Using Bernstein multi-scaling polynomials to obtain numerical solution of Volterra integral equations system
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Using Bernstein multi-scaling polynomials to obtain numerical solution of Volterra integral equations system A. R. Yaghoobnia1 · R. Ezzati1 Received: 2 November 2019 / Revised: 5 May 2020 / Accepted: 14 May 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper, first, Bernstein multi-scaling polynomials (BMSPs), which are generalization of Bernstein polynomials (BPs), are introduced and some of their properties are explained. Then, a new method based on BMSPs to achieved numerical solution for system of nonlinear integral equations is proposed. The proposed method converted the system of integral equations to a nonlinear system. To evaluate the efficiency of the proposed method, some systems of nonlinear integral equations are solved, and their numerical solutions are compared with other similar methods. Keywords Bernstein multi-scaling polynomials · Numerical solution · Volterra integral equation · System of integral equation · Operational matrix of integration Mathematics Subject Classification 45D05 · 45G15 · 65Rxx · 41A58
1 Introduction In many branches of sciences, modeling of many phenomena is performed through applying integral equations. For example, in different trends of physics such as theoretical physics, molecular physics, and nanoparticles, we encounter a variety of integral equations (Olshevsky 1930; Lee 1997; Borówko et al. 2017; Semkow and Li 2018). Also there are a variety of applications in engineering topics such as electromagnetic waves, cracking problems, etc., which are expressed by integral equations (Tsokos and Padgett 1974; Ladopoulos 1994; Sladek et al. 2003; Ladopoulos 2013). Due to the complexity of the research problems, these integral equations may appear in the form of systems of integral equations. For example, Dobri¸toiu and Serban ¸ (2010) and Debnath (2011) are applications of integral equations systems in the field of infectious diseases and wave propagation, respectively. Some of these
Communicated by Hui Liang.
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R. Ezzati [email protected] A. R. Yaghoobnia [email protected]
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Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran 0123456789().: V,-vol
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systems have analytical solutions (Shidfar and Molabahrami 2011), but, in many cases, their analytical solution cannot be obtained. Hence, it is necessary to calculate an approximation of such solutions through numerical methods. There are several methods to obtain numerical solutions of integral equations systems, that in Chen et al. (2005), some of these methods are classified. Also researchers presented various methods to obtain numerical solution of the integral equations system, namely the Haar wavelet method (Chen and Hsiao 1997), the Adomian method (Biazar et al. 2003; Goghary et al. 2005), the expansion method to solve the linear Volterra integral equations system (Rabbani et al. 2007), the block pulse functions (Maleknejad et al. 2005; Mirzaee 2016), Simpson’s rule method (Kılıçman et
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