In this paper, non – linear finite fuzzy Volterra integral equation of the second kind (NFVIEK2) is considered. The Homotopy analysis method will be used to solve it, and comparing with the exact solution and calculate the absolute error between them. Some numerical examples are prepared to show the efficiency and simplicity of the method.
| Published in | Applied and Computational Mathematics (Volume 4, Issue 4) |
| DOI | 10.11648/j.acm.20150404.13 |
| Page(s) | 245-257 |
| 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), 2015. Published by Science Publishing Group |
Fuzzy Number, Finite Level, Volterra Integral Equation of Second Kind, Homotopy Analysis Method, Fuzzy Integral
| [1] | C. T. H. Baker, A perspective on the numerical treatment of volterra equations, Journal of Computational and Applied Mathematics, 125 (2000), 217-249. |
| [2] | N.Boldik, M. Inc, Modified decomposition method for nonlinear Volterra-Fredholm integral equations, Chaos, Solution and Fractals 33 (2007) 308- 313. http://dr.doi.org/10.1016/j.chaos.2005.12.058 |
| [3] | M. I. Berenguer, D. Gamez, A. I. Garralda-Guillem, M. Ruiz Galan and M. C. Serrano Perez, Biorthogonal systems for solving volterra integral equation systems of the second kind, Journal of Computational and Applied Mathematics, 235 (2011), 1875-1883. |
| [4] | A. M. Bica, Error estimation in the approximation of the solution of nonlinear fuzzy fredholmintegral equations, Information Sciences, 178 (2008), 1279-1292. |
| [5] | A. H. Borzabadi and O. S. Fard, A numerical scheme for a class of nonlinear fredholmintegral equations of the second kind, Journal of Computational and Applied Mathematics, 232 (2009), 449-454. |
| [6] | S. S. L. Chang and L. Zadeh, On fuzzy mapping and control, IEEE Trans. System Man Cybernet, 2 (1972), 30-34. |
| [7] | Y. Chen and T. Tang, Spectral methods for weakly singular volterra integral equations with smooth solutions, Journal of Computational and Applied Mathematics, 233 (2009), 938-950. |
| [8] | D. Dubois and H. Prade, Operations on fuzzy numbers, International Journal of Systems Science, 9 (1978), 613-626. |
| [9] | D. Dubois and H. Prade, Towards fuzzy differential calculus, Fuzzy Sets and Systems, 8(1982), 1-7. |
| [10] | A. Kaufmann and M. M. Gupta, Introduction fuzzy arithmetic, Van Nostrand Reinhold, NewYork, 1985. |
| [11] | J. P. Kauthen, Continuous time collocation method for volterra-fredholm integral equations, Numerical Math., 56 (1989), 409-424. |
| [12] | P. Linz, Analytical and numerical methods for volterra equations, SIAM, Philadelphia, PA,1985. |
| [13] | O. Solaymani Fard and M. Sanchooli, Two successive schemes for numerical solution of linearfuzzy fredholm integral equations of the second kind, Australian Journal of Basic Applied Sciences, 4 (2010), 817-825. |
| [14] | L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, Information Sciences, 8 (1975), 199-249. |
| [15] | S. Abbasbandy, E. Babolian and M. Alavi, Numerical method for solving linear fredholm fuzzy integral equations of the second kind, Chaos Solitons & Fractals, 31 (2007), 138-146. |
| [16] | E. Babolian, H. S. Goghary and S. Abbasbandy, Numerical solution of linear fredholm fuzzy integral equations of the second kind by Adomian method, Applied Mathematics and Com-putation, 161 (2005), 733-744. |
| [17] | Alabdullatif, M., Abdusalam, H. A. and Fahmy, E. S., Adomain decomposition method for nonlinear reaction diffusion system of Lotka- Volterra type, International Mathematical Forum, 2, No. 2 (2007), pp. 87-96. |
| [18] | WazWaz, A.-M., The modified decomposition method for analytic treatment of nonlinearintegral equations and system of non-linear integral equations, International Journal of Computer Mathematics, Vol. 82, No. 9(2005), pp. 1107-1115. |
| [19] | R. Goetschel and W. Vaxman, Elementary fuzzy calculus, Fuzzy Sets and Systems, 18 (1986), 31-43. |
| [20] | T. Allahviranloo and M. Otadi, Gaussian quadratures for approximate of fuzzy multipleintegrals, Applied Mathematics and Computation, 172 (2006), 175-187. |
| [21] | Kaleva, O.: A note on fuzzy differential equations, Nonlinear Analysis 64, 895–900 (2006). |
| [22] | O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems, 24 (1987), 301-317. |
| [23] | M. Ma, M. Friedman and A. Kandel, A new fuzzy arithmetic, Fuzzy Sets and Systems, 108 (1999), 83-90 |
| [24] | G. J. Klir, U. S. Clair and B. Yuan, Fuzzy set theory: foundations and applications, Prentice-Hall, 1997. |
| [25] | W. Congxin and M. Ming, On embedding problem of fuzzy number spaces, Part 1, Fuzzy Setsand Systems, 44 (1991), 33-38. |
| [26] | M. L. Puri and D. Ralescu, Fuzzy random variables, Journal of Mathematical Analysis and Applications, 114 (1986), 409-422. |
| [27] | Eman A.hussain, Existence and uniqueness of the solution of nonlinear integral equation, Department of mathematics /college of science, university of Al-mustansiriyah Iraq/ Baghdad ,vol.26(2)2013. |
APA Style
Alan Jalal Abdulqader. (2015). Homotopy Method for Solving Finite Level Fuzzy Nonlinear Integral Equation. Applied and Computational Mathematics, 4(4), 245-257. https://doi.org/10.11648/j.acm.20150404.13
ACS Style
Alan Jalal Abdulqader. Homotopy Method for Solving Finite Level Fuzzy Nonlinear Integral Equation. Appl. Comput. Math. 2015, 4(4), 245-257. doi: 10.11648/j.acm.20150404.13
AMA Style
Alan Jalal Abdulqader. Homotopy Method for Solving Finite Level Fuzzy Nonlinear Integral Equation. Appl Comput Math. 2015;4(4):245-257. doi: 10.11648/j.acm.20150404.13
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Download@article{10.11648/j.acm.20150404.13,
author = {Alan Jalal Abdulqader},
title = {Homotopy Method for Solving Finite Level Fuzzy Nonlinear Integral Equation},
journal = {Applied and Computational Mathematics},
volume = {4},
number = {4},
pages = {245-257},
doi = {10.11648/j.acm.20150404.13},
url = {https://doi.org/10.11648/j.acm.20150404.13},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20150404.13},
abstract = {In this paper, non – linear finite fuzzy Volterra integral equation of the second kind (NFVIEK2) is considered. The Homotopy analysis method will be used to solve it, and comparing with the exact solution and calculate the absolute error between them. Some numerical examples are prepared to show the efficiency and simplicity of the method.},
year = {2015}
}
TY - JOUR T1 - Homotopy Method for Solving Finite Level Fuzzy Nonlinear Integral Equation AU - Alan Jalal Abdulqader Y1 - 2015/06/29 PY - 2015 N1 - https://doi.org/10.11648/j.acm.20150404.13 DO - 10.11648/j.acm.20150404.13 T2 - Applied and Computational Mathematics JF - Applied and Computational Mathematics JO - Applied and Computational Mathematics SP - 245 EP - 257 PB - Science Publishing Group SN - 2328-5613 UR - https://doi.org/10.11648/j.acm.20150404.13 AB - In this paper, non – linear finite fuzzy Volterra integral equation of the second kind (NFVIEK2) is considered. The Homotopy analysis method will be used to solve it, and comparing with the exact solution and calculate the absolute error between them. Some numerical examples are prepared to show the efficiency and simplicity of the method. VL - 4 IS - 4 ER -
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