The Modified Variational Iteration Method with Hermite Polynomials for the Numerical Solution of Tenth Order Boundary Value Problem

Otaide Ikechukwu Jackson; Ishaq Ajimoti Adam; John Obatarhe Emunefe; Ayinde Muhammed Abdullahi

doi:doi:10.11648/j.acm.20221104.12

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The Modified Variational Iteration Method with Hermite Polynomials for the Numerical Solution of Tenth Order Boundary Value Problem

Otaide Ikechukwu Jackson, Ishaq Ajimoti Adam, John Obatarhe Emunefe, Ayinde Muhammed Abdullahi

Published in Applied and Computational Mathematics (Volume 11, Issue 4)

Received: 1 July 2022 Accepted: 20 July 2022 Published: 17 August 2022

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Abstract

In this study, the numerical solution of tenth-order boundary value problems was obtained by employing the modified variational iteration method with Hermite polynomials. The correction functional is corrected for the boundary value problem (BVP) in this proposed method, and the Lagrange multiplier is optimally constructed using variational theory to reduce iteration on the integral operator while minimizing computational time. There was no need for any form of discretization or linearization with this method. The proposed modification also includes the generation of Hermite polynomials for the given boundary value problem and their use as the approximation's basis function. Four numerical examples were also provided to demonstrate the proposed method's effectiveness and reliability. Furthermore, we compared the results to some previously published findings. Tables 1, 2, and 3 show that our proposed method produces a better approximation to the exact solution than the Kasi Viswanadham & Sreenivasulu method, and Table 4 shows that our proposed method produces a better approximation to the exact solution in a few iterations than the Ali, Esra, Dumitru & Mustafa, and Iqbal et al. approaches, Rehman, Pervaiz, and Hakeem techniques (as can be seen from the tables of results). The calculations were carried out using the Maple 18 software.

Published in	Applied and Computational Mathematics (Volume 11, Issue 4)
DOI	10.11648/j.acm.20221104.12
Page(s)	95-101
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Modified Variational Iteration Method, Boundary Value Problems, Hermite Polynomials, Approximate Solutions

References

[1]	Kasi Viswanadham K. N. S & Sreenivasulu Ballem (2015). Numerical solution of tenth order boundary value problems by Galerkin method with septic B-splines, International Journal of Applied Science and Engineering. 13 (3), 247-260.
[2]	Ali, A., Esra, K. A., Dumitru, B., & Mustafa, I. (2018) New numerical method for solving tenth order boundary value problems, Mathematics, 6 (245), 2-9. doi: 10.3390/math6110245.
[3]	Iqbal, M. J., Rehman, S., Pervaiz, A., & Hakeem, A. (2015). Approximations for linear tenth-order boundary value problems through polynomial and non-polynomial cubic spline techniques, Proceedings of the Pakistan Academy of Sciences, 52 (4), 389–396.
[4]	Kasi Viswanadham K. N. S. & Reddy, S. M. (2015). Numerical solution of ninth order boundary value problems by Petrov- Galerkin method with Quintic B-splines as Basis Functions and Septic B-splines as Weight Functions, Procedia Engineering, 127, 1227-1234. doi: 10.1016/j.proeng.2015.11.470.
[5]	Reddy, S. N. (2016). Numerical solution of ninth order boundary value problems by Quintic B-splines. International Journal of Engineering Inventions, 5 (7), 38-47, ISSN: 2278- 7461.
[6]	Akram, G. & Sadaf, M. (2017). Application of Homotopy analysis method to the solution of ninth order boundary value problems in AFTI-F16 fighters, Journal of the Association of Arab Universities for Basic and Applied Sciences, 24, 149-155.
[7]	Noor, M. A & Mohyud-Din, S. T,. (2007). Variational iteration decomposition method for solving eight order boundary value problem, Hindawi Publishing Corporation Article ID 19529, 16 pages, doi: 10.1155/2007/19529.
[8]	Noor, M. A & Mohyud-Din, S. T. (2008). Variational iteration method for fifth order boundary value problem using He’s polynomials, Hindawi Publishing Corporation Article ID 954794, 12 pages doi: 10.1155/2008/954794.
[9]	Mohyud-Din, S. T. & Yildirim, A. (2010). Solutions of tenth and ninth-order boundary value problems by modified Variational iteration method, Application and Applied Mathematics, 5 (1), 11-25.
[10]	Noor, M. A. & Mohyud-Din, S. T. (2010). A new approach for solving fifth order boundary value problem, International Journal of Nonlinear Science, 9, 387-393.
[11]	Siddiqi, S. S. & Iftikhar, M. (2014). Variational iteration method for solution of seventh order boundary value problem using He’s polynomials, Journal of the Association of Arab Universities for Basic and Applied Sciences (2015) 18, 60–65.
[12]	Njoseh, I. N. & Mamadu, E. J. (2016) Numerical solutions of a generalized nth order boundary value problems using Power series approximation method. Applied Mathematics, 7, 1215-1224.http://dx.doi.org/10.4236/am.2016.711107
[13]	Mamadu, E. J. & Njoseh, I. N. (2016) Tau-Collocation approximation approach for solving first and second order ordinary differential equations. Journal of Applied Mathematics and Physics, 4: 383-390. doi.org/10.4236/jamp.2016.42045.
[14]	Caglar, H. N, Caglar, S. H, & Twizellll E. H (1999). The numerical solution of fifth-order value problems with sixth degree B-spline function, Applied Mathematics Letters, 12 (5): 25-30.
[15]	Adomian, G. (1990). A review of the decomposition method and some recent results for nonlinear equation, Math. Computer Modeling, 13 (7): 17-43. MR1071436 (92h: 00002a). Zbl 0713.65051.
[16]	Yahya Qaid Hasan & Liu Ming Zhu (2008), Modified Adomian decomposition method for singular initial value Problem in the second order ordinary differential equations, Surveys in Mathematics and its Applications, 3, 183-193.
[17]	Siddiqi, S. S. & Twizell, E. H. (1998). Spline solution of linear tenth order boundary value problems. International Journal of Computer Mathematics, 68, 3-4: 345-362.
[18]	Siddiqi, S. S. & Akram, G. (2007). Solutions of tenth order boundary value problems using the non-polynomial spline technique. Applied Mathematics and Computation, 185 (1), 115-127.
[19]	Abdulla, A. M. & Mohammed, A. D. (2019). Solutions of seventh order boundary value problem by using Variational Iteration method. Journal of Mathematics and Computational, 5 (1), 6-12.
[20]	Hamid, R. T. & Khadigeh, S. (2017). Bernstein polynomials basis for solution boundary value problems, Numerical Algorithms, 77, 211-228.

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APA Style

Otaide Ikechukwu Jackson, Ishaq Ajimoti Adam, John Obatarhe Emunefe, Ayinde Muhammed Abdullahi. (2022). The Modified Variational Iteration Method with Hermite Polynomials for the Numerical Solution of Tenth Order Boundary Value Problem. Applied and Computational Mathematics, 11(4), 95-101. https://doi.org/10.11648/j.acm.20221104.12

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ACS Style

Otaide Ikechukwu Jackson; Ishaq Ajimoti Adam; John Obatarhe Emunefe; Ayinde Muhammed Abdullahi. The Modified Variational Iteration Method with Hermite Polynomials for the Numerical Solution of Tenth Order Boundary Value Problem. Appl. Comput. Math. 2022, 11(4), 95-101. doi: 10.11648/j.acm.20221104.12

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AMA Style

Otaide Ikechukwu Jackson, Ishaq Ajimoti Adam, John Obatarhe Emunefe, Ayinde Muhammed Abdullahi. The Modified Variational Iteration Method with Hermite Polynomials for the Numerical Solution of Tenth Order Boundary Value Problem. Appl Comput Math. 2022;11(4):95-101. doi: 10.11648/j.acm.20221104.12

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@article{10.11648/j.acm.20221104.12,
  author = {Otaide Ikechukwu Jackson and Ishaq Ajimoti Adam and John Obatarhe Emunefe and Ayinde Muhammed Abdullahi},
  title = {The Modified Variational Iteration Method with Hermite Polynomials for the Numerical Solution of Tenth Order Boundary Value Problem},
  journal = {Applied and Computational Mathematics},
  volume = {11},
  number = {4},
  pages = {95-101},
  doi = {10.11648/j.acm.20221104.12},
  url = {https://doi.org/10.11648/j.acm.20221104.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20221104.12},
  abstract = {In this study, the numerical solution of tenth-order boundary value problems was obtained by employing the modified variational iteration method with Hermite polynomials. The correction functional is corrected for the boundary value problem (BVP) in this proposed method, and the Lagrange multiplier is optimally constructed using variational theory to reduce iteration on the integral operator while minimizing computational time. There was no need for any form of discretization or linearization with this method. The proposed modification also includes the generation of Hermite polynomials for the given boundary value problem and their use as the approximation's basis function. Four numerical examples were also provided to demonstrate the proposed method's effectiveness and reliability. Furthermore, we compared the results to some previously published findings. Tables 1, 2, and 3 show that our proposed method produces a better approximation to the exact solution than the Kasi Viswanadham & Sreenivasulu method, and Table 4 shows that our proposed method produces a better approximation to the exact solution in a few iterations than the Ali, Esra, Dumitru & Mustafa, and Iqbal et al. approaches, Rehman, Pervaiz, and Hakeem techniques (as can be seen from the tables of results). The calculations were carried out using the Maple 18 software.},
 year = {2022}
}

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TY  - JOUR
T1  - The Modified Variational Iteration Method with Hermite Polynomials for the Numerical Solution of Tenth Order Boundary Value Problem
AU  - Otaide Ikechukwu Jackson
AU  - Ishaq Ajimoti Adam
AU  - John Obatarhe Emunefe
AU  - Ayinde Muhammed Abdullahi
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PY  - 2022
N1  - https://doi.org/10.11648/j.acm.20221104.12
DO  - 10.11648/j.acm.20221104.12
T2  - Applied and Computational Mathematics
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PB  - Science Publishing Group
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UR  - https://doi.org/10.11648/j.acm.20221104.12
AB  - In this study, the numerical solution of tenth-order boundary value problems was obtained by employing the modified variational iteration method with Hermite polynomials. The correction functional is corrected for the boundary value problem (BVP) in this proposed method, and the Lagrange multiplier is optimally constructed using variational theory to reduce iteration on the integral operator while minimizing computational time. There was no need for any form of discretization or linearization with this method. The proposed modification also includes the generation of Hermite polynomials for the given boundary value problem and their use as the approximation's basis function. Four numerical examples were also provided to demonstrate the proposed method's effectiveness and reliability. Furthermore, we compared the results to some previously published findings. Tables 1, 2, and 3 show that our proposed method produces a better approximation to the exact solution than the Kasi Viswanadham & Sreenivasulu method, and Table 4 shows that our proposed method produces a better approximation to the exact solution in a few iterations than the Ali, Esra, Dumitru & Mustafa, and Iqbal et al. approaches, Rehman, Pervaiz, and Hakeem techniques (as can be seen from the tables of results). The calculations were carried out using the Maple 18 software.
VL  - 11
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Author Information

Otaide Ikechukwu Jackson

Department of Mathematics, Micheal and Cecilia Ibru University, Ughelli, Nigeria
Ishaq Ajimoti Adam

Department of Mathematics, Al-Hkmah University, Ilorin, Nigeria
John Obatarhe Emunefe

Department of General Studies, Mathematics and Statistics Unit, Petroleum Training Institute, Effurun, Nigeria
Ayinde Muhammed Abdullahi

Department of Mathematics, Modibbo Adama University, Yola, Nigeria

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APA Style

Otaide Ikechukwu Jackson, Ishaq Ajimoti Adam, John Obatarhe Emunefe, Ayinde Muhammed Abdullahi. (2022). The Modified Variational Iteration Method with Hermite Polynomials for the Numerical Solution of Tenth Order Boundary Value Problem. Applied and Computational Mathematics, 11(4), 95-101. https://doi.org/10.11648/j.acm.20221104.12

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ACS Style

Otaide Ikechukwu Jackson; Ishaq Ajimoti Adam; John Obatarhe Emunefe; Ayinde Muhammed Abdullahi. The Modified Variational Iteration Method with Hermite Polynomials for the Numerical Solution of Tenth Order Boundary Value Problem. Appl. Comput. Math. 2022, 11(4), 95-101. doi: 10.11648/j.acm.20221104.12

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AMA Style

Otaide Ikechukwu Jackson, Ishaq Ajimoti Adam, John Obatarhe Emunefe, Ayinde Muhammed Abdullahi. The Modified Variational Iteration Method with Hermite Polynomials for the Numerical Solution of Tenth Order Boundary Value Problem. Appl Comput Math. 2022;11(4):95-101. doi: 10.11648/j.acm.20221104.12

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@article{10.11648/j.acm.20221104.12,
  author = {Otaide Ikechukwu Jackson and Ishaq Ajimoti Adam and John Obatarhe Emunefe and Ayinde Muhammed Abdullahi},
  title = {The Modified Variational Iteration Method with Hermite Polynomials for the Numerical Solution of Tenth Order Boundary Value Problem},
  journal = {Applied and Computational Mathematics},
  volume = {11},
  number = {4},
  pages = {95-101},
  doi = {10.11648/j.acm.20221104.12},
  url = {https://doi.org/10.11648/j.acm.20221104.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20221104.12},
  abstract = {In this study, the numerical solution of tenth-order boundary value problems was obtained by employing the modified variational iteration method with Hermite polynomials. The correction functional is corrected for the boundary value problem (BVP) in this proposed method, and the Lagrange multiplier is optimally constructed using variational theory to reduce iteration on the integral operator while minimizing computational time. There was no need for any form of discretization or linearization with this method. The proposed modification also includes the generation of Hermite polynomials for the given boundary value problem and their use as the approximation's basis function. Four numerical examples were also provided to demonstrate the proposed method's effectiveness and reliability. Furthermore, we compared the results to some previously published findings. Tables 1, 2, and 3 show that our proposed method produces a better approximation to the exact solution than the Kasi Viswanadham & Sreenivasulu method, and Table 4 shows that our proposed method produces a better approximation to the exact solution in a few iterations than the Ali, Esra, Dumitru & Mustafa, and Iqbal et al. approaches, Rehman, Pervaiz, and Hakeem techniques (as can be seen from the tables of results). The calculations were carried out using the Maple 18 software.},
 year = {2022}
}

Copy | Download

TY  - JOUR
T1  - The Modified Variational Iteration Method with Hermite Polynomials for the Numerical Solution of Tenth Order Boundary Value Problem
AU  - Otaide Ikechukwu Jackson
AU  - Ishaq Ajimoti Adam
AU  - John Obatarhe Emunefe
AU  - Ayinde Muhammed Abdullahi
Y1  - 2022/08/17
PY  - 2022
N1  - https://doi.org/10.11648/j.acm.20221104.12
DO  - 10.11648/j.acm.20221104.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 95
EP  - 101
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20221104.12
AB  - In this study, the numerical solution of tenth-order boundary value problems was obtained by employing the modified variational iteration method with Hermite polynomials. The correction functional is corrected for the boundary value problem (BVP) in this proposed method, and the Lagrange multiplier is optimally constructed using variational theory to reduce iteration on the integral operator while minimizing computational time. There was no need for any form of discretization or linearization with this method. The proposed modification also includes the generation of Hermite polynomials for the given boundary value problem and their use as the approximation's basis function. Four numerical examples were also provided to demonstrate the proposed method's effectiveness and reliability. Furthermore, we compared the results to some previously published findings. Tables 1, 2, and 3 show that our proposed method produces a better approximation to the exact solution than the Kasi Viswanadham & Sreenivasulu method, and Table 4 shows that our proposed method produces a better approximation to the exact solution in a few iterations than the Ali, Esra, Dumitru & Mustafa, and Iqbal et al. approaches, Rehman, Pervaiz, and Hakeem techniques (as can be seen from the tables of results). The calculations were carried out using the Maple 18 software.
VL  - 11
IS  - 4
ER  -

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