In this research paper, we propose a numerical technique based on Chebyshev wavelets of the second kind for the investigation of numerical integrations. For this purpose, collocation points and the basis functions of Chebyshev wavelets of the second kind have been utilized. The results of numerical experiments are presented, and are compared with exact solutions to confirm the good accuracy of the proposed scheme.
You will need Adobe Acrobat reader. For more information and free download of the reader, please follow this link.
References
[1] C.F. Chen, C.H. Hsiao, Haar wavelet method for solving lumped and
distributed-parameter systems, IEEE Proceedings: Part D, 144, No 1
(1997), 87–94.
[2] U. Lepik, Numerical solution of differential equations using Haar wavelets,
Mathematics and Computers in Simulation, 68 (2005), 127–143.
[3] U. Lepik, Application of Haar wavelet transform to solving integral and differential equations, Proceedings of Estonian Academy of Science, Physics,
Mathematics, 56, No 1 (2007), 28–46.
[4] I. Singh, S. Arora, S. Kumar, Numerical solution of wave equation using
Haar wavelet, International Journal of Pure and Applied Mathematics, 98,
No 4 (2015), 457–469.
[5] I. Singh, S. Kumar, Haar wavelet collocation method for solving nonlinear Kuramoto–Sivashinsky equation, Italian Journal of Pure and Applied
Mathematics, 39 (2018), 373–384.
[6] I. Singh, Wavelet based method for solving generalized Burgers type equations, International Journal of Computational Materials Science and Engineering, 8, No 4 (2019), 1–24.
[7] I. Singh, S. Kumar, Modified wavelet method for solving two-dimensional
coupled system of evolution equations, Iranian Journal of Mathematical
Sciences and Informatics, 17, No 1 (2022), 239–259.
[8] A.K. Gupta, S.S. Ray, An investigation with Hermite wavelets for accurate solution of fractional Jaulent-Miodek equation associated with energy
dependent Schrodinger potential, Applied Mathematics and Computation,
270 (2015), 458–471.
[9] O. Oruc, A numerical procedure based on Hermite wavelets for two dimensional hyperbolic telegraph equation, Engineering with Computers, 34, No
4 (2018), 741–755.
[10] S.C. Shiralashetti, K. Srinivasa, Hermite wavelets operational matrix of
integration for the numerical solution of nonlinear singular initial value
problems, Alexandria Engineering Journal, 57 (2018), 2591–2600.
[11] Z. Avazzadeh, M. Heydari, Chebyshev polynomials for solving two dimensional linear and nonlinear integral equations of the second kind, Computational and Applied Mathematics, 31, No 1 (2012), 127–142.
[12] L. Zhu, Q. Fan, Solving fractional nonlinear Fredholm integro-differential
equations by the second kind Chebyshev wavelet, Communications in Nonlinear Science and Numerical Simulation, 17, No 6 (2012), 2333–2341.
[13] W. M. Abd-Elhameed, E.H. Doha, Y.H. Youssri, New spectral second kind
Chebyshev wavelets algorithm for solving linear and nonlinear second-order
differential equations involving singular and Bratu type equations, Abstract
and Applied Analysis, 2013 (2013), 1–9.
[14] E. Babolian, F. Fattahzadeh, Numerical solution of differential equations
by using Chebyshev wavelet operational matrix of integration, Applied
Mathematics and Computation, 188, No 1 (2007), 417–426.
[15] Y. Wang, Q. Fan, The second kind Chebyshev wavelet method for solving
fractional differential equations, Applied Mathematics and Computation,
218, No 17 (2012), 8592–8601.
[16] F. Zhou, X. Xu, Numerical solution of the convection diffusion equations
by the second kind Chebyshev wavelets, Applied Mathematics and Computation, 247 (2014), 353–367.
[17] E. H. Doha, W.M. Abd-Elhameed, Y. H. Youssri, Second kind Chebyshev
operational matrix algorithm for solving differential equations of LaneEmden type, New Astronomy, 23 (2013), 113–117.
[18] M.H. Heydari, M.R. Hooshmandasl, F. M. Ghaini, A new approach of the
Chebyshev wavelets method for partial differential equations with boundary conditions of the telegraph type, Applied Mathematical Modelling, 38,
No 5–6 (2014), 1597–1606.