Chebyshev wavelets operational matrices for solving nonlinear variable-order fractional integral equations
- PDF / 1,533,319 Bytes
- 24 Pages / 595.276 x 793.701 pts Page_size
- 88 Downloads / 266 Views
(2020) 2020:611
RESEARCH
Open Access
Chebyshev wavelets operational matrices for solving nonlinear variable-order fractional integral equations Y. Yang1 , M.H. Heydari2 , Z. Avazzadeh3*
and A. Atangana4,5
*
Correspondence: [email protected] 3 Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam Full list of author information is available at the end of the article
Abstract In this study, a wavelet method is developed to solve a system of nonlinear variable-order (V-O) fractional integral equations using the Chebyshev wavelets (CWs) and the Galerkin method. For this purpose, we derive a V-O fractional integration operational matrix (OM) for CWs and use it in our method. In the established scheme, we approximate the unknown functions by CWs with unknown coefficients and reduce the problem to an algebraic system. In this way, we simplify the computation of nonlinear terms by obtaining some new results for CWs. Finally, we demonstrate the applicability of the presented algorithm by solving a few numerical examples. MSC: 65T60; 33C47; 45G10; 26A33 Keywords: Chebyshev wavelets (CWs); Variable-order (V-O) fractional integral equations; Galerkin method; Operational matrix (OM); Hat functions
1 Introduction Fractional calculus is a useful extension of the classical calculus by allowing derivatives and integrals of arbitrary orders. It arose from a famous scientific discussion between Leibniz and L’Hopital in 1695 and was developed by other scientists like Laplace, Abel, Euler, Riemann, and Liouville [1]. In recent years, fractional calculus has become a popular topic for researchers in mathematics, physics, and engineering because the fractional differential (integral) equations govern the behavior of many physical systems with more precision [2]. We remind that the main advantage of using fractional differential (integral) equations for modeling applied problems is their nonlocal property [3], i.e., in a fractional dynamical system, the next state depends on all the previous situations so far [3]. Another interesting extension to fractional order calculus is considering the fractional order to be a known time-dependent function α(t) [4]. This generalization is called variable-order (V-O) fractional calculus. This subject finds enormous applications in science and engineering because the nonlocal property of fractional calculus becomes more evident [4]. Usually, the V-O fractional functional equations are difficult to solve, analytically. So, finding the exact solutions for these problems is impossible in most cases. Therefore, it is very important to propose approximation/numerical procedures to find © The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or ot
Data Loading...
