Optimum Solutions of Fredholm and Volterra Integro-differential Equations
International Journal of Theoretical and Applied Mathematics
Volume 5, Issue 6, December 2019, Pages: 100-112
Received: Apr. 5, 2019; Accepted: Nov. 29, 2019; Published: Dec. 6, 2019
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Authors
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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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.
Keywords
Integro-differential Equations, Approximate Solutions, Optimal Homotopy Asymptotic Method
To cite this article
Muhammad Akbar, Rashid Nawaz, Sumbal Ahsan, Optimum Solutions of Fredholm and Volterra Integro-differential Equations, International Journal of Theoretical and Applied Mathematics. Vol. 5, No. 6, 2019, pp. 100-112. doi: 10.11648/j.ijtam.20190506.14
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Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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