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Modified Adomian Decomposition Method for Solving Fuzzy Volterra-Fredholm Integral Equation


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1 Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad, 431004, India
     

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In this paper, a modied Adomian decomposition method has been applied to approximate the solution of the fuzzy Volterra-Fredholm integral equations of the first and second Kind. That, a fuzzy Volterra-Fredholm integral equation has been converted to a system of Volterra-Fredholm integral equations in crisp case. We use MADM to find the approximate solution of this system and hence obtain an approximation for the fuzzy solution of the Fuzzy Volterra-Fredholm integral equation. A nonlinear evolution model is investigated. Moreover, we will prove the existence, uniqueness of the solution and convergence of the proposed method. Also, some numerical examples are included to demonstrate the validity and applicability of the proposed technique.

Keywords

Modied Adomian Decomposition Method, Fuzzy Volterra-Fredholm Integral Equations, Approximate Solution, Existence and Convergence Analysis.
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  • Adomian G, A Review of The Decomposition Method in Applied Mathematics, J. Math. Anal. Appl., (1988), 501–44.

  • Allahviranloo T, The Adomian Decomposition Method for Fuzzy System of Linear Equations, J. Applied Mathematics and Computation, 163. (2005), 553–563.

  • Abbaoui K, Cherruault Y, Convergence of Adomian’s Method Applied to Nonlinear Equations, Math.Comput. Model, 20. (1994), 69–73.

  • Babolian E, Biazar J, Vahidi A, Solution of A System of Nonlinear Equations by Adomian Decomposition Method, Appl. Math. Comput., 150. (2004), 847–854.

  • Babolian E, Davari A, Numerical Implementation of Adomian Decomposition Method, Appl. Math. Comput, (2004), 301–305.

  • Babolian E, Goghary H, Abbasbandy S, Numerical Solution of Linear Fredholm Fuzzy Integral Equations of The Second kind by Adomian Method, Appl. Math.Comput., 161. (2005), 733–744.

  • Bede B, Gal S, Generalizations of The Differentiability of Fuzzy Number-Valued Functions With Applications to Fuzzy Differential Equations, Fuzzy Sets and Systems, 151. (2005), 581–599.

  • Bildik N, Inc M, Modified Decomposition Method for Nonlinear Volterra-Fredholm Integral Equations, Chaos, Solitons and Fractals, 33. (2007). 308–313.

  • Chalco-Cano Y, Roman-Flores H, On New Solutions of Fuzzy Differential Equations, Chaos, Solitons and Fractals, 38. (2008), 112–119.

  • Chang S, Zadeh L, On Fuzzy Mapping and Control, IEEE Trans. Systems, Man cybernet, 2. (1972), 30–34.

  • Dubois D, Prade H, Towards Fuzzy Differential Calculus. I: Integration of Fuzzy Mappings, Fuzzy Sets and Systems, 8. (1982), 1–17.

  • Dubois D, Prade H, Operations on Fuzzy Numbers, Int. J. systems Science, 9. (1978), 613–626.

  • El-Kalaa I, Convergence of The Adomian Method Applied to A Class of Nonlinear Integral Equations, App. Math. Comput., 21. (2008), 327–376.

  • Fariborzi Araghi M, Behzadi S, Solving Nonlinear Volterra-Fredholm Integro-Differential Equations Using The Modified Adomian Decomposition Method, Comput. Methods in Appl. Math., 9. (2009), 1–11.

  • Friedman M, Ming M, Kandel A, Numerical Solution of Fuzzy Differential and Integral Equations, Fuzzy Sets and System, 106. (1999), 35–48.

  • Goetschel R, Voxman W, Elementary Calculus, Fuzzy Sets and Systems, 18. (1986), 31–43.

  • Kaleva O, Fuzzy Differential Equation, Fuzzy Sets and Systems, 24. (1987). 301–317.

  • Kirtiwant G, Ahmed H, Study of the Approximate Solution of Fuzzy Volterra-Fredholm Integral Equations by using (ADM), Elixir Appl. Math., 98. (2016). 42567–42573.

  • Nanda S, On Integration of Fuzzy Mappings, Fuzzy Sets and Systems, 32. (1989), 95–101.

  • Shao Y, Zhang H, Fuzzy Integral Equations and Strong Fuzzy Henstock Integrals, Hindawi Publishing Corporation Abstract and Applied Analysis, (2014), 1–8.

  • Wazwaz A. M, A New Algorithm for Calculationg Adomian Polynomials for Nonlinear Operators, Appl. Math. Comput., 3. (2000), 53–69.


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  • Modified Adomian Decomposition Method for Solving Fuzzy Volterra-Fredholm Integral Equation

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Authors

Ahmed A. Hamoud
Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad, 431004, India
Kirtiwant P. Ghadle
Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad, 431004, India

Abstract


In this paper, a modied Adomian decomposition method has been applied to approximate the solution of the fuzzy Volterra-Fredholm integral equations of the first and second Kind. That, a fuzzy Volterra-Fredholm integral equation has been converted to a system of Volterra-Fredholm integral equations in crisp case. We use MADM to find the approximate solution of this system and hence obtain an approximation for the fuzzy solution of the Fuzzy Volterra-Fredholm integral equation. A nonlinear evolution model is investigated. Moreover, we will prove the existence, uniqueness of the solution and convergence of the proposed method. Also, some numerical examples are included to demonstrate the validity and applicability of the proposed technique.

Keywords


Modied Adomian Decomposition Method, Fuzzy Volterra-Fredholm Integral Equations, Approximate Solution, Existence and Convergence Analysis.

References