Numerical methods have essential role to approximate the solutions of Partial Differential Equations (PDEs). Spectral method is one of the best numerical methods of exponential order with high convergence rate to solve PDEs. In recent decades the Chebyshev Spectral Collocation (CSC) method has been used to approximate solutions of linear PDEs. In this paper, by using linear algebra operators, we implement Kronecker Chebyshev Spectral Collocation (KCSC) method for n-order linear PDEs. By statistical tools, we obtain that the Run times of KCSC method has polynomial growth, but the Run times of CSC method has exponential growth. Moreover, error upper bounds of KCSC and CSC methods are compared.
[1] S. Akbarpour, A. Shidfar, and H. Saberinajafi, A Shifted Chebyshev-Tau method for finding a time-dependent heat source in heat equation, Comput. Methods for Differ. Equations, 8(1) (2020), 1-13.
[2] K. E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge Univ. Press, New York, 1997.
[3] A. G. Atta, W. M. A. Elhameed, and Y. H. Youssri, Shifted fifth-kind Chebyshev Galerkin treatment for linear hyperbolic first-order partial differential equations, Appl. Numerical Math., 167 (2021), 237-256.
[4] I. Celik, Collocation method and residual correction using Chebyshev series, Appl. Math. Comput., 174 (2006), 910-920.
[5] M. Dehghan and M. Lakestani, The use of Chebyshev cardinal functions for solution of the second-order one- dimensional telegraph equation, Num. Methods for Partial Differ. Equations, 25(4) (2009), 931-938.
[6] W. M. A. Elhameed and Y. H. Youssri, New formulas of the high-order derivatives of fifth-kind Chebyshev polyno- mials: Spectral solution of the convectiondiffusion equation, Num. Methods for Partial. Differ. Equations, (2021).
[7] W. M. A. Elhameed, J. A. Tenreiro Machado, and Y. H. Youssri, Hypergeometric fractional derivatives formula of shifted Chebyshev polynomials: tau algorithm for a type of fractional delay differential equations, Comput. Methods for Differ. Equations, 2(3) (2014), 171-185.
[8] W. M. A. Elhameed, Y. Youssri, and E. H. Doha, High-order finite element methods for time-fractional partial differential equations , Comput. Methods for Differ. Equations, 2(3) (2014), 171-185.
[9] G. Evans, J. Blackledge, and P. Yardley, Analytic Methods for Partial Differential Equations, Springer, 1999.
[10] K. Goyal and M. Mehra, Fast diffusion wavelet method for partial differential equations. Appl. Math. Mod., 40 (2016), 5000-5025.
[11] A. S. Hadi, Regression Analysis By Example, WILEY, New Jersey, 2012.
[12] D. V. Hutton, Fundamentals of finite element analysis, Elizabeth A .Jones (2004) 721-732.
[13] Y. Jianga and J. Ma, High-order finite element methods for time-fractional partial differential equations, A Math. Journal, (2004), 1-32.
[14] C. Johnson and R. A. Horn, Matrix Analysis, Cambridge Univ. Press, 2013.
[15] D. Johnson, Chebyshev polynomial in the Tau spectral methods and applications to eigenvalue problems, National Aeronautics and Space Administration, 1996.
[17] J. Mason and C. Handscomb, Chebyshev polynomials, CRC Press, 2013.
[18] M. Izadi and M. Afshar, Solving the Basset equation via Chebyshev collocation and LDG methods, Journal of Math. Mod., 9 (2021), 61-79.
[19] M. Lakestani and M. Dehghan, Numerical solution of fourth-order integro-differential equations using Chebyshev cardinal functions, Inter. Journal of Com. Math., 87(6) (2008), 1389-1394.
[20] M. Lakestani and M. Dehghan, The use of Chebyshev cardinal functions for the solution of a partial differential equation with an unknown time-dependent coefficient subject to an extra measurement, Journal of Comput. App. Math., 235(3) (2020), 669-678.
[21] A. B. Orovio, V. M. P. Garcia, and F. H. Fenton, Spectral Methods for Partial Differential Equations in Irregular Domains: The Spectral Smoothed Boundary Method, SIAM Journal on Scientific Computing, 28(3) (2006), 886- 900.
[22] P. Pedersen, New Solutions for Singular Lane-Emden Equations Arising in Astrophysics Based on Shifted Ultra- spherical Operational Matrices of Derivatives, Differ. and Integral Equ., (1999), 721-732.
[23] M. Pourbabaee and A. Saadatmandi, Collocation method based on Chebyshev polynomials for solving distributed order fractional differential equations, Comput. Methods for Differ. Equations, 9(3) (2021), 858-873.
[24] J. O. Rawlings, S. G. Pantula, and D. K. Dickey, Applied Regression Analysis, Springer, 1998.
[25] C. K. San, Chebyshev polynomial solutions of second-order linear partial differential equations, Appl. Math. Comput., 134 (2003), 109-124.
[26] G. Yuksel, O. R. Isik, and M. Sezer, Error analysis of the Chebyshev collocation method for linear second order partial differential equations, Internat. Journal of Comput.Math., 43 (2015), 2261-2268.
[27] Y. H. Youssri, W. M. A. Elhameed, and M. Abdelhakem, A robust spectral treatment of a class of initial value problems using modified Chebyshev polynomials, Math. Methods in the Appl. Sciences, 44(11) (2021), 9224-9236.
[28] Y. H. Youssri and R. M. Hafez, Chebyshev collocation treatment of VolterraFredholm integral equation with error analysis, Arabian Journal of Math., 9 (2020), 471-480.
Razavi, M., Hosseini, M. M., & Salemi, A. (2022). Error analysis and Kronecker implementation of Chebyshev spectral collocation method for solving linear PDEs. Computational Methods for Differential Equations, 10(4), 914-927. doi: 10.22034/cmde.2021.46776.1966
MLA
Mehdi Razavi; Mohammad Mehdi Hosseini; Abbas Salemi. "Error analysis and Kronecker implementation of Chebyshev spectral collocation method for solving linear PDEs". Computational Methods for Differential Equations, 10, 4, 2022, 914-927. doi: 10.22034/cmde.2021.46776.1966
HARVARD
Razavi, M., Hosseini, M. M., Salemi, A. (2022). 'Error analysis and Kronecker implementation of Chebyshev spectral collocation method for solving linear PDEs', Computational Methods for Differential Equations, 10(4), pp. 914-927. doi: 10.22034/cmde.2021.46776.1966
VANCOUVER
Razavi, M., Hosseini, M. M., Salemi, A. Error analysis and Kronecker implementation of Chebyshev spectral collocation method for solving linear PDEs. Computational Methods for Differential Equations, 2022; 10(4): 914-927. doi: 10.22034/cmde.2021.46776.1966
)
