Convergence analysis of Jacobi pseudospectral method for delay fractional integral-differential equations in $L^2_{\omega^{\alpha ,\beta}}(I)$ space

Document Type : Original Paper

Authors

Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran

Abstract

In recent years, pseudospectral methods have been used for solving many classes of differential and integral equations due to their high accuracy and rate of convergence. In this paper, we present an efficient Jacobi pseudospectral method for solving a class of fractional delay integral-differential equations. Then, by presenting several lemmas and theorems, we investigate the convergence of the method on space

$ L ^ 2 _ {\ omega {alpha \ alpha, \ beta}} (I) $ and identify the error boundaries.

Keywords

Main Subjects


[1] K. K. Ali, E. M. Mohamed, K. S. Nisar, M. M. Khashan, M. Zakarya, A collocation approach for multiterm variable-order fractional delay-differential equations using shifted Chebyshev polynomials, Alexandria Engineering Journal. 61 (2022) 3511-3526 .
[2] P. Borisut, P. Kumam, I. Ahmed, K. Sitthithakerngkiet, Nonlinear Caputo fractional derivative with nonlocal riemann-liouville fractional integral condition via fixed point theorems, Symmetry. 11 (2019) 829.
[3] C. Canuto and M. Y. Hussaini and A. Quarteroni and T. A. Zang, Spectral methods: fundamentals in single domains, Springer, 2007.
[4] N. R. Gande, H. Madduri, Higher order numerical schemes for the solution of fractional delay differential equations, Journal of Computational and Applied Mathematics. 402 (2022) 113810.
[5] F. Hartung and M. Pituk, Recent Advances in Delay Differential and Difference Equations, Springer, 2014.
[6] G. Mastroianni, D. Occorsio, Optimal systems of nodes for Lagrange interpolation on bounded intervals. A survey, Journal of computational and applied mathematics. 134 (2001) 325-341.
[7] A. K. Mittal, Error analysis and approximation of Jacobi pseudospectral method for the integer and fractional order integro-differential equation, Applied Numerical Mathematics. 171 (2022) 249-268.
[8] P. Nevai, Mean convergence of Lagrange interpolation. III, Transactions of the American Mathematical Society. (1984) 669-698.
[9] N. Peykrayegan, M. Ghovatmand, M. H. Noori Skandari, On the convergence of Jacobi-Gauss collocation method for linear fractional delay differential equations, Mathematical Methods in the Applied Sciences. 44 (2021) 2237-2253.
[10] N. Peykrayegan, M. Ghovatmand, M. H. Skandari, An efficient method for linear fractional delay integro-differential equations, Computational and Applied Mathematics. 40 (2021) 1-33.
[11] I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Elsevier, 1998.
[12] D. L. Ragozin, Polynomial approximation on compact manifolds and homogeneous spaces, Transactions of the American Mathematical Society. 150 (1970) 41-53.
[13] D. L. Ragozin, Constructive polynomial approximation on spheres and projective spaces, Transactions of the American Mathematical Society. 162 (1971) 157-170.
[14] S. G. Samko and A. A. Kilbas and O. I. Marichev, Fractional integrals and derivatives (Vol. 1), Yverdon-les-Bains, Switzerland: Gordon and Breach Science Publishers, Yverdon, 1993.
[15] K. Saoudi, P. Agarwal, P. Kumam, A. Ghanmi, P. Thounthong, The Nehari manifold for a boundary value problem involving Riemann–Liouville fractional derivative, Advances in Difference Equations. 2018 (2018) 1-18.
[16] S. Shahmorad, M. H. Ostadzad, D. Baleanu, A Tau–like numerical method for solving fractional delay integro–differential equations, Applied Numerical Mathematics. 151 (2020) 322-336.
[17] J. Shen and T. Tang and L. L. Wang, Spectral methods: algorithms, analysis and applications (Vol. 41), Springer Science and Business Media, 2011.
[18] M. I. Syam, M. Sharadga, I. Hashim, A numerical method for solving fractional delay differential equations based on the operational matrix method, Chaos, Solitons and Fractals. 147 (2021) 110977.
[19] Y. Wei, Y. Chen, Convergence analysis of the spectral methods for weakly singular Volterra integrodifferential equations with smooth solutions, Advances in Applied Mathematics and Mechanics. 4 (2012) 1-20.
[20] Y. Yang, Y. Chen, Y. Huang, Spectral-collocation method for fractional Fredholm integro-differential equations, Journal of the Korean Mathematical Society. 51 (2014) 203-224.
[21] B. Yuttanan, M. Razzaghi, T. N. Vo, Legendre wavelet method for fractional delay differential equations, Applied Numerical Mathematics. 168 (2021) 127-142.