| spectral tau approach based on derivatives of Chebyshev polynomials with applications to high-order differential equations. |
| Paper ID : 1022-ISCH |
| Authors |
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Marwa Gamal Elazab *1, Mamdouh Metwally Elkady2, Mohamed G Abdelhakem2 1Helwan University 2Mathematics Department, Faculty of Science, Helwan University, Helwan 11795, Egypt |
| Abstract |
| We provide new basis functions, which are the second derivative of Chebyshev (2ndDCh) polynomials. These basis functions are used via the Tau spectral method as an efficient technique for finding approximation series solutions of linear and non-linear ordinary differential equations (ODEs). Linearization relation and some essential integration concerning the developed basis functions have been established to deal with Tau's method integrations. Unlike the regular weight function, another modified weight has been introduced. Also, different patterns and results have been obtained regarding the relation between the Jacobi polynomials, ultraspherical polynomials, Chebyshev polynomials, and their derivatives. In addition, the algebraic systems of the spectral expansion for solving the Riccati, Lane-Emden equations, and water contamination model have been determined explicitly. Also, convergence and error analysis have been introduced, studied, and proven. Moreover, the global error and upper bound of error are estimated. Finally, the discussed models and other IBVPs for real applications have been solved and approximated using 2ndDCh polynomials. The obtained results were compared favorably with different methods to confirm the accuracy and efficiency of the presented technique, which verified the error analysis discussion. |
| Keywords |
| Second derivative of Chebyshev polynomials, tau method, high-order differential equations, spectral methods |
| Status: Abstract Accepted (Oral Presentation) |