Journal of Applied Science and Engineering

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Nabaa R. Kareem1, Fadhel S. Fadhel2This email address is being protected from spambots. You need JavaScript enabled to view it., and Sadiq Al-Nassir3

1Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq

2Department of Mathematics and Computer Applications, College of Sciences, Al-Nahrain University, Jadriya, Baghdad, Iraq

3Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq



Received: December 28, 2023
Accepted: March 31, 2024
Publication Date: May 29, 2024

 Copyright The Author(s). This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are cited.

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The main difficulty of solving fuzzy stochastic ordinary differential equations is that they do not have closed form solution, which represents an exact solution. Therefore, the homotopy perturbation method is proposed in this paper in connection with the method of parametrizing the fuzzy differential equation using-level sets. Thus the problem is converted into two crisp or nonfuzzy stochastic differential equations. We prove that the obtained approximate solution converges to the exact solution as a fuzzy stochastic process and two illustrative examples are considered with fuzziness appears in the initial conditions to be either of triangular or trapezoidal fuzzy numbers. The obtained results show the efficiency and reliability of the followed approach for solving the model problem under consideration.

Keywords: Stochastic differential equations, Fuzzy differential equations, Homotopy perturbation method, Brownian motion, Weiner process

  1. [1] P.M.A.HasanandN.A.Sulaiman,(2020)“Homo topyperturbationmethodandconvergenceanalysisforthe linearmixedVolterra-Fredholmintegralequations"Iraqi JournalofScience61(2):409–415.DOI:10.24996/ijs. 2020.61.2.19.
  2. [2] J.-H. He, (1999) “Homotopy perturbation technique" Computer methods in applied mechanics and en gineering 178(3-4): 257–262. DOI: 10.1016/S0045-7825(99)00018-3.
  3. [3] A.Beléndez, C. Pascual, T. Beléndez, and A. Hernan dez, (2009) “Solution for an anti-symmetric quadratic nonlinear oscillator by a modified He’s homotopy perturba tion method" Nonlinear Analysis: Real World Appli cations 10(1): 416–427. DOI: 10.1016/j.nonrwa.2007.10.002.
  4. [4] M. Chowdhury and I. Hashim, (2009) “Application of multistage homotopy-perturbation method for the so lutions of the Chen system" Nonlinear Analysis: Real World Applications 10(1): 381–391. DOI: 10.1016/j.nonrwa.2007.09.014.
  5. [5] M. Miansari, D. Ganji, and M. Miansari, (2008) “Ja cobi elliptic function solutions of the (1+1)-dimensional dispersive long wave equation by homotopy perturbation method" Numerical Methods for Partial Differen tial Equations: An International Journal 24(6): 1361 1370. DOI: 10.1002/num.20321.
  6. [6] S. H. Nia, A. Ranjbar, D. Ganji, H. Soltani, and J. Ghasemi, (2008) “Maintaining the stability of nonlin ear differential equations by the enhancement of HPM" Physics Letters A 372(16): 2855–2861. DOI: 10.1016/j.physleta.2007.12.054.
  7. [7] O.Abdulaziz, I. Hashim, and M. Chowdhury, (2008) “Solving variational problems by homotopy–perturbation method" International journal for numerical meth ods inengineering 75(6): 709–721. DOI: 10.1002/nme.2279.
  8. [8] X.-J. Yang, H. Srivastava, and C. Cattani, (2015) “Lo cal fractional homotopy perturbation method for solving fractal partial differential equations arising in mathemati cal physics" Romanian Reports in Physics 67(3): 752–761.
  9. [9] J.-H. He, (2006) “Homotopy perturbation method for solving boundary value problems" Physics letters A 350(1-2): 87–88. DOI: 10.1016/j.physleta.2005.10.005.
  10. [10] A. Yildirim, (2009) “Homotopy perturbation method for the mixed Volterra–Fredholm integral equations" Chaos, Solitons & Fractals 42(5): 2760–2764. DOI: 10.1016/j.chaos.2009.03.147.
  11. [11] J.-H. He, (1999) “Variational iteration method–a kind of non-linear analytical technique: some examples" In ternational journal of non-linear mechanics 34(4): 699–708. DOI: 10.1016/S0020-7462(98)00048-1.
  12. [12] M.Inokuti, H. Sekine, and T. Mura, (1978) “General use of the Lagrange multiplier in nonlinear mathematical physics" Variational method in the mechanics of solids 33(5): 156–162.
  13. [13] F. S. Fadhel and H. M. Sagban, (2021) “Approximate solution of linear fuzzy initial value problems using mod ified variaional iteration method" Al-Nahrain Journal of Science 24(4): 32–39. DOI: 10.22401/ANJS.24.4.05.
  14. [14] X.-J. Yang and D. Baleanu, (2013) “Fractal heat conduc tion problem solved by local fractional variation iteration method" Therm. Sci. 17(2): 625–628. DOI: 10.2298/TSCI121124216Y.
  15. [15] S. Chakraverty, N. Mahato, P. Karunakar, and T. D. Rao. Advanced numerical and semi-analytical methods for differential equations. John Wiley & Sons, 2019.
  16. [16] M.S. Ismael, F. S. Fadhel, and A. Al-Fayadh, (2020) “Approximate Solution of Multi-Term Fractional Order Delay Differential Equations Using Homotopy Perturba tion Method" Al-Nahrain Journal of Science 23(2): 60–66. DOI: 10.22401/ANJS.23.2.09.
  17. [17] F. Shakeri and M. Dehghan, (2008) “Solution of de lay differential equations via a homotopy perturbation method" Mathematical and computer Modelling 48(3-4): 486–498. DOI: 10.1016/j.mcm.2007.09.016.
  18. [18] A. A.Abdulsahib, F. S. Fadhel, and J. H. Eidi, (2024) “Approximate Solution of Linear Fuzzy Random Ordinary Differential Equations Using Laplace Variational Iteration Method" Iraqi Journal of Science: 804–817. DOI: 10.24996/ijs.2024.65.2.18.
  19. [19] O. M.Atyia, F. S. Fadhel, and M. H. Alobaidi, (2023) “Using Variational Iteration Method for Solving Linear Fuzzy Random Ordinary Differential Equations" Mathe matical Modelling of Engineering Problems 10(4): 1457–1466. DOI: 10.18280/mmep.100442.
  20. [20] A. Kareem and S. Al-Azzawi, (2021) “A stochastic differential equations model for the spread of coronavirus COVID-19: (the case of Iraq)" Iraqi Journal of Science 62(3): 1025–1035. DOI: 10.24996/ijs.2021.62.3.31.
  21. [21] A. Abdulsahib, F. Fadhel, and S. Abid, (2019) “Mod ified approach for solving random ordinary differential equations" Journal of Theoretical and Applied Infor mation Technology 97(13): 3574–3584.
  22. [22] A.K.Hussain, N. Rusli, F. S. Fadhel, and Z. R. Yahya. “Solution of one-dimensional fractional order partial integro-differential equations using variational itera tion method”. In: AIP Conference Proceedings. 1775. 1. AIP Publishing. 2016. DOI: 10.1063/1.4965216.
  23. [23] S. Abbasbandy, M. Otadi, and M. Mosleh, (2008) “Minimal solution of general dual fuzzy linear systems" Chaos, Solitons & Fractals 37(4): 1113–1124. DOI: 10.1016/j.chaos.2006.10.045.
  24. [24] S. Mauthner, (1998) “Step size control in the numerical solution of stochastic differential equations" Journal of computational and applied mathematics 100(1): 93 109. DOI: 10.1016/S0377-0427(98)00139-3.
  25. [25] H.M.Srivastava, R. Chaharpashlou, R. Saadati, and C. Li, (2022) “A fuzzy random boundary value problem" Axioms 11(8): 414. DOI: 10.3390/axioms11080414.
  26. [26] M.T. Malinowski, (2016) “Stochastic fuzzy differential equations of a nonincreasing type" Communications in Nonlinear Science and Numerical Simulation 33: 99–117. DOI: 10.1016/j.cnsns.2015.07.001.
  27. [27] A. Kandel. Fuzzy mathematical techniques with applica tions. Addison-Wesley Longman Publishing Co., Inc., 1986.
  28. [28] L. Arnold. Stochastic differential equations: theory and applications. John Wiley and Sons, Inc., 1974.
  29. [29] R. Moore. Interval analysis. 4. Englewood Cliffs: Prentice-Hall, 1966, 8–13.
  30. [30] M. T. Malinowski, (2013) “Some properties of strong solutions to stochastic fuzzy differential equations" Infor mation Sciences 252: 62–80. DOI: 10.1016/j.ins.2013.02.053.
  31. [31] M. B. Khan, H. M. Srivastava, P. O. Mohammed, J. Guirao, and T. M. Jawa, (2022) “Fuzzy-interval inequal ities for generalized preinvex fuzzy interval valued func tions" Math. Biosci. Eng. 19: 812–835. DOI: 10.3934/mbe.2022037.
  32. [32] A. Alderremy, J. Gómez-Aguilar, S. Aly, and K. M. Saad, (2021) “A fuzzy fractional model of coronavirus (COVID-19) and its study with Legendre spectral method" Results in Physics 21: 103773. DOI: 10.1016/j.rinp.2020.103773.
  33. [33] H.M.Srivastava, K. M. Saad, and W. M. Hamanah, (2022) “Certain new models of the multi-space fractal fractional Kuramoto-Sivashinsky and Korteweg-de Vries equations" Mathematics 10(7): 1089. DOI: 10.3390/math10071089.
  34. [34] M.Alqhtani,K.M.Saad,R.Zarin,A.Khan,andW.M. Hamanah, (2024) “Qualitative behavior of a highly non linear Cutaneous Leishmania epidemic model under con vex incidence rate with real data" Mathematical Bio sciences and Engineering 21(2): 2084–2120. DOI: 10.3934/mbe.2024092.
  35. [35] S. Panda, J. K. Dash, and G. B. Panda, (2023) “Stochas tic differential equation with fuzzy coefficients" IAENG International Journal of Applied Mathematics53(1): 66–75.
  36. [36] F. S. Fadhel, J. H. Eidi, H. M. Wali, et al., (2021) “Con traction mapping theorem in partial fuzzy metric spaces" Journal of Applied Science and Engineering 25(2): 353–360. DOI: 10.6180/jase.202204_25(2).0020.
  37. [37] J. Kider and N. Kadhum, (2019) “Properties of fuzzy compact linear operators on fuzzy normed spaces, Baghdad Sci" J 16(1): 104–110. DOI: 10.21123/bsj.2019.16.1. 0104.
  38. [38] R. I. Ali and E. A. Hussein, (2020) “Some Properties of Fuzzy Anti-Inner Product Spaces" Iraqi Journal of Science 61(11): 3053–3058. DOI: 10.24996/ijs.2020.61. 11.26.
  39. [39] R. E. Moore, R. B. Kearfott, and M. J. Cloud. Introduc tion to interval analysis. SIAM, 2009.
  40. [40] S. Chakraverty, S. Tapaswini, and D. Behera. Fuzzy differential equations and applications for engineers and scientists. CRC Press, 2016. DOI: 10.1201/9781315372853.
  41. [41] L.Stefanini and B.Bede, (2009) “Generalized Hukuhara differentiability of interval-valued functions and interval differential equations" Nonlinear Analysis: Theory, Methods & Applications 71(3-4): 1311–1328. DOI: 10.1016/
  42. [42] M. H. Suhhiem, (2017) “Artificial Neural network for solving fuzzy differential equations under generalized H derivation" International Journal 5(1): 1–9. DOI: 10.12691/ijpdea-5-1-1.
  43. [43] Y. Chalco-Cano, A. Rufián-Lizana, H. Román-Flores, and M.-D. Jiménez-Gamero, (2013) “Calculus for interval-valued functions using generalized Hukuhara derivative and applications" Fuzzy Sets and Systems 219: 49–67. DOI: 10.1016/j.fss.2012.12.004.
  44. [44] M.T. Malinowski, (2012) “Itô type stochastic fuzzy dif ferential equations with delay" Systems & Control Let ters 61(6): 692–701. DOI: 10.1016/j.sysconle.2012.02.012.
  45. [45] R. I. Sabri and B. A. Ahmed, (2023) “On α − φ − Fuzzy Contractive Mapping in Fuzzy Normed Space" Baghdad Science Journal: DOI: 10.21123/bsj.2023.2509.
  46. [46] N. R. Kareem, F. S. Fadhel, and S. Al-Nassir, (2023) “Existence and Uniqueness Theorem of Fuzzy Stochastic Ordinary Differential Equations" Iraqi Journal of Sci ence 64(11): 5878–5886. DOI: 10.24996/ijs.2023.64.11.33.
  47. [47] R. I. Sabri and B. A. Ahmed, (2023) “Another Type of Fuzzy Inner Product Space" Iraqi Journal of Science 64(4): 1853–1861. DOI: 10.24996/ijs.2023.64.4.25.
  48. [48] J.-C. Cortés, J.-V. Romero, M.-D. Roselló, and C. San tamaría, (2011) “Solving random diffusion models with nonlinear perturbations by the Wiener–Hermite expan sion method" Computers & Mathematics with Appli cations 61(8): 1946–1950. DOI: 10.1016/j.camwa.2010.07.057.
  49. [49] J. C. Butcher. The numerical analysis of ordinary differ ential equations: Runge-Kutta and general linear methods. Wiley-Interscience, 1987.