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Approximate solution of second-order fuzzy boundary value problem

Year 2017, Volume: 5 Issue: 3, 7 - 21, 01.07.2017

Mine Aylin Bayrak

Abstract

In this paper, a new approach is proposed based on the Adomian Decomposition Method(ADM) with Green's function in order to find a solution for the second-order fuzzy boundary value problem under generalized H-differentiability. The proposed technique divides the domain and builds on Green's function before installing the modified recursive scheme. Some examples are presented to illustrate the efficiency of the proposed technique.

Keywords

Fuzzy second-order boundary value problem Adomian decomposition method Green function's

References

  • Akın Ö, Oruc¸ Ö. A prey-predator model with fuzzy initial values. Hacet J Math Stat 2012; 41: 387-395.
  • Allahviranloo T, Abbasbandy S, Sedaghgatfar O, Darabi P. A new method for solving fuzzy integro-differential equation under generalized differentiability. Neural Comput Appl 2012; 1: 191-196.
  • Allahviranloo T, Amirteimoori A, Khezerloo M, Khezeloo S. A new method for solving fuzzy Volterra integro-differential equations. Aust J Basic Appl Sci 2011; 5: 154-164.
  • Allahviranloo T, KhezerlooM, GhanbariM, Khezerloo S. The homotopy perturbation method for fuzzy Volterra inetgral equations. Intern J Comp Cogn 2010; 8: 31-37.
  • Babolian E, Sadeghi G. H, Abbasbandy S. Numerical solution of linear Fredholm fuzzy integral equations of the second kind by Adomian method. Appl Math Comput 2005; 161: 733-744.
  • Bede B. A note on ”Two-point boundary value problems associated with nonlinear fuzzy diffential equations”. Fuzzy Set Syst 2006; 157: 986-989.
  • Bede B, Gal S. G. Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations. Fuzzy Set Syst 2005; 151: 581-599.
  • Behzadi S. S, Allahviranloo T, Abbasbandy S. Solving fuzzy second-order nonlinear Volterra-Fredholm integro-differential equqtions by using Picard method. Neural Comput Appl 2012; 1: 337-346.
  • Can E, Bayrak M. A. A New Method For Solution Of Fuzzy Reaction Equation. MATCH-Commun Math Co 2015; 74: 649-661.
  • Chalco-Cano Y, Romacutean-Flores H. On new solutions of fuzzy differential equations. Chaos Soliton Fract 2008; 38: 112-119.
  • Chang S. S. L, Zadeh LA. On fuzzy mapping and control. IEEE T Syst Man Cyb 1972; 2: 30-34.
  • Chen M, Fu Y, Xue X, Wu C. Two-point boundary value problems of undamped uncertain dynamical systems. Fuzzy Set Syst 2008; 159: 2077-2089.
  • Chen M, Fu Y, Xue X, Wu C. On fuzzy boundary value problems. Inform Sciences 2008; 178: 1877-1892.
  • Fatullayev A. G, Koroglu C. Numerical solving a boundary value problem for fuzzy differential equations. CMES-Comput Model Eng 2012; 8: 39-52.
  • Friedman M, Ma M, Kandel A. Numerical methods for calculating the fuzzy integral. Fuzzy Set Syst 1996; 83: 57-62.
  • Gasilov N. A, Hashimoglu IF, Amrahov SE, Fatullayev AG. A new approach to non-homogeneous fuzzy initial value problem. CMES-Comp Model Eng 2012; 85: 367-378.
  • Goetschel R, Voxman W. Elementary calculus . Fuzzy Set Syst 1986; 18: 31-43.
  • Jahantigh M, Allahviranloo T, Otadi M. Numerical solution of fuzzy integral equation. Appl Math Sci 2008; 2: 33-46.
  • Kaleva O. Fuzzy Differential Equations. Fuzzy Set Syst 1987; 24: 301-317.
  • Khastan A, Ivaz K. Numerical solution of fuzzy differential equations by Nystr¨om method. Chaos Soliton Fract 2009; 41: 859-868.
  • Khastan A, Nieto J. J. A boundary value problem for second-order fuzzy differential equations. Nonlinear Anal 2010; 72: 3583-3593.
  • Khastan A, Bahrami F, Ivaz K. New results on multiple solutions for Nth-order fuzzy differential equations under generalized differentiability. Bound Value Probl 13p,ARTICLE ID 395714, 2009.
  • Khastan A, Nieto J. J, Lopes RR. Variation of constant formula for first order fuzzy differential equations. Fuzzy Set Syst 2011;177: 20-33.
  • Molabahrami A, Shidfar A, Ghyasi A. A An analytical method for solving linear Fredholm fuzzy integral equations of the second kind. Comput Math Appl 2011; 61: 2754-2761.
  • Park J. Y, Lee S. Y, Jeong J. On the existence and uniqueness of solutions of fuzzy Volterra-Fredholm integral equations. Fuzzy Set Syst 2000; 115: 425-431.
  • Puri M, Ralescu D. Differential and fuzzy functions. J Math Anal Appl 1983; 91: 552-558.
  • Puri M, Ralescu D. Fuzzy Random variables. J Math Anal Appl 1986; 114: 409-422.
  • Rach R. C. A new definition of the Adomian polynomials. Kybernetes 2008; 37: 910-955.
  • Singh R, Wazwaz A. An efficient approach for solving second-order nonlinear differential equation with Neumann boundary conditions. J Math Chem 2015; 53: 767-790.
  • Zadeh L. A. The concept of linguistic variable and its application to approximate reasoning. Inform Sciences 1975; 8: 199-249.

Year 2017, Volume: 5 Issue: 3, 7 - 21, 01.07.2017

Mine Aylin Bayrak

Abstract

References

  • Akın Ö, Oruc¸ Ö. A prey-predator model with fuzzy initial values. Hacet J Math Stat 2012; 41: 387-395.
  • Allahviranloo T, Abbasbandy S, Sedaghgatfar O, Darabi P. A new method for solving fuzzy integro-differential equation under generalized differentiability. Neural Comput Appl 2012; 1: 191-196.
  • Allahviranloo T, Amirteimoori A, Khezerloo M, Khezeloo S. A new method for solving fuzzy Volterra integro-differential equations. Aust J Basic Appl Sci 2011; 5: 154-164.
  • Allahviranloo T, KhezerlooM, GhanbariM, Khezerloo S. The homotopy perturbation method for fuzzy Volterra inetgral equations. Intern J Comp Cogn 2010; 8: 31-37.
  • Babolian E, Sadeghi G. H, Abbasbandy S. Numerical solution of linear Fredholm fuzzy integral equations of the second kind by Adomian method. Appl Math Comput 2005; 161: 733-744.
  • Bede B. A note on ”Two-point boundary value problems associated with nonlinear fuzzy diffential equations”. Fuzzy Set Syst 2006; 157: 986-989.
  • Bede B, Gal S. G. Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations. Fuzzy Set Syst 2005; 151: 581-599.
  • Behzadi S. S, Allahviranloo T, Abbasbandy S. Solving fuzzy second-order nonlinear Volterra-Fredholm integro-differential equqtions by using Picard method. Neural Comput Appl 2012; 1: 337-346.
  • Can E, Bayrak M. A. A New Method For Solution Of Fuzzy Reaction Equation. MATCH-Commun Math Co 2015; 74: 649-661.
  • Chalco-Cano Y, Romacutean-Flores H. On new solutions of fuzzy differential equations. Chaos Soliton Fract 2008; 38: 112-119.
  • Chang S. S. L, Zadeh LA. On fuzzy mapping and control. IEEE T Syst Man Cyb 1972; 2: 30-34.
  • Chen M, Fu Y, Xue X, Wu C. Two-point boundary value problems of undamped uncertain dynamical systems. Fuzzy Set Syst 2008; 159: 2077-2089.
  • Chen M, Fu Y, Xue X, Wu C. On fuzzy boundary value problems. Inform Sciences 2008; 178: 1877-1892.
  • Fatullayev A. G, Koroglu C. Numerical solving a boundary value problem for fuzzy differential equations. CMES-Comput Model Eng 2012; 8: 39-52.
  • Friedman M, Ma M, Kandel A. Numerical methods for calculating the fuzzy integral. Fuzzy Set Syst 1996; 83: 57-62.
  • Gasilov N. A, Hashimoglu IF, Amrahov SE, Fatullayev AG. A new approach to non-homogeneous fuzzy initial value problem. CMES-Comp Model Eng 2012; 85: 367-378.
  • Goetschel R, Voxman W. Elementary calculus . Fuzzy Set Syst 1986; 18: 31-43.
  • Jahantigh M, Allahviranloo T, Otadi M. Numerical solution of fuzzy integral equation. Appl Math Sci 2008; 2: 33-46.
  • Kaleva O. Fuzzy Differential Equations. Fuzzy Set Syst 1987; 24: 301-317.
  • Khastan A, Ivaz K. Numerical solution of fuzzy differential equations by Nystr¨om method. Chaos Soliton Fract 2009; 41: 859-868.
  • Khastan A, Nieto J. J. A boundary value problem for second-order fuzzy differential equations. Nonlinear Anal 2010; 72: 3583-3593.
  • Khastan A, Bahrami F, Ivaz K. New results on multiple solutions for Nth-order fuzzy differential equations under generalized differentiability. Bound Value Probl 13p,ARTICLE ID 395714, 2009.
  • Khastan A, Nieto J. J, Lopes RR. Variation of constant formula for first order fuzzy differential equations. Fuzzy Set Syst 2011;177: 20-33.
  • Molabahrami A, Shidfar A, Ghyasi A. A An analytical method for solving linear Fredholm fuzzy integral equations of the second kind. Comput Math Appl 2011; 61: 2754-2761.
  • Park J. Y, Lee S. Y, Jeong J. On the existence and uniqueness of solutions of fuzzy Volterra-Fredholm integral equations. Fuzzy Set Syst 2000; 115: 425-431.
  • Puri M, Ralescu D. Differential and fuzzy functions. J Math Anal Appl 1983; 91: 552-558.
  • Puri M, Ralescu D. Fuzzy Random variables. J Math Anal Appl 1986; 114: 409-422.
  • Rach R. C. A new definition of the Adomian polynomials. Kybernetes 2008; 37: 910-955.
  • Singh R, Wazwaz A. An efficient approach for solving second-order nonlinear differential equation with Neumann boundary conditions. J Math Chem 2015; 53: 767-790.
  • Zadeh L. A. The concept of linguistic variable and its application to approximate reasoning. Inform Sciences 1975; 8: 199-249.
There are 30 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

Mine Aylin Bayrak

Publication Date July 1, 2017
Published in Issue Year 2017 Volume: 5 Issue: 3

Cite

APA Bayrak, M. A. (2017). Approximate solution of second-order fuzzy boundary value problem. New Trends in Mathematical Sciences, 5(3), 7-21.
AMA Bayrak MA. Approximate solution of second-order fuzzy boundary value problem. New Trends in Mathematical Sciences. July 2017;5(3):7-21.
Chicago Bayrak, Mine Aylin. “Approximate Solution of Second-Order Fuzzy Boundary Value Problem”. New Trends in Mathematical Sciences 5, no. 3 (July 2017): 7-21.
EndNote Bayrak MA (July 1, 2017) Approximate solution of second-order fuzzy boundary value problem. New Trends in Mathematical Sciences 5 3 7–21.
IEEE M. A. Bayrak, “Approximate solution of second-order fuzzy boundary value problem”, New Trends in Mathematical Sciences, vol. 5, no. 3, pp. 7–21, 2017.
ISNAD Bayrak, Mine Aylin. “Approximate Solution of Second-Order Fuzzy Boundary Value Problem”. New Trends in Mathematical Sciences 5/3 (July2017), 7-21.
JAMA Bayrak MA. Approximate solution of second-order fuzzy boundary value problem. New Trends in Mathematical Sciences. 2017;5:7–21.
MLA Bayrak, Mine Aylin. “Approximate Solution of Second-Order Fuzzy Boundary Value Problem”. New Trends in Mathematical Sciences, vol. 5, no. 3, 2017, pp. 7-21.
Vancouver Bayrak MA. Approximate solution of second-order fuzzy boundary value problem. New Trends in Mathematical Sciences. 2017;5(3):7-21.