Optimum Solutions of Fredholm and Volterra Integro-differential Equations

Muhammad Akbar; Rashid Nawaz; Sumbal Ahsan

doi:doi:10.11648/j.ijtam.20190506.14

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Optimum Solutions of Fredholm and Volterra Integro-differential Equations

Muhammad Akbar, Rashid Nawaz, Sumbal Ahsan

Published in International Journal of Theoretical and Applied Mathematics (Volume 5, Issue 6)

Received: 5 April 2019 Accepted: 29 November 2019 Published: 6 December 2019

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Abstract

Integro-differential equations arise in modeling various physical and engineering problems. Several numerical and analytical methods have been developed for solving integro-differential equations. In this paper, a powerful semi analytical technique known as Optimal Homotopy Asymptotic Method (OHAM) has been used for finding the approximate solutions of Fredholm type integro-differential equations and Volterra type integro-differential equations. The proposed method does not required discretization like other numerical and approximate method, and it is also free from any small/large parameters. The presented technique provides better accuracy at lower order of approximation, the accuracy of the method can further be increases with higher order of approximation. Moreover, we can easily adjust and control the convergence region. The ability of the method is checked with different problems in literature. The results obtained through OHAM are compared with solutions of Adomian Decomposition Method. It is observed that solutions obtained through the proposed method is more accurate than existing techniques, which proves the validity and stability of the proposed method for solving integro-differential equations. The presented technique is more consistent, effective, suitable and rapidly convergent. The use of Optimal Homotopy Asymptotic Method is simple and straight forward. For the computation of problems, we have used Mathematica 9.0.

Published in	International Journal of Theoretical and Applied Mathematics (Volume 5, Issue 6)
DOI	10.11648/j.ijtam.20190506.14
Page(s)	100-112
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

Integro-differential Equations, Approximate Solutions, Optimal Homotopy Asymptotic Method

References

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[2]	Wazwaz, A. M. (2014). The variational iteration method for solving the Volterra integro-differential forms of the Lane–Emden equations of the first and the second kind. Journal of Mathematical Chemistry, 52 (2), 613-626.
[3]	Eshkuvatov, Z. K., Zulkarnain, F. S., Muminov, Z., & Long, N. M. A. N. (2017). Convergence of modified homotopy perturbation method for Fredholm-Volterra integro-differential equation of order m. Malaysian Journal of Fundamental and Applied Sciences, 13 (4-1), 340-345.
[4]	Elbeleze, A. A., Kılıçman, A., & Taib, B. M. (2016). Modified Homotopy Perturbation Method for Solving Linear Second-Order Fredholm Integro–Differential Equations. Filomat, 30 (7), 1823-1831.
[5]	Shah, k., & Singh, T. (2015). • Solution of second kind Volterra Integral and Integro-defferential equation by homotopy analysis method. International Journal of Mathematical Archive EISSN 6 (4), 2229-5046.
[6]	Mohamed, M. S., Gepreel, K. A., Alharthi, M. R., & Alotabi, R. A. (2016). Homotopy analysis transform method for integro-differential equations. General Mathematics Notes, 32 (1), 32-48.
[7]	Abbas, Z., Vahdati, S., Ismail, F., & Dizicheh, A. K. (2010). Application of homotopy analysis method for linear integro-differential equations. In International Mathematical Forum. 5 (5), 237-249.
[8]	Saeidy, M., Matinfar, M., & Vahidi, J. (2010). Analytical solution of BVPs for fourth-order integro-differential equations by using homotopy analysis method. International Journal of Nonlinear Science, 9 (4), 414-421.
[9]	Aruchunan, E., & Sulaiman, J. (2011). Half-sweep conjugate gradient method for solving first order linear Fredholm integro-differential equations. Australian Journal of Basic and Applied Sciences, 5 (3), 38-43.
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[12]	Rahmani, L., Rahimi, B., & Mordad, M. (2011). Numerical Solution of Volterra-Fredholm Integro-Differential Equation by Block Pulse Functions and Operational Matrices. General Mathematics Note, 4 (2), 37-48.
[13]	Marinca, V., & Herişanu, N. (2008). Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer. International Communications in Heat and Mass Transfer, 35 (6), 710-715.
[14]	Herişanu, N., Marinca, V., Dordea, T., & Madescu, G. (2008). A new analytical approach to nonlinear vibration of an electrical machine. Proceedings of the Romanian Academy-Series A, 9 (3), 229-236.
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APA Style

Muhammad Akbar, Rashid Nawaz, Sumbal Ahsan. (2019). Optimum Solutions of Fredholm and Volterra Integro-differential Equations. International Journal of Theoretical and Applied Mathematics, 5(6), 100-112. https://doi.org/10.11648/j.ijtam.20190506.14

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ACS Style

Muhammad Akbar; Rashid Nawaz; Sumbal Ahsan. Optimum Solutions of Fredholm and Volterra Integro-differential Equations. Int. J. Theor. Appl. Math. 2019, 5(6), 100-112. doi: 10.11648/j.ijtam.20190506.14

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AMA Style

Muhammad Akbar, Rashid Nawaz, Sumbal Ahsan. Optimum Solutions of Fredholm and Volterra Integro-differential Equations. Int J Theor Appl Math. 2019;5(6):100-112. doi: 10.11648/j.ijtam.20190506.14

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@article{10.11648/j.ijtam.20190506.14,
  author = {Muhammad Akbar and Rashid Nawaz and Sumbal Ahsan},
  title = {Optimum Solutions of Fredholm and Volterra Integro-differential Equations},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {5},
  number = {6},
  pages = {100-112},
  doi = {10.11648/j.ijtam.20190506.14},
  url = {https://doi.org/10.11648/j.ijtam.20190506.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20190506.14},
  abstract = {Integro-differential equations arise in modeling various physical and engineering problems. Several numerical and analytical methods have been developed for solving integro-differential equations. In this paper, a powerful semi analytical technique known as Optimal Homotopy Asymptotic Method (OHAM) has been used for finding the approximate solutions of Fredholm type integro-differential equations and Volterra type integro-differential equations. The proposed method does not required discretization like other numerical and approximate method, and it is also free from any small/large parameters. The presented technique provides better accuracy at lower order of approximation, the accuracy of the method can further be increases with higher order of approximation. Moreover, we can easily adjust and control the convergence region. The ability of the method is checked with different problems in literature. The results obtained through OHAM are compared with solutions of Adomian Decomposition Method. It is observed that solutions obtained through the proposed method is more accurate than existing techniques, which proves the validity and stability of the proposed method for solving integro-differential equations. The presented technique is more consistent, effective, suitable and rapidly convergent. The use of Optimal Homotopy Asymptotic Method is simple and straight forward. For the computation of problems, we have used Mathematica 9.0.},
 year = {2019}
}

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TY - JOUR
T1 - Optimum Solutions of Fredholm and Volterra Integro-differential Equations
AU - Muhammad Akbar
AU - Rashid Nawaz
AU - Sumbal Ahsan
Y1 - 2019/12/06
PY - 2019
N1 - https://doi.org/10.11648/j.ijtam.20190506.14
DO - 10.11648/j.ijtam.20190506.14
T2 - International Journal of Theoretical and Applied Mathematics
JF - International Journal of Theoretical and Applied Mathematics
JO - International Journal of Theoretical and Applied Mathematics
SP - 100
EP - 112
PB - Science Publishing Group
SN - 2575-5080
UR - https://doi.org/10.11648/j.ijtam.20190506.14
AB - Integro-differential equations arise in modeling various physical and engineering problems. Several numerical and analytical methods have been developed for solving integro-differential equations. In this paper, a powerful semi analytical technique known as Optimal Homotopy Asymptotic Method (OHAM) has been used for finding the approximate solutions of Fredholm type integro-differential equations and Volterra type integro-differential equations. The proposed method does not required discretization like other numerical and approximate method, and it is also free from any small/large parameters. The presented technique provides better accuracy at lower order of approximation, the accuracy of the method can further be increases with higher order of approximation. Moreover, we can easily adjust and control the convergence region. The ability of the method is checked with different problems in literature. The results obtained through OHAM are compared with solutions of Adomian Decomposition Method. It is observed that solutions obtained through the proposed method is more accurate than existing techniques, which proves the validity and stability of the proposed method for solving integro-differential equations. The presented technique is more consistent, effective, suitable and rapidly convergent. The use of Optimal Homotopy Asymptotic Method is simple and straight forward. For the computation of problems, we have used Mathematica 9.0.
VL - 5
IS - 6
ER -

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Author Information

Muhammad Akbar

Department of Mathematics, Abdul Wali Khan University, Mardan, Pakistan
Rashid Nawaz

Department of Mathematics, Abdul Wali Khan University, Mardan, Pakistan
Sumbal Ahsan

Department of Mathematics, Abdul Wali Khan University, Mardan, Pakistan

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Plain Text BibTeX RIS

APA Style

Muhammad Akbar, Rashid Nawaz, Sumbal Ahsan. (2019). Optimum Solutions of Fredholm and Volterra Integro-differential Equations. International Journal of Theoretical and Applied Mathematics, 5(6), 100-112. https://doi.org/10.11648/j.ijtam.20190506.14

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ACS Style

Muhammad Akbar; Rashid Nawaz; Sumbal Ahsan. Optimum Solutions of Fredholm and Volterra Integro-differential Equations. Int. J. Theor. Appl. Math. 2019, 5(6), 100-112. doi: 10.11648/j.ijtam.20190506.14

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AMA Style

Muhammad Akbar, Rashid Nawaz, Sumbal Ahsan. Optimum Solutions of Fredholm and Volterra Integro-differential Equations. Int J Theor Appl Math. 2019;5(6):100-112. doi: 10.11648/j.ijtam.20190506.14

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@article{10.11648/j.ijtam.20190506.14,
  author = {Muhammad Akbar and Rashid Nawaz and Sumbal Ahsan},
  title = {Optimum Solutions of Fredholm and Volterra Integro-differential Equations},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {5},
  number = {6},
  pages = {100-112},
  doi = {10.11648/j.ijtam.20190506.14},
  url = {https://doi.org/10.11648/j.ijtam.20190506.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20190506.14},
  abstract = {Integro-differential equations arise in modeling various physical and engineering problems. Several numerical and analytical methods have been developed for solving integro-differential equations. In this paper, a powerful semi analytical technique known as Optimal Homotopy Asymptotic Method (OHAM) has been used for finding the approximate solutions of Fredholm type integro-differential equations and Volterra type integro-differential equations. The proposed method does not required discretization like other numerical and approximate method, and it is also free from any small/large parameters. The presented technique provides better accuracy at lower order of approximation, the accuracy of the method can further be increases with higher order of approximation. Moreover, we can easily adjust and control the convergence region. The ability of the method is checked with different problems in literature. The results obtained through OHAM are compared with solutions of Adomian Decomposition Method. It is observed that solutions obtained through the proposed method is more accurate than existing techniques, which proves the validity and stability of the proposed method for solving integro-differential equations. The presented technique is more consistent, effective, suitable and rapidly convergent. The use of Optimal Homotopy Asymptotic Method is simple and straight forward. For the computation of problems, we have used Mathematica 9.0.},
 year = {2019}
}

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