AN ENHANCED WAVELET BASED METHOD FOR NUMERICAL SOLUTION OF HIGH ORDER BOUNDARY VALUE PROBLEMS
Main Article Content
Abstract
The Legendre wavelet collocation method (LWCM) is suggested in this study for solving high-order boundary value problems numerically. Eighth, tenth, and twelfth-order examples are used as test problems to ensure that the technique is efficient and accurate. In comparison to other approaches, the numerical results obtained using LWCM demonstrate that the method's accuracy is very good. The results indicate that the method requires less computational effort to achieve better results.
Article Details
How to Cite
HAQ, Sirajul; SOHAIB, Muhammad.
AN ENHANCED WAVELET BASED METHOD FOR NUMERICAL SOLUTION OF HIGH ORDER BOUNDARY VALUE PROBLEMS.
Journal of Mountain Area Research, [S.l.], v. 6, p. 63-76, dec. 2021.
ISSN 2518-850X.
Available at: <https://journal.kiu.edu.pk/index.php/JMAR/article/view/109>. Date accessed: 26 apr. 2024.
doi: https://doi.org/10.53874/jmar.v6i0.109.
Issue
Section
Mathematical Sciences
This work is licensed under a Creative Commons Attribution 4.0 International License.
Papers accepted for publication become copyright of the Editorial Office, Journal of Mountain Area Research, Karakoram International University Gilgit, Gilgit-15100, Pakistan and authors will be asked to sign a copyright transfer agreement. Articles cannot be published until a signed copyright transfer agreement form has been received.
The work published in this journal is licensed under a Creative Commons Attribution 4.0 International License.
References
[1] R. P. Agrawal, Boundary Value Problems for Higher Ordinary Differential Equations, World Scientific, Singapore, (1986).
[2] G. Akram and H. Rehman, Numerical solution of Eighth order boundary value problems in reproducing kernel space, Numer. Algorithms 62 (2013), pp. 527-540.
[3] A. Boutayeb and E. H. Twizell, Finite difference methods for Twelfth-order, J. Comput. Appl. Math. 35 (1991), pp. 133-138.
[4] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover, New York, (1981).
[5] A. K. Dizicheh, F. Ismail, M. T. Kajani, and M. Maleki, A Legendre wavelet spectral collocation method for solving Oscillatory initial value problems, J. Appl. Math. http://dx.doi.org/10.1155/2013/591636.
[6] F. Geng and X. Li, Variational iteration method for solving Tenth-order boundary value problems, Math. Sci. 3 (2009), pp. 161-172.
[7] Sohaib M, Haq S. An efficient wavelet-based method for numerical solution of nonlinear integral and integro-differential equations. Math Meth Appl Sci. (2020), pp. 1-15.
[8] A. Golbabai and M. Javidi, Application of Homotopy perturbation method for solving Eighth-order boundary value problems, Appl. Math. Comput. 191 (2007), pp. 334-346.
[9] J. S. Guf and W. S. Jiang, The Haar wavelets operational matrix of integration, Internat. J. Systems Sci. 27 (1996), pp. 623-628.
[10] M. S. Hafshejani, S. K. Vanani, and J. S. Hafshejani, Numerical solution of Delay differential equations using Legendre wavelet method, World Appl. Sci. J. 13 (2011), pp. 27-33.
[11] S. Haq, M. Idrees, and S. U. Islam, Application of Optimal Homotopy asymptotic method to Eighth order initial and boundary value problems, Int. J. Appl. Math. Comput. 2 (2010), pp. 73-80.
[12] J. H. He, The variational iteration method for eighth-order initial-boundary value problems, Phys. Scr. 76 (2007), pp. 680-682.
[13] S. U. Islam, S. Haq, and J. Ali, Numerical solution of special 12th-order boundary value problems using Differential transform method, Commun. Nonlinear Sci. Numer. Simul. 14 (2009), pp. 1132-1138.
[14] M. Inc, D.J. Evans, An efficient approach to approximate solutions of eighth order boundary value problems, Int. J. Comput. Math. 81 (2004), pp. 685-692.
[15] A. S. V. R. Kanth and K. Aruna, Variational iteration method for Twelfth-order boundary-value problems, Comput. Math. Appl. 58 (2009), pp. 2360-2364.
[16] S. Hussain, A. Shah, S. Ayub and A. Ullah, An approximate analytical solution of the Allen-Cahn equation using homotopy perturbation method and homotopy analysis method, Helion 5 (12), (2019), https://doi.org/10.1016/j.heliyon.2019.e03060.
[17] A. A. Kurdi and S. Mulhem, Solution of Twelfth order boundary value problems using Adomian decomposition method, J. Appl. Sci. Res. 7 (2011), pp. 922-934.
[18] G. R. Liu and T. Y. Wu, Differential quadrature solutions of Eighth-order boundary-value differential equations, J. Comput. Appl. Math. 145 (2002), pp. 223-235.
[19] H. Mirmoradi, H. Mazaheripour, S. Ghanbarpour, and A. Barari, Homotopy perturbation method for solving Twelfth order boundary value problems, Int. J. of Res. and Rev. in Appl. Sci. 1 (2009), pp. 163-173.
[20] S. Hussain and A. Shah, Solution of generalized Drinfeld-Sokolov equations by using homotopy perturbation method and variational iteration method, Math. Rep., pp. 49-58, (2013).
[21] M. A. Noor, S. T. Mohyud-Din, Variational iteration decomposition method for solving eighth-order boundary value problems, Differ. Equat. Nonlinear Mech. (2007), doi:10.1155/2007/19529.
[22] M. I. A. 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 (2010), pp. 14-27.
[23] J. Rashidinia, R. Jalilian, and K. Farajeyan, Non polynomial spline solutions for special linear Tenth-order boundary value problems, W. J. Mod. Simul 7 (2011), pp. 40-51.
[24] A. Shah and A. A. Siddiqui, Variational iteration method for the solution of viscous Cahn-Hilliard equation, World Appl. Sci. J., (2012), pp. 1589-1592.
[25] M. Razzaghi and S. Yousefi, The Legendre wavelets operational matrix of integration, Int. J. Sys. Sci. 32 (2001), pp. 495-502.
[26] S. S. Siddiqi and M. Iftikhar, Numerical solution of higher order boundary value problems, Abstr. Appl. Anal. (2013), http://dx.doi.org/10.1155/2013/427521.
[27] S. S. Siddiqi, G. Akram, Solution of eighth-order boundary value problems using the nonpolynomial spline technique, Int. J. Comput. Math. 84 (3) (2007), pp. 347-368.
[28] S. S. Siddiqi and G. Akram, Solution of 10th-order boundary value problems using non-polynomial spline technique, Appl. Math. Comput. 190 (2007), pp. 641-651.
[29] S. S. Siddiqi and G. Akram, Solutions of Tenth-order boundary value problems using Eleventh degree spline, Appl. Math. Comput. 185 (2007), pp. 115-127.
[30] S. S. Siddiqi and A. Ghazala, Solutions of Twelfth order boundary value problems using Thirteen-degree spline, Appl. Math. Comput. 182 (2006), pp. 1443-1453.
[31] I. Ullah, H. Khan, and M. T. Rahim, Numerical solutions of higher order non-linear boundary value problems by New iterative method, Appl. Math. Sci. 7 (2013), pp. 2429-2439.
[32] S. G. Venkatesh, S. K. Ayyaswamy, and S. R. Balachandar, Legendre wavelets-based approximation method for solving Advection problems, A. S. Eng. J. 4 (2013), pp. 925-932.
[33] K. N. S. K. Viswanadham and Y.S. Raju, Quintic B-spline collocation method for Eighth order boundary value problems, Adv. Comput. Math. Appli. 1 (2012).
[34] A. M. Wazwaz, Approximate solutions to boundary value problems of higher order by the Modified decomposition method, Compu. Math Appli. 40 (2000), pp. 679-691.
[35] S. A. Yousefi, Legendre wavelets method for solving differential equations of Lane Emden type, Appl. Math. Comput. 181 (2006), pp. 1417-1422.
[36] E. Yusufoglu, New solitonary solutions for the MBBN equations using Exp function method, Phys. Lett. A 2007.
[37] S. S. Bayin, Mathematical Methods in Science and Engineering. Wiley. ch. 2. ISBN 978-0-470-04142-0 (2006).
[38] S. Singh, V. K. Patel, V. Singh, Application of wavelet collocation method for hyperbolic partial differential equations via matrices, Appl. Math. Comput. 320: 407-424, (2018).
[39] N. Khorrami, A. S. Shamloo, and P. B. Moghaddam, Numerical Solution of Interval Volterra-Fredholm-Hammerstein Integral Equations via Interval Legendre Wavelets Method. International Journal of Industrial Mathematics, 13(1), pp.15-28 (2021).
[40] Yuttanan, Boonrod, Razzaghi, Mohsen & Vo, Thieu. Legendre wavelet method for fractional delay differential equations. Appl Numer Math. 168. 10.1016/j.apnum.2021.05.024 (2021).
[41] M. Paliivets, E. Andreeve, A. Bakshtanin, D. Benin, and V. Snezhko, New iterative method for solving nonlinear equations in fluid mechanics, Int. J. Appl. Mech. Engg. pp. 163-174, DOI: 10.2478/ijame-2021-0042.
[42] A. Shah, S. Khlil, and S. Hussain, An efficient iterative technique for solving some nonlinear problems, Int. J. of Nonlinear Science, 13 (1), DOI: IJNS.2012.02.15/583.
[43] A. M. Siddiqui, A. Shah, and Q. K. Ghori, Homotopy Analysis of Slider bearing Lubricated with Powell-Eyring fluid, J. Appl. Sci., (2006), pp. 2358-2367.
[2] G. Akram and H. Rehman, Numerical solution of Eighth order boundary value problems in reproducing kernel space, Numer. Algorithms 62 (2013), pp. 527-540.
[3] A. Boutayeb and E. H. Twizell, Finite difference methods for Twelfth-order, J. Comput. Appl. Math. 35 (1991), pp. 133-138.
[4] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover, New York, (1981).
[5] A. K. Dizicheh, F. Ismail, M. T. Kajani, and M. Maleki, A Legendre wavelet spectral collocation method for solving Oscillatory initial value problems, J. Appl. Math. http://dx.doi.org/10.1155/2013/591636.
[6] F. Geng and X. Li, Variational iteration method for solving Tenth-order boundary value problems, Math. Sci. 3 (2009), pp. 161-172.
[7] Sohaib M, Haq S. An efficient wavelet-based method for numerical solution of nonlinear integral and integro-differential equations. Math Meth Appl Sci. (2020), pp. 1-15.
[8] A. Golbabai and M. Javidi, Application of Homotopy perturbation method for solving Eighth-order boundary value problems, Appl. Math. Comput. 191 (2007), pp. 334-346.
[9] J. S. Guf and W. S. Jiang, The Haar wavelets operational matrix of integration, Internat. J. Systems Sci. 27 (1996), pp. 623-628.
[10] M. S. Hafshejani, S. K. Vanani, and J. S. Hafshejani, Numerical solution of Delay differential equations using Legendre wavelet method, World Appl. Sci. J. 13 (2011), pp. 27-33.
[11] S. Haq, M. Idrees, and S. U. Islam, Application of Optimal Homotopy asymptotic method to Eighth order initial and boundary value problems, Int. J. Appl. Math. Comput. 2 (2010), pp. 73-80.
[12] J. H. He, The variational iteration method for eighth-order initial-boundary value problems, Phys. Scr. 76 (2007), pp. 680-682.
[13] S. U. Islam, S. Haq, and J. Ali, Numerical solution of special 12th-order boundary value problems using Differential transform method, Commun. Nonlinear Sci. Numer. Simul. 14 (2009), pp. 1132-1138.
[14] M. Inc, D.J. Evans, An efficient approach to approximate solutions of eighth order boundary value problems, Int. J. Comput. Math. 81 (2004), pp. 685-692.
[15] A. S. V. R. Kanth and K. Aruna, Variational iteration method for Twelfth-order boundary-value problems, Comput. Math. Appl. 58 (2009), pp. 2360-2364.
[16] S. Hussain, A. Shah, S. Ayub and A. Ullah, An approximate analytical solution of the Allen-Cahn equation using homotopy perturbation method and homotopy analysis method, Helion 5 (12), (2019), https://doi.org/10.1016/j.heliyon.2019.e03060.
[17] A. A. Kurdi and S. Mulhem, Solution of Twelfth order boundary value problems using Adomian decomposition method, J. Appl. Sci. Res. 7 (2011), pp. 922-934.
[18] G. R. Liu and T. Y. Wu, Differential quadrature solutions of Eighth-order boundary-value differential equations, J. Comput. Appl. Math. 145 (2002), pp. 223-235.
[19] H. Mirmoradi, H. Mazaheripour, S. Ghanbarpour, and A. Barari, Homotopy perturbation method for solving Twelfth order boundary value problems, Int. J. of Res. and Rev. in Appl. Sci. 1 (2009), pp. 163-173.
[20] S. Hussain and A. Shah, Solution of generalized Drinfeld-Sokolov equations by using homotopy perturbation method and variational iteration method, Math. Rep., pp. 49-58, (2013).
[21] M. A. Noor, S. T. Mohyud-Din, Variational iteration decomposition method for solving eighth-order boundary value problems, Differ. Equat. Nonlinear Mech. (2007), doi:10.1155/2007/19529.
[22] M. I. A. 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 (2010), pp. 14-27.
[23] J. Rashidinia, R. Jalilian, and K. Farajeyan, Non polynomial spline solutions for special linear Tenth-order boundary value problems, W. J. Mod. Simul 7 (2011), pp. 40-51.
[24] A. Shah and A. A. Siddiqui, Variational iteration method for the solution of viscous Cahn-Hilliard equation, World Appl. Sci. J., (2012), pp. 1589-1592.
[25] M. Razzaghi and S. Yousefi, The Legendre wavelets operational matrix of integration, Int. J. Sys. Sci. 32 (2001), pp. 495-502.
[26] S. S. Siddiqi and M. Iftikhar, Numerical solution of higher order boundary value problems, Abstr. Appl. Anal. (2013), http://dx.doi.org/10.1155/2013/427521.
[27] S. S. Siddiqi, G. Akram, Solution of eighth-order boundary value problems using the nonpolynomial spline technique, Int. J. Comput. Math. 84 (3) (2007), pp. 347-368.
[28] S. S. Siddiqi and G. Akram, Solution of 10th-order boundary value problems using non-polynomial spline technique, Appl. Math. Comput. 190 (2007), pp. 641-651.
[29] S. S. Siddiqi and G. Akram, Solutions of Tenth-order boundary value problems using Eleventh degree spline, Appl. Math. Comput. 185 (2007), pp. 115-127.
[30] S. S. Siddiqi and A. Ghazala, Solutions of Twelfth order boundary value problems using Thirteen-degree spline, Appl. Math. Comput. 182 (2006), pp. 1443-1453.
[31] I. Ullah, H. Khan, and M. T. Rahim, Numerical solutions of higher order non-linear boundary value problems by New iterative method, Appl. Math. Sci. 7 (2013), pp. 2429-2439.
[32] S. G. Venkatesh, S. K. Ayyaswamy, and S. R. Balachandar, Legendre wavelets-based approximation method for solving Advection problems, A. S. Eng. J. 4 (2013), pp. 925-932.
[33] K. N. S. K. Viswanadham and Y.S. Raju, Quintic B-spline collocation method for Eighth order boundary value problems, Adv. Comput. Math. Appli. 1 (2012).
[34] A. M. Wazwaz, Approximate solutions to boundary value problems of higher order by the Modified decomposition method, Compu. Math Appli. 40 (2000), pp. 679-691.
[35] S. A. Yousefi, Legendre wavelets method for solving differential equations of Lane Emden type, Appl. Math. Comput. 181 (2006), pp. 1417-1422.
[36] E. Yusufoglu, New solitonary solutions for the MBBN equations using Exp function method, Phys. Lett. A 2007.
[37] S. S. Bayin, Mathematical Methods in Science and Engineering. Wiley. ch. 2. ISBN 978-0-470-04142-0 (2006).
[38] S. Singh, V. K. Patel, V. Singh, Application of wavelet collocation method for hyperbolic partial differential equations via matrices, Appl. Math. Comput. 320: 407-424, (2018).
[39] N. Khorrami, A. S. Shamloo, and P. B. Moghaddam, Numerical Solution of Interval Volterra-Fredholm-Hammerstein Integral Equations via Interval Legendre Wavelets Method. International Journal of Industrial Mathematics, 13(1), pp.15-28 (2021).
[40] Yuttanan, Boonrod, Razzaghi, Mohsen & Vo, Thieu. Legendre wavelet method for fractional delay differential equations. Appl Numer Math. 168. 10.1016/j.apnum.2021.05.024 (2021).
[41] M. Paliivets, E. Andreeve, A. Bakshtanin, D. Benin, and V. Snezhko, New iterative method for solving nonlinear equations in fluid mechanics, Int. J. Appl. Mech. Engg. pp. 163-174, DOI: 10.2478/ijame-2021-0042.
[42] A. Shah, S. Khlil, and S. Hussain, An efficient iterative technique for solving some nonlinear problems, Int. J. of Nonlinear Science, 13 (1), DOI: IJNS.2012.02.15/583.
[43] A. M. Siddiqui, A. Shah, and Q. K. Ghori, Homotopy Analysis of Slider bearing Lubricated with Powell-Eyring fluid, J. Appl. Sci., (2006), pp. 2358-2367.