Numerical and analytic method for solvingproposal New Type for fuzzy nonlinear volterra integral equation
DOI:
https://doi.org/10.24297/jam.v12i5.4009Keywords:
Fuzzy Number, Volterra nonlinear Integral equation, fuzzy integral, Homotopy analysis methodAbstract
In this paper, we proved the existence and uniqueness and convergence of the solution of new type for nonlinear fuzzy volterra integral equation . The homotopy analysis method are proposed to solve the new type fuzzy nonlinear Volterra integral equation . We convert a fuzzy volterra integral equation for new type of kernel for integral equation, to a system of crisp function nonlinear volterra integral equation . We use the homotopy analysis method to find the approximate solution of the system and hence obtain an approximation for fuzzy solution of the nonlinear fuzzy volterra integral equation . Some numerical examples is given and results reveal that homotopy analysis method is very effective and compared with the exact solution and calculate the absolute error between the exact and AHM .Finally using the MAPLE program to solve our problem .Downloads
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References
[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.
[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.
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Published
2016-06-30
How to Cite
Hasan, S. Q., & jalal, A. A. (2016). Numerical and analytic method for solvingproposal New Type for fuzzy nonlinear volterra integral equation. JOURNAL OF ADVANCES IN MATHEMATICS, 12(5), 6207–6231. https://doi.org/10.24297/jam.v12i5.4009
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