APPROXIMATION OF FUNCTIONS USING EXTENDED CHEBYSHEV WAVELETS AND ITS APPLICATION IN SOLVING INTEGRAL EQUATIONS

Harish Chandra Yadav     (Department of Mathematics, School of Sciences, Noida International University, Gautam Budh Nagar-201308, Uttar Pradesh, India)
Abhilasha Abhilasha     (Institute of Integrated and Honors Studies, Kurukshetra University, Kurukshetra-136118, Haryana, India)

Abstract


This paper presents a novel method in wavelet analysis for approximating solution functions of Fredholm integral equations. The proposed approach uses partial sums derived from this expansion on the interval \([0,\gamma)\), where \(\gamma>0\), resulting in accurate approximations of the solution function. The paper shows that the results obtained by applying the Chebyshev wavelet method coincide with the exact solutions of Fredholm integral equations. This agreement confirms the efficacy and reliability of the proposed approach as a tool for obtaining accurate solutions to complex problems.


Keywords


Hölder’s class, Chebyshev wavelet of the first kind, Fredholm integral equation

Full Text:

PDF

References


  1. Bahmanpour M., Araghi M.A.F. A method for solving Fredholm integral equation of the first kind based on Chebyshev wavelets. Anal. Theory Appl., 2013. Vol. 29, No. 3. P. 197–207. DOI: 10.4208/ata.2013.v29.n3.1
  2. Bin Jebreen H. On the multiwavelets Galerkin solution of the Volterra–Fredholm integral equations by an efficient algorithm. J. Math., 2020. Vol. 2020. Art. no. 2672683. P. 1–10. DOI: 10.1155/2020/2672683
  3. Černá D., Finěk V. Galerkin method with new quadratic spline wavelets for integral and integro-differential equations. J. Comput. Appl. Math., 2020. Vol. 363. P. 426–443. DOI: 10.1016/j.cam.2019.06.033
  4. Cuchta T., Pavelites M., Tinney R. The Chebyshev difference equation. Mathematics, 2020. Vol. 8, No. 1. Art. no. 74. DOI: 10.3390/math8010074
  5. Debnath L. Wavelet Transforms and Their Applications. Boston: Birkhäuser, 2002. 565 p. DOI: 10.1007/978-1-4612-0097-0Â
  6. Emamzadeh M.J., Kajani M.T. Nonlinear Fredholm integral equation of the second kind with quadrature methods. J. Math. Extension, 2010. Vol. 4, No. 2. P. 51–58.
  7. Heydari M.H., Hooshmandasl M.R., Shakiba A., Cattani C. Legendre wavelets Galerkin method for solving nonlinear stochastic integral equations. Nonlinear Dyn., 2016. Vol. 85. P. 1185–1202. DOI: 10.1007/s11071-016-2753-x
  8. Golbabai A., Seifollahi S. Numerical solution of the second kind integral equations using radial basis function networks. Appl. Math. Comput., 2006. Vol. 174, No. 2. P. 877–883. DOI: 10.1016/j.amc.2005.05.034
  9. Gonzalez O., Li J. A convergence theorem for a class of Nyström methods for weakly singular integral equations on surfaces in R3R3. Math. Comp., 2015. Vol. 84. P. 675–714. DOI: 10.1090/S0025-5718-2014-02869-X
  10. Guo R. et al. Physics embedded deep neural network for solving volume integral equation: 2-D Case. IEEE Trans. Antennas and Propagation, 2022. Vol. 70, No. 8. P. 6135–6147. DOI: 10.1109/TAP.2021.3070152
  11. Kong F., Li S., Zhou W. Wavelet–Galerkin approach for power spectrum determination of nonlinear oscillators. Mech. Syst. Signal Process., 2014. Vol. 48, No. 1. P. 300–324. DOI: 10.1016/j.ymssp.2014.02.015
  12. Lepik Ãœ. Application of the Haar wavelet transform to solving integral and differential equations. Proc. Estonian Acad. Sci. Phys. Math., 2007. Vol. 56, No. 1. P. 28–46. DOI: 10.3176/phys.math.2007.1.03Â
  13. Lin E.B., Al-Jarrah Y. A wavelet based method for the solution of Fredholm integral equations. Am. J. Comput. Math., 2012. Vol. 2, No. 2. P. 114–117. DOI: 10.4236/ajcm.2012.22015
  14. Mason J.C., Handscomb D.C. Chebyshev Polynomials. 1st ed. New York: Chapman and Hall/CRC, 2003. 360 p. DOI: 10.1201/9781420036114
  15. Meyer Y. Wavelets: their past and their Future. In: Progress in Wavelet Analysis and Applications: Proc. of the 3rd Int. Conf. on Wavelets and Applications (Toulouse, France, June 8–13, 1992). Frontiers, Gif-sur-Yvette, 1993. P. 9–18.
  16. Mahmoudi Y. Wavelet Galerkin method for numerical solution of nonlinear integral equation. Appl. Math. Comput., 2005. Vol. 167, No. 2. P. 1119–1129. DOI: 10.1016/j.amc.2004.08.004
  17. Noeiaghdam S., Sidorov D. Integral equations: theories, approximations and applications. Symmetry, 2021. Vol. 13, No. 8. Art. no. 1402. DOI: 10.3390/sym13081402Â
  18. Poljak D. et al. Integral equation formulations and related numerical solution methods in some biomedical applications of electromagnetic fields: transcranial magnetic stimulation (TMS), Nerve Fiber Stimulation. Int. J. E-Health Medical Commun., 2018. Vol. 9, No. 1. P. 65–84. DOI: 10.4018/IJEHMC.2018010105
  19. Rathsfeld A. Quadrature methods for 2D and 3D problems. J. Comput. Appl. Math., 2000. Vol. 125, No. 1–2. P. 439–460. DOI: 10.1016/S0377-0427(00)00484-2
  20. Shamooshaky M.M., Assari P., Adibi H. The Numerical solution of nonlinear Fredholm-Hammerstein integral equations of the second kind utilizing Chebyshev wavelets. J. Math. Comput. Sci., 2014. Vol. 10, No. 2. P. 235–246.
  21. Wilson J.H. Solution of integral equation important in signal processing of radar and sonar hydrophone array outputs. SIAM J. Appl. Math., 1980. Vol. 39, No. 1. P. 1–7. DOI: 10.1137/0139001
  22. Zygmund A.Trigonometric Series. Vol. I. Cambridge: Cambridge University Press, 1959. 383 p.




DOI: http://dx.doi.org/10.15826/umj.2026.1.012

Article Metrics

Metrics Loading ...

Refbacks

  • There are currently no refbacks.