A tau method based on Jacobi operational matrix for solving fractional telegraph equation with Riesz-space derivative
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A tau method based on Jacobi operational matrix for solving fractional telegraph equation with Riesz-space derivative Samira Bonyadi1 · Yaghoub Mahmoudi1 · Mehrdad Lakestani2 · Mohammad Jahangiri Rad1 Received: 7 February 2020 / Revised: 29 June 2020 / Accepted: 11 July 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper, we have presented an accurate and impressive spectral algorithm for solving fractional telegraph equation with Riesz-space derivative and Dirichlet boundary conditions. The proposed method is based on Jacobi tau spectral procedure together with the Jacobi operational matrices of Riemann–Liouville fractional integral and left- and right-sided Caputo fractional derivatives. Primarily, we implement the proposed algorithm in both temporal and spatial discretizations. This algorithm reduces the problem to a system of algebraic equations which considerably simplifies the problem. In addition, an error bound is established in the L ∞ -norm for the suggested spectral Jacobi tau method. Illustrative examples are included to demonstrate the validity and accuracy of the presented technique. Keywords Fractional telegraph equation · Operational matrix · Shifted Jacobi tau method · Riesz fractional derivative · Error bound
1 Introduction In the last decades, fractional calculus has been utilized in science and engineering such as signal processing, electromagnetism, diffusion processes, fluid mechanics and electrochem-
Communicated by José Tenreiro Machado.
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Yaghoub Mahmoudi [email protected] Samira Bonyadi [email protected] Mehrdad Lakestani [email protected] Mohammad Jahangiri Rad [email protected]
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Mathematics Department, Tabriz Branch, Islamic Azad University, Tabriz, Iran
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Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran 0123456789().: V,-vol
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istry (Bhrawy and Zaky 2016; Doha et al. 2012; Nikan et al. 2019a, 2020; Kilbas et al. 2006; Podlubny 1999). Introduction theory of fractional derivatives and fractional differential equations (FDEs) have been introduced in Kilbas et al. (2006), Podlubny (1999) and Changpin and Fanhai (2015). So far, many authors derived that fractional order of differential equations are more suitable than integer order ones, because fractional derivatives describe the memory and hereditary properties of diverse materials and processes. Recently, FDEs have gained much interest in many research areas such as engineering, physics, chemistry and other branches of science (Nikan et al. 2019a, b; Kilbas et al. 2006; Changpin and Fanhai 2015; Machado et al. 2011). In the last few years, several procedures have been used for solving FDEs such as finite differences, finite elements (Atangana 2015; Chen et al. 2008; Sweilam et al. 2011; Deng 2008; Zhao and Li 2012) and spectral methods (Safdari et al. 2020; Bhrawy et al. 2015, 2016b, a, 2017; Bhrawy and Zaky 2015; Borhanifar and Sadri 2014; Golbabai et al. 2019). Spe
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