Numerical Solution of the Two-Sided Space–Time Fractional Telegraph Equation Via Chebyshev Tau Approximation
The operational matrices of left Caputo fractional derivative, right Caputo fractional derivative, and Riemann–Liouville fractional integral, for shifted Chebyshev polynomials, are presented and derived. We propose an accurate and efficient spectral algorithm for the numerical solution of the two-si...
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| Vydáno v: | Journal of optimization theory and applications Ročník 174; číslo 1; s. 321 - 341 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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New York
Springer US
01.07.2017
Springer Nature B.V |
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| ISSN: | 0022-3239, 1573-2878 |
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| Abstract | The operational matrices of left Caputo fractional derivative, right Caputo fractional derivative, and Riemann–Liouville fractional integral, for shifted Chebyshev polynomials, are presented and derived. We propose an accurate and efficient spectral algorithm for the numerical solution of the two-sided space–time Caputo fractional-order telegraph equation with three types of non-homogeneous boundary conditions, namely, Dirichlet, Robin, and non-local conditions. The proposed algorithm is based on shifted Chebyshev tau technique combined with the derived shifted Chebyshev operational matrices. We focus primarily on implementing the novel algorithm both in temporal and spatial discretizations. This algorithm reduces the problem to a system of algebraic equations greatly simplifying the problem. This system can be solved by any standard iteration method. For confirming the efficiency and accuracy of the proposed scheme, we introduce some numerical examples with their approximate solutions and compare our results with those achieved using other methods. |
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| AbstractList | The operational matrices of left Caputo fractional derivative, right Caputo fractional derivative, and Riemann-Liouville fractional integral, for shifted Chebyshev polynomials, are presented and derived. We propose an accurate and efficient spectral algorithm for the numerical solution of the two-sided space-time Caputo fractional-order telegraph equation with three types of non-homogeneous boundary conditions, namely, Dirichlet, Robin, and non-local conditions. The proposed algorithm is based on shifted Chebyshev tau technique combined with the derived shifted Chebyshev operational matrices. We focus primarily on implementing the novel algorithm both in temporal and spatial discretizations. This algorithm reduces the problem to a system of algebraic equations greatly simplifying the problem. This system can be solved by any standard iteration method. For confirming the efficiency and accuracy of the proposed scheme, we introduce some numerical examples with their approximate solutions and compare our results with those achieved using other methods. |
| Author | Machado, José A. Tenreiro Bhrawy, Ali H. Zaky, Mahmoud A. |
| Author_xml | – sequence: 1 givenname: Ali H. surname: Bhrawy fullname: Bhrawy, Ali H. email: alibhrawy@yahoo.co.uk organization: Department of Mathematics, Faculty of Science, Beni-Suef University – sequence: 2 givenname: Mahmoud A. surname: Zaky fullname: Zaky, Mahmoud A. organization: Department of Applied Mathematics, National Research Centre – sequence: 3 givenname: José A. Tenreiro surname: Machado fullname: Machado, José A. Tenreiro organization: Department of Electrical Engineering, Institute of Engineering, Polytechnic of Porto |
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| Keywords | Operational matrix Fractional telegraph equation Riesz fractional derivative Fractional Klein–Gordon equation Shifted Chebyshev Tau method |
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B20062007791803221525210.1142/S02179792060336201101.37025 AH Bhrawy (863_CR20) 2015 A Atangana (863_CR15) 2015; 293 EH Doha (863_CR44) 2011; 62 M Dehghan (863_CR32) 2011; 27 S Shen (863_CR38) 2011; 56 F Liu (863_CR39) 2012; 64 YX Zhang (863_CR40) 2014; 260 AH Bhrawy (863_CR19) 2015 WY Tian (863_CR25) 2014; 30 ASVRavi Kanth (863_CR37) 2009; 180 E Orsingher (863_CR29) 2003; 24B E Orsingher (863_CR28) 2004; 128 D Kumar (863_CR4) 2015 E Shivanian (863_CR24) 2015 C Li (863_CR5) 2006; 20 XL Ding (863_CR26) 2013; 14 WH Deng (863_CR14) 2008; 47 W Jiang (863_CR27) 2011; 16 J Chen (863_CR31) 2008; 338 AAA Kilbas (863_CR2) 2006 L Wei (863_CR18) 2014; 51 K Diethelm (863_CR6) 2010 AH Bhrawy (863_CR8) 2015; 81 VR Hosseini (863_CR34) 2014; 38 VR Hosseini (863_CR23) 2015; 130 863_CR17 WY Tian (863_CR12) 2015; 294 EH Doha (863_CR35) 2011; 35 J Chen (863_CR42) 2012; 219 K Moaddya (863_CR33) 2011; 61 C Li (863_CR16) 2014; 255 K Miller (863_CR7) 1993 AH Bhrawy (863_CR36) 2013; 26 WH Deng (863_CR13) 2013; 47 AH Bhrawy (863_CR43) 2015 A Atangana (863_CR10) 2015; 19 Z Zhao (863_CR9) 2012; 219 S Chen (863_CR11) 2015; 278 S Das (863_CR1) 2008 JAT Machado (863_CR3) 2011; 16 OP Agrawal (863_CR21) 2002; 29 S Momani (863_CR30) 2005; 170 Y Luchko (863_CR41) 2013; 54 W Zhang (863_CR22) 2015; 107 |
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