Applying the fuzzy CESTAC method to find the optimal shape parameter in solving fuzzy differential equations via RBF-mes
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METHODOLOGIES AND APPLICATION
Applying the fuzzy CESTAC method to find the optimal shape parameter in solving fuzzy differential equations via RBF-meshless methods Hasan Barzegar Kelishami1 · Mohammad Ali Fariborzi Araghi1 · Majid Amirfakhrian1
© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract 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. Keywords Fuzzy differential equation (FDE) · Radial basis function (RBF) · Stochastic arithmetic · CESTAC method · CADNA library
1 Introduction The concept of fuzzy derivative was first offered by Chang and Zadeh in Chang and Zadeh (1972). It was followed up by Dubois and Prade in Dubois and Prade (1982), who defined and used the extension principle. A comprehensive approach to FIVPs has been the works of Seikkala (1987), Kaleva (1990) and Buckley and Feuring (2001). Recently, Bede has presented a strongly generalized differentiability of fuzzy functions (Bede and Gal 2005). Then, Abbasbandy and Allahviranloo in Abbasbandy et al. (2004), Abbasbandy and Allahviranloo (2004), Abbasbandy and Allahviranloo (2002a), Abbasbandy and Allahviranloo (2002b), Allahviranloo et al. (2007), Allahviranloo et al. (2008) have proposed a variety of methods to solving FIVPs. More specifically, in order to obtain numerical solutions of fuzzy differential equations under Hukuhara differentiability, it is not necessary to rewrite the whole literature on numerical solutions of ODEs in the fuzzy setting, but instead we can use any Communicated by V. Loia.
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Mohammad Ali Fariborzi Araghi [email protected]; [email protected] Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran 19558-47881, Iran
numerical method for the ODEs directly (Bede 2008). Since the RBF-meshless method is used in this work, it is natural to begin by presenting a background of RBF method, in particular determining the shape parameter . The RBFmeshless method, first presented by Kansa (1990a, b) , is well known for solving systems of partial differential equations with excellent accuracy. Sarra and Sturgill (2009) presented a random variable shape parameter strategy and applied it to interpolations and two-dimensional linear elliptic boundary value problems. Also, Luh (2014) worked on finding optimal shape parameter in the interpolation functions. The current authors in Barzegar Kelishami et al. (2020) applied the CESTAC method to find the optimal shape pa
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