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Numerical Solution of Algebraic Fuzzy Complex Equations with Complex Fuzzy Coefficients

Journal of Interdisciplinary Mathematics, Accepted.

In this research, we solve fuzzy complex equations with complex fuzzy coefficients by a numerical method. We transform fuzzy complex equations into two separate nonlinear systems of equations that both can be solved by the proposed algorithms. At the end of this paper, we illustrate our method with some numerical examples.

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Applying the Fuzzy CESTAC Method to Find the Optimal Shape Parameter in Solving Fuzzy Differential Equations via RBF-Meshless Method

Soft Computing, 2020, In Press

In this paper, by using the CESTAC method and the CADNA library a procedure is proposed to solve a fuzzy initial value problem based on RBF-meshless methods under generalized H-differentiability. So a reliable approach is presented to determine optimal shape parameter and number of points for RBF-meshless methods. The results reveal that the proposed method is very effective and simple. Also, the numerical accuracy of the method is shown in the tables and figures, and algorithms are given based on the stochastic arithmetic. The examples illustrate the efficiency and importance of using the stochastic arithmetic in place of the floating-point arithmetic.

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The use of CADNA Library to Find Optimal Shape Parameter and Optimal Number of Points in RBF-Meshless Method to Solve Differential Equations

Computational Methods for Differential Equations, 2020, In Press

One of the schemes to find the optimal shape parameter and optimal number of points in the radial basis function (RBF) methods is to apply the stochastic arithmetic (SA) in place of the common floating-point arithmetic (FPA). The main purpose of this work is to introduce a reliable approach based on this new arithmetic to compute the optimal shape parameter and number of points in multiquadric and Gaussian RBF-meshless methods for solving differential equations, in the iterative process. To this end, the CESTAC method is applied. Also, in order to implement the proposed algorithms, the CADNA library is performed. The examples illustrate the efficiency and importance of using this library to validate the results.

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A Fuzzy System Based Active Set Algorithm for the Numerical Solution of the Optimal Control Problem Governed by Partial Differential Equations

European Journal of Control, 2019, In Press

A computational indirect method based on a fuzzy system technique and active set strategy is presented. This new method has been used for the numerical solution of an optimal control problem governed by elliptic partial differential equation. To solve this problem, we start with obtaining the first-order necessary optimality conditions, which contain a mixed variational inequality in function space. Then a fuzzy basis functions technique combined with a primal-dual active set method to derive a new efficient algorithm for solving the given problem. Finally, some numerical examples are given to illustrate the accuracy and efficiency of the proposed technique.

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The Reverse Interpolation and its Application in the Numerical Solutions of Fredholm Integral Equations of the Second Kind

Computational and Applied Mathematics, Vol. 38, Article Number: 179, (2019).

In this paper, we introduce a new interpolation method that is easy to use and its interpolating function can be explicitly expressed. This interpolation method can be used for a wide spectral type of functions, since it has the ability to change the form of the interpolating function. This interpolation inspired by the Shepard type interpolation. We examine the ability of this interpolation to piecewise functions and rational functions and compare it with Lagrange interpolation and linear interpolation by using Bernstein polynomials basis. We applied this interpolation for an integral equation, and we presented a linear system of equations to approximate the solution of a Fredholm integral equation by collocation method. In the collocation method, the Reverse Interpolation expresses the approximate solution in the form of a linear combination of some basic functions. Several examples are presented to illustrate the effectiveness of the proposed method and the numerical results confirm the desired accuracy.

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Developing Two Efficient Adaptive Newton-type Methods with Memory

Mathematical Methods in the Applied Sciences, Vol. , No. 17, (2019) 5687-5695.

In this paper, we derive two general adaptive methods with memory in the class of Newton-type methods by modifying and introducing one and two self accelerators over a variant of Ostrowski’s method. The idea of introducing adaptive self-accelerator (via all the old information for Newton-type methods) is new and efficient in order to obtain a higher high efficiency index. Finally, we provide convergence analysis and numerical implementations to show the feasibility and applicability of the proposed methods.

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Application of Radial Basis Functions in Solving Fuzzy Integral Equations

Neural Computing and Applications, Vol. 31, (2019), 6373–6381.

In the present paper, a numerical method based on radial basis functions (RBFs) is proposed to approximate the solution of fuzzy integral equations. By applying RBF in fuzzy integral equation, a linear system ΨC=G is obtained. Then target function would be approximated by defining coefficient vector C. Error estimation of the method has been shown which is based on exponential convergence rates of RBFs. Finally, validity of the method is illustrated by some examples.

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Scattered Data Approximation of Fully Fuzzy Data by Quasi-interpolation

Iranian Journal of Fuzzy Systems, Vol. 16, No. 3, (2019), 63-72.

Fuzzy quasi-interpolations help to reduce the complexity of solving a linear system of equations compared with fuzzy interpolations. Almost all fuzzy quasi-interpolations are focused on the form of $\widetilde{f}^{*}:\mathbb{R}\rightarrow F(\mathbb{R})$ or $\widetilde{f}^{*}:F(\mathbb{R})\rightarrow \mathbb{R}$.  In this paper, we intend to offer a novel fuzzy radial basis function by the concept of source distance. Then, we will construct a fuzzy linear combination of such basis functions in order to introduce a fully fuzzy quasi-interpolation in the form of $\widetilde{f}^{*}:F(\mathbb{R})\rightarrow F(\mathbb{R})$. Also the error estimation of the proposed method is proved in terms of the fully fuzzy modulus of continuity which will be introduced in this paper. Finally some examples have been given to emphasize the acceptable accuracy of our method.

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An Approximation of Fuzzy Numbers Based on Polynomial Form Fuzzy Numbers

International Journal of Analysis and Applications, ‎ Vol. 16 No. 2, (2018), 290-305.

In this paper, we approximate an arbitrary fuzzy number by a polynomial fuzzy number through minimizing the distance between them. Throughout this work, we used a distance that is a meter on the set of all fuzzy numbers with continuous left and right spread functions. To support our claims analytically, we have proven some theorems and given supplementary corollaries.

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‎Fuzzy Semi-Numbers and a Distance on Them with a Case Study in Medicine

Mathematical Sciences, Vol. 12 No. 1, (2018) 41-52.

In this paper, we present the novel concept of fuzzy semi-numbers. Then, a method for assigning distance between every pair of fuzzy semi-numbers is given. Moreover, it is shown that this distance is a metric on the set of all trapezoidal fuzzy semi-numbers with the same height and is a pseudo-metric on the set of all fuzzy semi-numbers. Also, by utilizing this distance, we propose an approximation of a fuzzy semi-number with given height and apply this approximation method in a medical case study.

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