In this paper, we presented the convergence of the solution for the nonlinear fuzzy volterra integral equation with high computational and complexity to find the solution in analytical method, so we describable this solution by using Homotopy analysis method, by using Banach fixed point theory for existence and uniqueness. That with explained numerical examples. Finally using the MAPLE program to solve our problem.
| Published in | American Journal of Applied Mathematics (Volume 4, Issue 3) |
| DOI | 10.11648/j.ajam.20160403.15 |
| Page(s) | 142-157 |
| 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), 2016. Published by Science Publishing Group |
Fuzzy Number, Volterra nonlinear Integral Equation, Operator of Fuzzy Number, Fuzzy Integral, Homotopy Analysis Method
| [1] | Nor hanim abd. rahman, arsmah Ibrahim, mohd idris jays, newton homotopy solution for nonlinear equations using maple 14, faculty of computer and mathematical scienes university teknologi mara malyusia (uitm) 40450 Shah Alam, Selangor, Malaysia volume. 3, number 2 Decamber (2011). |
| [2] | M. Ghanbari, numerical solution of fuzzy linear volterra integral equation of the second kind by homotopy analysis method, Department of mathematics, science and research branch, Islamic Azad University, Tehran, Iran, Received 10 january 2010. |
| [3] | Eman A. Hussain, Ayad W. Ali, Homotopy Analysis Method for solving Non linear fuzzy integral equation, Department of Mathematics, College of Science, Al-Mustansiriyah Baghdad, Iraq (2011). |
| [4] | Edyta H. Damian Slota, Tomasz. T, Roman Wituda. Usage of the Homotopy analysis method for solving the Non Linear and linear integral equation of the second kind (2013). Northeland. |
| [5] | R. Goetschel and Vaxman, Elementry fuzzy clculus, Fuzzy Sets and Systems, 18 (1986), 31-43. |
| [6] | Sarmad A. Altari, Numerical Solutions of Fuzzy Fredholm Integral Equation of the second kind using Bernstein Polynomials, Department of Computer Engineering and Information Technology University of Technology, Baghdad – Iraq (2012). |
| [7] | Hany. N. Magdy. A, A new technique of using Homotopy Analysis Method for Second Order Non Linear Differential Equation, Department of Basic Science, Faculty of Engineering of Branch Benha University, Egypt (2012). |
| [8] | N. A. Rajab, A. M. Ahmad, O. M. Alfaour, Reduction Formula for Linear Fuzzy Equation, Applied Science Department, University of Technology Baghdad – Iraq (2013). |
| [9] | Sushila Rathora, Devendra Kumar, Jagdev Singh, Sumit Gapta, Homotopy Analysis Method for Non Linear Equation, Department of Mathematics, Jagon Meth, University Village – Rampun Tehsil Chaksu, Jaipur – 303901, Ragashtan – India (2012). |
| [10] | S. Abbasbandy, E. Babolian, M. Alavi, Numerical method for solving linear Fredholm fuzzy integral equations of the second kind, Chaos Soliton and Fractals 31 (2007) 138146. |
| [11] | Abbasbandy, S., Msgyai, E., & Shivanian, E. (2009). The homotopy analysis method for multiple solutions of nonlinear boundary value problems. Commun Nonlinear Sci. Num. Simulat., 14(9-10), 3530-3536. |
| [12] | H.C. Wu, The improper fuzzy Riemann integral and its numerical integration, Information Science 111 (1999) 109-137. |
| [13] | S. Abbasbandy and A. Jafarian, “Steepest descent method for solving fuzzy nonlinear equations,” Applied Mathematics and Computation, vol. 174, no. 1, pp. 669–675, 2006. |
| [14] | .H. C. Wu, The proper fuzzy Riemann integral and its numerical integration, Information Science 111(1999) 109-137. |
| [15] | G. J. Klir, U. St. Clair, and B. Yuan, Fuzzy Set Theory: Foundations and Applications, Prentice-Hall, Eaglewood Cliffs, NJ, USA, 1997. |
| [16] | J. Y. Park, Y. C. Kwan, J. V. Jeong, Existence of solutions of fuzzy integral equationsin Banach spaces, Fuzzy Sets and System 72 (1995) 373-378. |
| [17] | Chang, S. S. L., & Zadah, L. A. (1972). On fuzzy mapping and control. Trans. Systems, Man Cyberetics, SMC-2(1), 30-34. |
| [18] | S. Abbasbandy, “The application of homotopy nanlysis mthod to nonlinear equations arising in heat transfer,” Phys. Lett. A, vol.360, pp 109-113, 2006. |
| [19] | S. J. Liao, Notes on the homotopy analysis method: Some definitions and theorems, Commun Nonlinear Sci. Numer. Simulat. 14 (2009) 983-997. |
| [20] | J. Y. Park, Y. C. Kwan, J. V. Jeong, Existence of solution of fuzzy integral equations in Banach spaces, Fuzzy Sets and System 72 ( 1995) 373- 378. |
| [21] | D. Dubois, H. Prade, Operation on fuzzy numbers, Int. J. system Science 9 (1978) 613-626 http://dx.doi.org/10.1080/00207727808941724 |
| [22] | Nanda, S. (1989). On integration of fuzzy mappings. Fuzzy Sets and Systems, 32, 95-101. |
| [23] | R. Goetschel and W. Vaxman, Elementary fuzzy calculus, Fuzzy Sets and Systems, 18 (1986), 31-43 |
| [24] | M. Mizumoto, K. Tanaka, The four operations of arithmetic on fuzzy numbers, Systems Comput. Controls 7(1976) 73-81. |
| [25] | O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems 24 (1987) 301-317 http://dx.doi.org/10.1016/0165-0114(87)90029-7. |
APA Style
Sameer Qasim Hasan, Alan Jalal Abdulqader. (2016). Numerical and Analytic Method for Solving Proposal Fuzzy Nonlinear Volterra Integral Equation by Using Homotopy Analysis Method. American Journal of Applied Mathematics, 4(3), 142-157. https://doi.org/10.11648/j.ajam.20160403.15
ACS Style
Sameer Qasim Hasan; Alan Jalal Abdulqader. Numerical and Analytic Method for Solving Proposal Fuzzy Nonlinear Volterra Integral Equation by Using Homotopy Analysis Method. Am. J. Appl. Math. 2016, 4(3), 142-157. doi: 10.11648/j.ajam.20160403.15
AMA Style
Sameer Qasim Hasan, Alan Jalal Abdulqader. Numerical and Analytic Method for Solving Proposal Fuzzy Nonlinear Volterra Integral Equation by Using Homotopy Analysis Method. Am J Appl Math. 2016;4(3):142-157. doi: 10.11648/j.ajam.20160403.15
Share This Article
Twitter
Linked In
Facebook
Copy
Download@article{10.11648/j.ajam.20160403.15,
author = {Sameer Qasim Hasan and Alan Jalal Abdulqader},
title = {Numerical and Analytic Method for Solving Proposal Fuzzy Nonlinear Volterra Integral Equation by Using Homotopy Analysis Method},
journal = {American Journal of Applied Mathematics},
volume = {4},
number = {3},
pages = {142-157},
doi = {10.11648/j.ajam.20160403.15},
url = {https://doi.org/10.11648/j.ajam.20160403.15},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20160403.15},
abstract = {In this paper, we presented the convergence of the solution for the nonlinear fuzzy volterra integral equation with high computational and complexity to find the solution in analytical method, so we describable this solution by using Homotopy analysis method, by using Banach fixed point theory for existence and uniqueness. That with explained numerical examples. Finally using the MAPLE program to solve our problem.},
year = {2016}
}
TY - JOUR T1 - Numerical and Analytic Method for Solving Proposal Fuzzy Nonlinear Volterra Integral Equation by Using Homotopy Analysis Method AU - Sameer Qasim Hasan AU - Alan Jalal Abdulqader Y1 - 2016/05/28 PY - 2016 N1 - https://doi.org/10.11648/j.ajam.20160403.15 DO - 10.11648/j.ajam.20160403.15 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 142 EP - 157 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20160403.15 AB - In this paper, we presented the convergence of the solution for the nonlinear fuzzy volterra integral equation with high computational and complexity to find the solution in analytical method, so we describable this solution by using Homotopy analysis method, by using Banach fixed point theory for existence and uniqueness. That with explained numerical examples. Finally using the MAPLE program to solve our problem. VL - 4 IS - 3 ER -
Email: