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Crisp Function of Integral Nonlinaer Equation of the Second Kind over the Fuzzy Interval with Application

Received: 27 June 2015     Accepted: 8 July 2015     Published: 18 July 2015
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Abstract

In this paper, the basic principle and definitions for nonlinear integral equation of a crisp function over a fuzzy interval have been discussed. a numerical technique method and some algorithm for solving non-linear of crisp valued function over fuzzy interval using the domain and range partitions of the membership functions of the fuzzy interval . the numerical solution of the crisp function over the fuzzy interval using the LR-type representation of fuzzy interval. Some numerical examples are prepared to show the efficiency and accuracy of the methods.

Published in American Journal of Applied Mathematics (Volume 3, Issue 4)
DOI 10.11648/j.ajam.20150304.15
Page(s) 189-200
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

Keywords

Fuzzy Number, Volterra Non-linear Integral Equation of Second Kind, Trapezoidal Quadrature Method, Fuzzy Interval, LR-type of Fuzzy Interval

References
[1] K. E. Atkinson, The Numerical solution of Integral Equation of the Second Cambridge University Press, 1997.
[2] A. Alipanah and M. Dehghan, Numerical solution of the nonlinear Fredholm integral equations by positive definitions. Appl. Math. Comput., 190(2007), 1754-176.
[3] J. Store and R. Bulirsch, Introduction to Numerical Analysis, Second ed, Springer-Verlag, 1993.
[4] Chen, S. H. 1985. Operations on fuzzy numbers with function principle. Tamkang journal of Management Science, 6:13-25
[5] R. Goetschel, W. Voxman, Elementary calculate, fuzzy Sets System 18 (1986)31-43
[6] C. T. H. Baker, A perspective on the numerical treatment of volterra equations, Journal of Computational and Applied Mathematics, 125 (2000), 217-249.
[7] D. Dubois and H. Prade, Operations on fuzzy numbers, International Journal of Systems Science, 9 (1978), 613-626.
[8] A. Kaufmann and M. M. Gupta, Introduction fuzzy arithmetic, Van Nostrand Reinhold, New York, 1985
[9] 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.
[10] T. Allahviranloo and M. Otadi, Gaussian quadratures for approximate of fuzzy multiple integrals, Applied Mathematics and Computation, 172 (2006), 175-187.
[11] M. Ma, M. Friedman and A. Kandel, A new fuzzy arithmetic, Fuzzy Sets and Systems, 108 (1999), 83-90
[12] C. T. H. Baker, A perspective on the numerical treatment of volterra equations, Journal of Computational and Applied Mathematics, 125 (2000), 217-249
[13] A. M. Bica, Error estimation in the approximation of the solution of nonlinear fuzzy fredholm integral equations, Information Sciences, 178 (2008), 1279-1292
[14] D. Dubois and H. Prade, Operations on fuzzy numbers, International Journal of Systems Science, 9 (1978), 613-626.
[15] G. J. Klir, U. S. Clair and B. Yuan, Fuzzy set theory: foundations and applications, Prentice-Hall, 1997.
[16] W. Congxin and M. Ming, On embedding problem of fuzzy number spaces, Part 1, Fuzzy Sets and Systems, 44 (1991), 33-38.
[17] M. L. Puri and D. Ralescu, Fuzzy random variables, Journal of Mathematical Analysis and Applications, 114 (1986), 409-422.
[18] [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
[19] Kandel, A., “Fuzzy Mathematical Techniques with applications”, Addison Wsely publishing Company, Inc., (1986)
[20] Negotia, C. V.m Ralescu, D. A., “Application of Fuzzy Sets to System Analysis “Basel, Stuttgart, (1975)
[21] Zadeh, L. A., “fuzzy Sets”, Information Control, Vol.8, (1965), pp338-353
[22] Dubois, D. and Prade, H., “Fuzzy Sets and System: Theory and Application”, Academic Press, Inc., (1908).
Cite This Article
  • APA Style

    Alan Jalal Abdulqader. (2015). Crisp Function of Integral Nonlinaer Equation of the Second Kind over the Fuzzy Interval with Application. American Journal of Applied Mathematics, 3(4), 189-200. https://doi.org/10.11648/j.ajam.20150304.15

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    ACS Style

    Alan Jalal Abdulqader. Crisp Function of Integral Nonlinaer Equation of the Second Kind over the Fuzzy Interval with Application. Am. J. Appl. Math. 2015, 3(4), 189-200. doi: 10.11648/j.ajam.20150304.15

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    AMA Style

    Alan Jalal Abdulqader. Crisp Function of Integral Nonlinaer Equation of the Second Kind over the Fuzzy Interval with Application. Am J Appl Math. 2015;3(4):189-200. doi: 10.11648/j.ajam.20150304.15

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  • @article{10.11648/j.ajam.20150304.15,
      author = {Alan Jalal Abdulqader},
      title = {Crisp Function of Integral Nonlinaer Equation of the Second Kind over the Fuzzy Interval with Application},
      journal = {American Journal of Applied Mathematics},
      volume = {3},
      number = {4},
      pages = {189-200},
      doi = {10.11648/j.ajam.20150304.15},
      url = {https://doi.org/10.11648/j.ajam.20150304.15},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20150304.15},
      abstract = {In this paper, the basic principle and definitions for nonlinear integral equation of a crisp function over a fuzzy interval have been discussed. a numerical technique method and some algorithm for solving non-linear of crisp valued function over fuzzy interval using the domain and range partitions of the membership functions of the fuzzy interval . the numerical solution of the crisp function over the fuzzy interval using the LR-type representation of fuzzy interval. Some numerical examples are prepared to show the efficiency and accuracy of the methods.},
     year = {2015}
    }
    

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    T1  - Crisp Function of Integral Nonlinaer Equation of the Second Kind over the Fuzzy Interval with Application
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    T2  - American Journal of Applied Mathematics
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    AB  - In this paper, the basic principle and definitions for nonlinear integral equation of a crisp function over a fuzzy interval have been discussed. a numerical technique method and some algorithm for solving non-linear of crisp valued function over fuzzy interval using the domain and range partitions of the membership functions of the fuzzy interval . the numerical solution of the crisp function over the fuzzy interval using the LR-type representation of fuzzy interval. Some numerical examples are prepared to show the efficiency and accuracy of the methods.
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Author Information
  • Department of Mathematics and Natural Science, Faculty MIPA, University Gadjah Mada, Yogyakarta, Indonesia

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