IJPAM: Volume 118, No. 1 (2018)




A.F. Jameel1, A. Saaban2, S.A. Altaie3,
N.R. Anakira4, A.K. Alomari5, N. Ahmad6
1,2,3,6School of Quantitative Sciences
Universiti Utara Malaysia (UUM)
Kedah, Sintok, 06010, MALAYSIA
3Computer Engineering Department
University of Technology
Baghdad, IRAQ
4,5Department of Mathematics
Faculty of Science and Technology
Irbid National University
2600 Irbid, JORDAN


In this paper, we discuss the approximate solution of first order nonlinear fuzzy initial value problems (FIVP) by formulate and analyze the use of the Optimal Homotopy Asymptotic Method (OHAM). OHAM allows for the solution of the fuzzy differential equation to be calculated in the form of an infinite series in which the components can be easily computed. This method provides us with a convenient way to control the convergence of approximation series. Numerical examples using the well-known nonlinear FIVP are presented to show the capability of the this method in this regard and the results are satisfied the convex triangular fuzzy number.


Received: 2017-04-21
Revised: 2017-10-11
Published: February 15, 2018

AMS Classification, Key Words

AMS Subject Classification: 03A72, 34A07, 14F35
Key Words and Phrases: fuzzy numbers, fuzzy differential equations, first order fuzzy initial value problems, optimal homotopy asymptotic method

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G. Gasper, M. Rahman, Basic Hyper-geometric Series, Cambridge University Press, Cambridge (1990).

M. Rosenblum, Generalized Hermite polynomials and the Bose-like oscillator calculus, In: Operator Theory: Advances and Applications, Birkhäuser, Basel (1994), 369-396.

D.S. Moak, The $q$-analogue of the Laguerre polynomials, J. Math. Anal. Appl., 81 (1981), 20-47.

M. F. Abbod, D. G. Von Keyserlingk, D. A. Linkens, M. Mahfouf, Survey of Utilization of Fuzzy Technology in Medicine and Healthcare, Fuzzy Sets and Systems, 120 (2001), 331-349.

A. Omer, O. Omer. A Pray and Pretdour Model with Fuzzy Initial Values. Hacettepe Journal of Mathematics and Statistics, 41 (2013), 387-395.

M. S. El Naschie, From Experimental Quantum Optics to Quantum Gravity Via a Fuzzy Kahler Manifold, Chaos, Solitons & Fractals, 25 (2005), 969-977.

S. Seikkala, On the Fuzzy Initial Value Problem, Fuzzy Sets and Systems, 24 (1987), 319-330.

M. Ghanbari, Numerical Solution of Fuzzy Initial Value Problems under Generalization Differentiability by HPM, International Journal of Industrial Mathematics, 1 (2009), 19-39.

A. F. Jameel, A. I. Md Ismail, Approximate Solution of First Order Nonlinear Fuzzy Initial Value Problem with Two Different Fuzzifications, Journal of Uncertain Systems, 9 (2015), 221-229.

E. Babolian, H. Sadeghi, S Javadi, Numerically Solution of Fuzzy Differential Equations by Adomian Method, Applied Mathematics and Computation, 149 (2004), 547-557.

T. Allahviranlo, S. Khezerloo, M, Mohammadzaki, Numerical Solution for Differential Inclusion by Adomian Decomposition Method, Journal of Applied Mathematics, 5 (2008), 51-62.

T. Allahviranloo, A. Panahi, H, Rouhparvar, A Computational Method To Find An Approximate Analytical Solution For Fuzzy Differential Equations, Analele stiintifice ale Universitatii Ovidius Constanta, 17 (2009), 5-14.

V. Marinca, N. Herisanu, I, Nemes, An optimal homotopy asymptotic method with application to thin film flow, Central European Journal of Physics, 6 (2008), 648-653.

N. Herisanu, V. Marinca, T. Dordea, G. Madescu, A new analytical approach to nonlinear vibration of an electric machine, Proc. Romanian. Acad. Series A: Mathematics. Physics, 9 (2008), 229-236.

V. Marinca, N. Herisanu, I. Nemes. An optimal homotopy asymptotic method with application to thin film flow, Central European Journal of Physics, 6 (2008), 648-653.

V. Marinca, N. Herisanu, C Bota, B. Marinca, An optimal homotopy asymptotic method applied to steady flow of a fourth-grade fluid past a porous plate, Applied Mathematics Letters, 22 (2009), 245-251.

A. K. Alomari, N. R. Anakira, A. S. Bataineh, I. Hashim. Approximate solution of nonlinear system of BVP arising in fluid flow problem, Mathematical Problems in Engineering, 2013 (2013), 1-7.

A. K. Alomari, F. Awawdeh, N. Tahat, F. B. Ahmad, W. Shatanawi, Multiple solutions for fractional differential equations: Analytic approach, Applied Mathematics and Computation, 219 (2013), 8893-8903.

N. R. Anakira, A. K. Alomari, I. Hashim. Optimal homotopy asymptotic method for solving delay differential equations, Mathematical Problems in Engineering, 2013 (2013), 1-11.

F. Mabood, Comparison of optimal homotopy asymptotic method and homotopy perturbation method for strongly nonlinear equation, Journal of the Association of Arab Universities for Basic and Applied Sciences, 16 (2014), 21-26.

S, Bodjanova, Median Alpha-Levels of A Fuzzy Number, Fuzzy Sets and Systems, 157 (2006), 879-891.

D, Dubois, H, Prade, Towards Fuzzy Differential Calculus, Part 3, Differentiation Fuzzy Sets and Systems, 8 (1982), 225-333.

S, Mansouri, N, Ahmady, A Numerical Method For Solving Nth-Order Fuzzy Differential Equation by using Characterization Theorem, Communication in Numerical Analysis, 2012 (2012),1-12.

O. Kaleva. Fuzzy Differential Equations. Fuzzy Sets and Systems, 24 (1987), 301-317.

G. Xiaobin, S, Dequan, Approximate solution of nth-order fuzzy linear differential equations, Mathematical Problems in Engineering, 2013 (2013), 1-12.

L. A. Zadeh, Toward a generalized theory of uncertainty, Information Sciences, 172 (2005), 1-40.

S. Salahshour, Nth-order fuzzy differential equations under generalized differentiability, Journal of Fuzzy Set Valued Analysis, 2011 (2011), 1-14.

F. Mabood, A. I. M. Ismail, I. Hashim. Application of optimal homotopy asymptotic method for the approximate solution of Riccati equation. Sains Malaysiana, 42 (2013), 863-867.

A. F. Jameel, A. I. M. Ismail and F Mabood. Optimal homotopy asymptotic method for solving nth order linear fuzzy initial value problems, Journal of the Association of Arab Universities for Basic and Applied Sciences, 21 (2016), 77-85.

S. Liang, J. J. David, Comparison of Homotopy Analysis Method and Technique, Applied Mathematics and Computation, 135 (2003), 73-79.

N.R.Anakira, A. K. Alomari, and I. Hashim. Numerical scheme for solving singular two-point boundary value problems. Journal of Applied Mathematics, 2013 (2013).

A.F. Jameel, N. R. Anakira, A.K. Alomari, I. Hashim, and S. Momani. A New Approximation Method for Solving Fuzzy Heat Equations. Journal of Computational and Theoretical Nanoscience, 13 2016, 7825-7832.

How to Cite?

DOI: 10.12732/ijpam.v118i1.5 How to cite this paper?

International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2018
Volume: 118
Issue: 1
Pages: 49 - 64

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