ArticleOpen Access

HERMITE WAVELET METHOD FOR APPROXIMATE SOLUTION OF HIGHER ORDER BOUNDARY VALUE PROBLEMS OF ORDINARY DIFFERENTIAL EQUATIONS

AMANULLAH

Department of Mathematics, University of Malakand, Chakdara, Dir Lower 18800, Pakistan

E-mail Address: amanshawar22@gmail.com

Search for more papers by this author

MUHAMMAD YOUSAF

Department of Mathematics, University of Malakand, Chakdara, Dir Lower 18800, Pakistan

E-mail Address: myousafuom@gmail.com

Search for more papers by this author

SALMAN ZEB

Department of Mathematics, University of Malakand, Chakdara, Dir Lower 18800, Pakistan

E-mail Address: salmanzeb@uom.edu.pk

Search for more papers by this author

MOHAMMAD AKRAM

Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 170, Saudi Arabia

E-mail Address: akramkhan_20@rediffmail.com

Search for more papers by this author

SARDAR MUHAMMAD HUSSAIN

Department of Mathematical Sciences, Balochistan University of Information Technology, Engineering and Management Sciences (BUITEMS), Quetta 87300, Pakistan

E-mail Address: smhussain01@gmail.com

Search for more papers by this author

, and

JONG-SUK RO

School of Electrical and Electronics Engineering, Chung-Ang University, Dongjak-gu, Seoul 06974, Republic of Korea

Department of Intelligent Energy and Industry, Chung-Ang University, Dongjak-gu, Seoul, 06974, Republic of Korea

E-mail Address: jongsukro@gmail.com

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S0218348X23400327Cited by:0 (Source: Crossref)

This article is part of the issue:

Special Issue on Applications of Wavelets and Fractals in Engineering Sciences
Guest Editors: K. S. Nisar, F. A. Shah, S. K. Upadhyay and P. E. T. Jorgensen

Abstract

In this paper, Hermite wavelet method (HWM) is considered for numerical solution of 12- and 13-order boundary value problems (BVPs) of ordinary differential equations (ODEs). The proposed algorithm for HWM developed in Maple software converts the ODEs into an algebraic systems of equations. These algebraic equations are then solved by evaluating the unknown constants present in the system of equations and the approximate solution of the problem is obtained. Test problems are considered and their solutions are investigated using HWM-based algorithm. The obtained results from the test problems are compared with exact solution, and with other numerical methods solution in the existing literature. Results comparison are presented both graphically and in tabular form showing close agreement with exact solution, and greater accuracy than homotopy perturbation method (HPM) and differential transform method (DTM).

Keywords:

References

1. R. Polikar and N. Mastorakis, The story of wavelets, in Physics and Modern Topics in Mechanical and Electrical Engineering (World Scientific and Engineering Society Press, 1999), pp. 192–197. Google Scholar
2. W. C. Shih, Time Frequency Analysis and Wavelet Transform Tutorial, Wavelet for Music Signals Analysis (Graduate Institute of Communication Engineering National Taiwan University, Taipei, Taiwan, ROC, 2006). Google Scholar
3. P. Sandoz, Wavelet transform as a processing tool in white-light interferometry, Opt. Lett. 22(14) (1997) 1065–1067. Crossref, Web of Science, Google Scholar
4. A. Bultheel, Wavelets with Applications in Signal and Image Processing (Course Material University Leuven, Belgium, 2014) Google Scholar
5. V. V. Zhirnov, S. V. Solonskaya and I. I. Zima, Application of wavelet transform for generation of radar virtual images, Telecomm. Radio Eng. 73(17) (2014) 1533–1539. Crossref, Google Scholar
6. T. Cong, G. Su, S. Qiu and W. Tian, Applications of ANNs in flow and heat transfer problems in nuclear engineering: A review work, Prog. Nucl. Energy 62 (2013) 54–71. Crossref, Web of Science, Google Scholar
7. A. Ali, R. Ghazali and M. M. Deris, The wavelet multilayer perceptron for the prediction of earthquake time series data, in Proceedings of 13th International Conference on Information Integration and Web-based Applications and Services (ACM, New York, NY, USA, 2011), pp. 138–143. Crossref, Google Scholar
8. E. F. Georgiou and P. Kumar, Wavelets in Geophysics (Academic Press, UK, 2014). Google Scholar
9. L. I. U. Kui, L. I. U. Zhaojun and Z. H. U. Jianwei, Application of time-frequency analysis in geology, World Geol. 3 (2000) 282–285. Google Scholar
10. J. L. Starck, Multiscale methods in astronomy: Beyond wavelets, in Astronomical Data Analysis Software and Systems XI, Vol. 281 (ASP Conference Proceedings, San Francisco, California, USA, 2002), pp. 391–399. Google Scholar
11. K. K. Viswanadham and S. Ballem, Numerical solution of tenth order boundary value problems by Galerkin method with septic B-splines, Int. J. Appl. Sci. Eng. 13(3) (2015) 247–260. Google Scholar
12. R. Crochiere, A weighted overlap-add method of short-time Fourier analysis/synthesis, IEEE Trans. Acoust. Speech Signal Process. 28(1) (1980) 99–102. Crossref, Google Scholar
13. M. Sifuzzaman, M. R. Islam and M. Z. Ali, Application of wavelet transform and its advantages compared to Fourier transform, J. Phys. Sci. 13 (2009) 121–134. Google Scholar
14. F. A. Shah, M. Irfan, K. S. Nisar, R. T. Matoog and E. E. Mahmoud, Fibonacci wavelet method for solving time-fractional telegraph equations with Dirichlet boundary conditions, Results Phys. 24 (2021) 104123. Crossref, Web of Science, Google Scholar
15. O. Ahmad, N. A. Sheikh, K. S. Nisar and F. A. Shah, Biorthogonal wavelets on the spectrum, Math. Methods Appl. Sci. 44(6) (2021) 4479–4490. Crossref, Web of Science, Google Scholar
16. K. S. Nisar, K. Jothimani, K. Kaliraj and C. Ravichandran, An analysis of controllability results for nonlinear Hilfer neutral fractional derivatives with non-dense domain, Chaos Solitons Fractals 146 (2021) 110915. Crossref, Web of Science, Google Scholar
17. C. Ravichandran, K. Jothimani, K. S. Nisar. E. E. Mahmoudd and I. S. Yahia, An interpretation on controllability of Hilfer fractional derivative with nondense domain, Alex. Eng. J. 61(12) (2022) 9941–9948. Crossref, Web of Science, Google Scholar
18. V. Vijayakumar, Approximate controllability results for abstract neutral integro-differential inclusions with infinite delay in Hilbert spaces, IIMA J. Math. Control. Inf. 35(1) (2018) 297–314. Web of Science, Google Scholar
19. W. K. Williams, V. Vijayakumar, U. Ramalingam, S. K. Panda and K. S. Nisar, Existence and controllability of nonlocal mixed Volterra–Fredholm type fractional delay integro-differential equations of order $1 < r < 2$ , Numer. Methods Partial Differential Equations (2020), https://doi.org/10.1002/num.22697. Web of Science, Google Scholar
20. K. Kavitha, V. Vijayakumar, A. Shukla, K. S. Nisar and R. Udhayakumar, Results on approximate controllability of Sobolev-type fractional neutral differential inclusions of Clarke subdifferential type, Chaos Solitons Fractals 151 (2021) 111264. Crossref, Web of Science, Google Scholar
21. K. Kavitha, V. Vijayakumar, R. Udhayakumar and C. Ravichandran, Results on controllability of Hilfer fractional differential equations with infinite delay via measures of noncompactness, Asian J. Control 24(3) (2022) 1406–1415. Crossref, Web of Science, Google Scholar
22. V. Vijayakumar, S. K. Panda, K. S. Nisar and H. M. Baskonus, Results on approximate controllability results for second-order Sobolev-type impulsive neutral differential evolution inclusions with infinite delay, Numer. Methods Partial Differential Equations 37(2) (2021) 1200–1221. Crossref, Web of Science, Google Scholar
23. M. Irfan, F. A. Shah and K. S. Nisar, Fibonacci wavelet method for solving Pennes bioheat transfer equation, Int. J. Wavelets Multiresolut. Inf. Process. 19(6) (2021) 2150023. Link, Web of Science, Google Scholar
24. S. Kumar, A. Ahmadian, R. Kumar, D. Kumar, J. Singh, D. Baleanu and M. Salimi, An efficient numerical method for fractional SIR epidemic model of infectious disease by using Bernstein wavelets, Mathematics 8(4) (2020) 558. Crossref, Web of Science, Google Scholar
25. S. Kumar. R. Kumar. C. Cattani and B. Samet, Chaotic behaviour of fractional predator-prey dynamical system, Chaos Solitons Fractals 135 (2020) 109811. Crossref, Web of Science, Google Scholar
26. K. S. Nisar and F. A. Shah, A numerical scheme based on Gegenbauer wavelets for solving a class of relaxation-oscillation equations of fractional order, Math. Sci. (2022) 1–13, https://doi.org/10.1007/s40096-022-00465-1. Google Scholar
27. F. A. Shah, M. Irfan and K. S. Nisar, Gegenbauer wavelet quasi-linearization method for solving fractional population growth model in a closed system, Math. Methods Appl. Sci. 45(7) (2022) 3605–3623. Crossref, Web of Science, Google Scholar
28. S. C. Shiralashetti, B. S. Hoogar and S. Kumbinarasaiah, Hermite wavelet based method for the numerical solution of linear and nonlinear delay differential equations, Int. J. Eng. Sci. Math. 6(8) (2017) 71–79. Google Scholar
29. W. Bellil, C. B. Amar and A. M. Alimi, Beta wavelet networks for function approximation, in Adaptive and Natural Computing Algorithms, eds. B. Ribeiro, R. F. Albrecht, A. Dobnikar, D. W. Pearson, N. C. Steele (Springer, Vienna, 2005), pp. 18–21, https://doi.org/10.1007/3-211-27389-1_5. Crossref, Google Scholar
30. G. G. Walter and X. Shen, Deconvolution using Meyer wavelets, J. Integral Equ. Appl. 11(4) (1999) 515–534. Crossref, Google Scholar
31. R. Bssow, An algorithm for the continuous Morlet wavelet transform, Mech. Syst. Signal Process. 21(8) (2007) 2970–2979. Crossref, Web of Science, Google Scholar
32. H. H. Szu, C. C. Hsu, L. D. Sa and W. Li, Hermitian hat wavelet design for singularity detection in the Paraguay river-level data analyses, in Wavelet Applications IV, SPIE, Vol. 3078 (SPIE, Orlando, FL, USA, 1997), pp. 96–115. Crossref, Google Scholar
33. Z. Masood, K. Majeed, R. Samar and M. A. Z. Raja, Design of Mexican Hat Wavelet neutral networks for solving Bratu type nonlinear systems, Neurocomputing 221 (2017) 1–14. Crossref, Web of Science, Google Scholar
34. J. G. Han, W. X. Ren and Y. Huang, A spline wavelet finite-element method in structural mechanics, Int. J. Numer. Methods Eng. 66(1) (2006) 166–190. Crossref, Web of Science, Google Scholar
35. U. Lepik, Numerical solution of differential equations using Haar wavelets, Math. Comput. Simul. 68(2) (2005) 127–143. Crossref, Web of Science, Google Scholar
36. F. Mohammadi and M. M. Hosseini, A new Legendre wavelet operational matrix of derivative and its applications in solving the singular ordinary differential equations, J. Franklin Inst. 348(8) (2011) 1787–1796. Crossref, Web of Science, Google Scholar
37. D. U. Lee, L. W. Kim and J. D. Villasenor, Precision-aware self-quantizing hardware architectures for the discrete wavelet transform, IEEE Trans. Image Process. 21(2) (2011) 768–777. Crossref, Web of Science, Google Scholar
38. S. Khalid, U. Jamil, K. Saleem, M. U. Akram, W. Manzoor, W. Ahmed and A. Sohail, Segmentation of skin lesion using Cohen–Daubechies–Feauveau biorthogonal wavelet, Springerplus 5(1) (2016) 1–17. Crossref, Google Scholar
39. C. Vonesch, T. Blu and M. Unser, Generalized Daubechies wavelet families, IEEE Trans. Signal Process. 55(9) (2007) 4415–4429. Crossref, Web of Science, Google Scholar
40. S. Kumbinarasaiah and K. R. Raghunatha, The applications of hermite wavelet method to nonlinear differential equations arising in heat transfer, Int. J. Thermofluid 9 (2021) 100066. Crossref, Google Scholar
41. E. Babolian and A. Shahsavaran, Numerical solution of nonlinear Fredholm integral equations of the second kind using Haar wavelets, J. Comput. Appl. Math. 225 (2009) 87–95. Crossref, Web of Science, Google Scholar
42. M. I. Othman, A. M. S. Mahdy and R. M. Farouk, Numerical solution of 12th order boundary value problems by using homotopy perturbation method, J. Math. Comput. Sci. 1(1) (2010) 14–27. Crossref, Google Scholar
43. M. Iftikhar, H. U. Rehman and M. Younis, Solution of thirteenth order boundary value problems by differential transformation method, Asian J. Math. 2014 (2014) 1–11. Google Scholar

Vol. 31, No. 02

Metrics

Downloaded 485 times

History

Received 24 March 2022

Accepted 12 July 2022

Published: March 4, 2023

Information

This is an Open Access article in the “Special Issue on Applications of Wavelets and Fractals in Engineering Sciences”, edited by K. S. Nisar (Prince Sattam bin Abdulaziz University, Saudi Arabia), F. A. Shah (University of Kashmir, India), S. K. Upadhyay (Indian Institute of Technology, BHU, India), P. E. T. Jorgensen (University of Iowa, USA) published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution-NonCommercial 4.0 (CC BY-NC) License which permits use, distribution and reproduction in any medium, provided that the original work is properly cited and is used for non-commercial purposes.

Keywords

PDF download

System Upgrade on Tue, May 28th, 2024 at 2am (EDT)

HERMITE WAVELET METHOD FOR APPROXIMATE SOLUTION OF HIGHER ORDER BOUNDARY VALUE PROBLEMS OF ORDINARY DIFFERENTIAL EQUATIONS

Abstract

Recommended