Convergence analysis of an efficient Chebyshev wavelet and its applications to differential equations via operational matrices of integration

Article Sidebar

PDF
Published: Mar 24, 2026
DOI: https://doi.org/10.5556/j.tkjm.57.2026.5958
Keywords:
Chebyshev wavelet, convergence analysis, operation matrix of integration, product operation matrix, Lane–Emden type, third order singular differential equation

Main Article Content

Hare Krishna Nigam
Central University of South Bihar, India
Md Mahtab Alam
Central University of South Bihar, India

Abstract

In this paper, the convergence analysis of the Chebyshev wavelet of the second kind is thoroughly carried out. Operational matrices for integration and product operations of the second kind Chebyshev wavelet are constructed, and these matrices are utilized to obtain solutions to the differential equations. A theorem related to the proposed operational matrix method is established. Solutions of the differential equations considered in this paper resemble their exact solutions. The characteristics of second kind Chebyshev wavelet are utilized to transform differential equations into systems of algebraic equations, which are solved very efficiently using a suitable method.

Article Details

How to Cite
Nigam, H. K., & Alam, M. M. (2026). Convergence analysis of an efficient Chebyshev wavelet and its applications to differential equations via operational matrices of integration. Tamkang Journal of Mathematics, 57(3), 171–193. https://doi.org/10.5556/j.tkjm.57.2026.5958
Issue
Vol. 57 No. 3 (2026)
Section
Papers
Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

Author Biography

Hare Krishna Nigam, Central University of South Bihar, India

Associate Professor and Head, Department of Mathematics

References

[1] Berry, M. V., Lewis, Z. V., & Nye, J. F. (1980). On the Weierstrass-Mandelbrot fractal function. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 370(1743), 459-484.

[2] Beylkin, G., Coifman, R., & Rokhlin, V. (1991). Fast wavelet transforms and numerical algorithms I. Communications on pure and applied mathematics, 44(2), 141-183.

[3] Chen, C. F., & Hsiao, C. H. (1997). Haar wavelet method for solving lumped and distributedparameter systems. IEE Proceedings-Control Theory and Applications, 144(1), 87-94.

[4] Guariglia, E., & Guido, R. C. (2022). Chebyshev wavelet analysis. Journal of Function Spaces, 2022(1), 5542054.

[5] Guariglia, E., & Silvestrov, S. (2017). Fractional-wavelet analysis of positive definite distributions and wavelet on $D′(C)$. In Engineering mathematics II: Algebraic, stochastic and analysis structures for networks, data classification and optimization (pp. 337–353). Cham: Springer International Publishing.

[6] Guariglia, E. (2018). Harmonic Sierpinski gasket and applications. Entropy, 20(9), 714.

[7] Guariglia, E. (2019). Primality, fractality, and image analysis. Entropy, 21(3), 304.

[8] Guido, R. C., Pedroso, F., Contreras, R. C., Rodrigues, L. C., Guariglia, E., & Neto, J.S. (2021). Introducing the Discrete Path Transform (DPT) and its applications in signal analysis, artefact removal, and spoken word recognition. Digital Signal Processing, 117,103158.

[9] Hwang, C., & Shih, Y. P. (1983). Laguerre series direct method for variational problems. Journal of optimization theory and applications, 39, 143-149.

[10] Kajani, M. T., & Vencheh, A. H. (2004). Solving linear integro-differential equation with Legendre wavelet. International Journal of Computer Mathematics, 81(6), 719-726.

[11] K. Maleknejad, M. Tavassoli Kajani, .Y.(2003). Mahmoudi, Numerical solution of linear Fredholm and Volterra integral equation of the second kind by using Legendre wavelet, Kybernetes, Int. J. Syst. Math. 32 1530–1539.

[12] Mahalakshmi, M., & Hariharan, G. (2014). An efficient wavelet based approximation method to steady state reaction–diffusion model arising in mathematical chemistry. The Journal of Membrane Biology, 247, 263-271.

[13] Maleknejad, K., & Kajani, M. T. (2003). Solving integro-differential equation by using Hybrid Legendre and block-pulse functions. International Journal of Applied Mathematics, 11(1), 67-76.

[14] Maleknejad, K., & Kajani, M. T. (2003). Solving second kind integral equations by Galerkin methods with hybrid Legendre and Block-Pulse functions. Applied Mathematics and Computation, 145(2-3), 623-629.

[15] Maleknejad, K., Tavassoli Kajani, M., & Mahmoudi, Y. (2003). Numerical solution of linear Fredholm and Volterra integral equation of the second kind by using Legendre wavelet. Kybernetes, 32(9/10), 1530-1539.

[16] Marzban, H. R., & Razzaghi, M. (2003). Hybrid functions approach for linearly constrained quadratic optimal control problems. Applied Mathematical Modelling, 27(6), 471-485.

[17] Marzban, H. R., & Razzaghi, M. (2004). Solution of time-varying delay systems by hybrid functions. Mathematics and Computers in Simulation, 64(6), 597-607.

[18] Mason, J. C., & Handscomb, D. C. (2002). Chebyshev polynomials. Chapman and Hall/CRC.

[19] M. Razzaghi, M. Razzaghi, (1988). Fourier series direct method for variational problems, Int. J. Control 48 887–895.

[20] Nigam, H. K., & Alam, M. (2025). wavelet-based approach for approximating Jacobi polynomial via characterized Hausdorff matrix. Filomat, 39(10), 3527-3536.

[21] Nigam, H. K., & Alam, M. (2024). An analysis of best wavelet approximation problem of a function using Hermite wavelet. Mathematical Methods in the Applied Sciences, 47(12), 10268-10279.

[22] Nigam, H. K., Hazarika, B., & Alam, M. (2024). An analysis of best wavelet approximation problem of a function using Laguerre wavelet. Filomat, 38(21), 7399-7411.

[23] Nigam, H. K., & Alam, M. M. (2025). Solution of differential equations via Chebyshev Operational matrix of integration. Boletim da Sociedade Paranaense de Matem´atica (Accepted).

[24] Nigam, H. K., Mohapatra, R. N., & Murari, K. (2020). wavelet approximation of a function using Chebyshev wavelet. Journal of Inequalities and Applications, 2020, 1-14.

[25] Nigam, H. K., & Murari, K. (2023). Approximation of functions by wavelet expansions with dilation matrix. Filomat, 37(22), 7589-7598.

[26] Nigam, H. K. , Mohapatra, R. N., & Murari,, K. (2020). wavelet approximation of a function in weighted Lipschitz class by Haar wavelet PanAmerican Mathematical Journal,30(1),39-50,

[27] Nigam, H. K., & Srivastava, H. M. (2023). Filtering of audio signals using discrete wavelet transforms. Mathematics, 11(19), 4117.

[28] Postnikov, E. B., Lebedeva, E. A., & Lavrova, A. I. (2016). Computational implementation of the inverse continuous wavelet transform without a requirement of the admissibility condition. Applied Mathematics and Computation, 282, 128-136.

[29] Razzaghi, M., & Yousefi, S. (2000). Legendre wavelet direct method for variational problems. Mathematics and computers in simulation, 53(3), 185-192.

[30] Sahu, P. K., & Ray, S. S. (2015). Legendre wavelet operational method for the numerical solutions of nonlinear Volterra integro-differential equations system. Applied mathematics and computation, 256, 715-723.

[31] Sahu, P. K., & Ray, S. S. (2016). Legendre spectral collocation method for the solution of the model describing biological species living together. Journal of Computational and Applied Mathematics, 296, 47-55.

[32] Yang, L., Su, H., Zhong, C., Meng, Z., Luo, H., Li, X., ... & Lu, Y. (2019). Hyperspectral image classification using wavelet transform-based smooth ordering. International Journal of wavelet, Multiresolution and Information Processing, 17(06), 1950050.

[33] Zheng, X., Tang, Y. Y., & Zhou, J. (2019). A framework of adaptive multiscale wavelet decomposition for signals on undirected graphs. IEEE Transactions on Signal Processing, 67(7), 1696-1711.