perturbation Method

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    The main purpose of this paper is to present an approximation method for solving fuzzy integral equation. The solution of various types of non-linear fuzzy integral equations like non-linear fuzzy Volterra integral equation, non-linear fuzzy Fredholm integral equation and non-linear able fuzzy integral equation is determined by an advanced iterative approach the homotopy perturbation method. The method is discussed in details and it is illustrated by solving some numerical examples. Keywords: Homotopy perturbation method, non-linear fuzzy Volterra integral equations, non-linear fuzzy Fredholm integral equation, non-linear Abel fuzzy integral equations

    On Some Linear and Non-linear Fuzzy Integral Equations by Homotopy Perturbation Method

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    Many mathematical models are contributed to give rise to of linear and nonlinear integral equations. In this paper, we study the performance of recently developed technique homotopy perturbation method by implement on various types of linear and non-linear fuzzy Volterra integral equations of second kind, mixed fuzzy volterra fredholm integral equation and singular fuzzy integral equations. Obtained results show that technique is reliable, efficient and easy to use through recursive relations that involve integrals. Moreover, these particular examples show the reliability and the performance of proposed modifications. Keywords: Homotopy perturbation method, linear fuzzy integral equations, non-linear fuzzy integral equations

    Fuzzification of Fractal Calculus

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    In this manuscript, fractal and fuzzy calculus are summarized. Fuzzy calculus in terms of fractal limit, continuity, its derivative, and integral are formulated. The fractal fuzzy calculus is a new framework that includes fractal fuzzy derivatives and fractal fuzzy integral. In this framework, fuzzy number-valued functions with fractal support are the solutions of fractal fuzzy differential equations. Different kinds of fractal fuzzy differential equations are given and solved

    Fuzzy Stochastic Differential Equations Driven by Semimartingales-Different Approaches

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    The first aim of the paper is to present a survey of possible approaches for the study of fuzzy stochastic differential or integral equations. They are stochastic counterparts of classical approaches known from the theory of deterministic fuzzy differential equations. For our aims we present first a notion of fuzzy stochastic integral with a semimartingale integrator and its main properties. Next we focus on different approaches for fuzzy stochastic differential equations. We present the existence of fuzzy solutions to such equations as well as their main properties. In the first approach we treat the fuzzy equation as an abstract relation in the metric space of fuzzy sets over the space of square integrable random vectors. In the second one the equation is interpreted as a system of stochastic inclusions. Finally, in the last section we discuss fuzzy stochastic integral equations with solutions being fuzzy stochastic processes. In this case the notion of the stochastic Itî’s integral in the equation is crisp; that is, it has single-valued level sets. The second aim of this paper is to show that there is no extension to more general diffusion terms

    On the Lp-spaces techniques in the existence and uniqueness of the fuzzy fractional Korteweg-de Vries equation’s solution

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    In this paper, is proposed the existence and uniqueness of the solution of all fuzzy fractional differential equations, which are equivalent to the fuzzy integral equation. The techniques on LP-spaces are used, defining the LpF F ([0; 1]) for 1≀P≀∞, its properties, and using the functional analysis methods. Also the convergence of the method of successive approximations used to approximate the solution of fuzzy integral equation be proved and an iterative procedure to solve such equations is presented

    Nth-order Fuzzy Differential Equations Under Generalized Differentiability

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    In this paper, the multiple solutions of Nth-order fuzzy differential equations by the equivalent integral forms are considered. Also, an Existence and uniqueness theorem of solution of Nth-order fuzzy differential equations is proved under Nth-order generalized differentiability in Banach space

    Solution of Fuzzy Fredholm Integral Equation via Modified Homotopy Method

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    In this paper, we proposed a modification to the Homotopy method by introducing accelerating parameters for solving fuzzy integral equations.The modified method is employed to find exact solutions for fuzzy Fredholm integral equations . The results imply that the modified method is very simple and effective

    HÀgusad teist liiki integraalvÔrrandid

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    KĂ€esolevas doktoritöös on uuritud hĂ€gusaid teist liiki integraalvĂ”rrandeid. Need vĂ”rrandid sisaldavad hĂ€gusaid funktsioone, s.t. funktsioone, mille vÀÀrtused on hĂ€gusad arvud. Me tĂ”estasime tulemuse sileda tuumaga hĂ€gusate Volterra integraalvĂ”rrandite lahendite sileduse kohta. Kui integraalvĂ”rrandi tuum muudab mĂ€rki, siis integraalvĂ”rrandi lahend pole ĂŒldiselt sile. Nende vĂ”rrandite lahendamiseks me vaatlesime kollokatsioonimeetodit tĂŒkiti lineaarsete ja tĂŒkiti konstantsete funktsioonide ruumis. Kasutades lahendi sileduse tulemusi tĂ”estasime meetodite koonduvuskiiruse. Me vaatlesime ka nĂ”rgalt singulaarse tuumaga hĂ€gusaid Volterra integraalvĂ”rrandeid. Uurisime lahendi olemasolu, ĂŒhesust, siledust ja hĂ€gusust. Ülesande ligikaudseks lahendamiseks kasutasime kollokatsioonimeetodit tĂŒkiti polĂŒnoomide ruumis. TĂ”estasime meetodite koonduvuskiiruse ning uurisime lĂ€hislahendi hĂ€gusust. Nii analĂŒĂŒs kui ka numbrilised eksperimendid nĂ€itavad, et gradueeritud vĂ”rke kasutades saame parema koonduvuskiiruse kui ĂŒhtlase vĂ”rgu korral. Teist liiki hĂ€gusate Fredholmi integraalvĂ”rrandite lahendamiseks pakkusime uue lahendusmeetodi, mis pĂ”hineb kĂ”igi vĂ”rrandis esinevate funktsioonide lĂ€hendamisel TĆĄebÔƥovi polĂŒnoomidega. Uurisime nii tĂ€pse kui ka ligikaudse lahendi olemasolu ja ĂŒhesust. TĂ”estasime meetodi koonduvuse ja lĂ€hislahendi hĂ€gususe.In this thesis we investigated fuzzy integral equations of the second kind. These equations contain fuzzy functions, i.e. functions whose values are fuzzy numbers. We proved a regularity result for solution of fuzzy Volterra integral equations with smooth kernels. If the kernel changes sign, then the solution is not smooth in general. We proposed collocation method with triangular and rectangular basis functions for solving these equations. Using the regularity result we estimated the order of convergence of these methods. We also investigated fuzzy Volterra integral equations with weakly singular kernels. The existence, regularity and the fuzziness of the exact solution is studied. Collocation methods on discontinuous piecewise polynomial spaces are proposed. A convergence analysis is given. The fuzziness of the approximate solution is investigated. Both the analysis and numerical methods show that graded mesh is better than uniform mesh for these problems. We proposed a new numerical method for solving fuzzy Fredholm integral equations of the second kind. This method is based on approximation of all functions involved by Chebyshev polynomials. We analyzed the existence and uniqueness of both exact and approximate fuzzy solutions. We proved the convergence and fuzziness of the approximate solution.https://www.ester.ee/record=b539569

    H ? filtering for stochastic singular fuzzy systems with time-varying delay

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    This paper considers the H? filtering problem for stochastic singular fuzzy systems with timevarying delay. We assume that the state and measurement are corrupted by stochastic uncertain exogenous disturbance and that the system dynamic is modeled by Ito-type stochastic differential equations. Based on an auxiliary vector and an integral inequality, a set of delay-dependent sufficient conditions is established, which ensures that the filtering error system is e?t - weighted integral input-to-state stable in mean (iISSiM). A fuzzy filter is designed such that the filtering error system is impulse-free, e?t -weighted iISSiM and the H? attenuation level from disturbance to estimation error is belowa prescribed scalar.Aset of sufficient conditions for the solvability of the H? filtering problem is obtained in terms of a new type of Lyapunov function and a set of linear matrix inequalities. Simulation examples are provided to illustrate the effectiveness of the proposed filtering approach developed in this paper
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