Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Numerical solutions of fuzzy differential equations using reproducing kernel Hilbert space method

  • 784 Accesses

  • 141 Citations


Modeling of uncertainty differential equations is very important issue in applied sciences and engineering, while the natural way to model such dynamical systems is to use fuzzy differential equations. In this paper, we present a new method for solving fuzzy differential equations based on the reproducing kernel theory under strongly generalized differentiability. The analytic and approximate solutions are given with series form in terms of their parametric form in the space \(W_2^2 [a,b]\oplus W_2^2 [a,b].\) The method used in this paper has several advantages; first, it is of global nature in terms of the solutions obtained as well as its ability to solve other mathematical, physical, and engineering problems; second, it is accurate, needs less effort to achieve the results, and is developed especially for the nonlinear cases; third, in the proposed method, it is possible to pick any point in the interval of integration and as well the approximate solutions and their derivatives will be applicable; fourth, the method does not require discretization of the variables, and it is not effected by computation round off errors and one is not faced with necessity of large computer memory and time. Results presented in this paper show potentiality, generality, and superiority of our method as compared with other well-known methods.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5


  1. Abbasbandy S, Nieto JJ, Alavi M (2005) Tuning of reachable set in one dimensional fuzzy differential inclusions. Chaos, Solitons Fractals 26:1337–1341. doi:10.1016/j.chaos.2005.03.018

  2. Abu Arqub O (2013) Series solution of fuzzy differential equations under strongly generalized differentiability. J Adv Res Appl Math 5:31–52. doi:10.5373/jaram.1447.051912

  3. Abu Arqub O, Al-Smadi M, Shawagfeh N (2013a) Solving Fredholm integro-differential equations using reproducing kernel Hilbert space method. Appl Math Comput 219:8938–8948. doi:10.1016/j.amc.2013.03.006

  4. Abu Arqub O, El-Ajou A, Momani S, Shawagfeh N (2013b) Analytical solutions of fuzzy initial value problems by HAM. Appl Math Inf Sci 7:1903–1919. doi:10.12785/amis/070528

  5. Abu Arqub O, Al-Smadi M (2014) Numerical algorithm for solving two-point, second-order periodic boundary value problems for mixed integro-differential equations. Appl Math Comput 243:911–922. doi:10.1016/j.amc.2014.06.063

  6. Abu Arqub O (2015) An iterative method for solving fourth-order boundary value problems of mixed type integro-differential equations. J Comput Anal Appl 8:857–874

  7. Abu Arqub O, Al-Smadi M, Momani S (2012) Application of reproducing kernel method for solving nonlinear Fredholm–Volterra integro-differential equations. Abstr Appl Anal. doi:10.1155/2012/839836

  8. Abu Arqub O, Momani S, Al-Mezel S, Kutbi M (2015) Existence, uniqueness, and characterization theorems for nonlinear fuzzy integrodifferential equations of Volterra type. Math Probl Eng. doi:10.1155/2015/835891

  9. Ahmad MZ, Hasan MK, De Baets B (2013) Analytical and numerical solutions of fuzzy differential equations. Inf Sci 236:156–167. doi:10.1016/j.ins.2013.02.026

  10. Allahviranloo T, Abbasbandy S, Ahmady N, Ahmady E (2009) Improved predictor–corrector method for solving fuzzy initial value problems. Inf Sci 179:945–955. doi:10.1016/j.ins.2008.11.030

  11. Allahviranloo T, Kiani NA, Motamedi N (2009) Solving fuzzy differential equations by differential transformation method. Inf Sci 179:956–966. doi:10.1016/j.ins.2008.11.016

  12. Al-Smadi M, Abu Arqub O, El-Ajuo A (2014) A numerical method for solving systems of first-order periodic boundary value problems. J Appl Math. doi:10.1155/2014/135465

  13. Al-Smadi M, Abu Arqub O, Momani S (2013) A computational method for two-point boundary value problems of fourth-order mixed integro-differential equations. Math Probl Eng. doi:10.1155/2013/832074

  14. Bede B, Gal SG (2004) Almost periodic fuzzy-number-valued functions. Fuzzy Sets Syst 147:385–403. doi:10.1016/j.fss.2003.08.004

  15. Bede B, Gal SG (2005) Generalizations of the differentiability of fuzzy number value functions with applications to fuzzy differential equations. Fuzzy Sets Syst 151:581–599. doi:10.1016/j.fss.2004.08.001

  16. Bede B, Rudas IJ, Bencsik AL (2007) First order linear fuzzy differential equations under generalized differentiability. Inf Sci 177:1648–1662. doi:10.1016/j.ins.2006.08.021

  17. Berlinet A, Agnan CT (2004) Reproducing kernel Hilbert Space in probability and statistics. Kluwer Academic Publishers, Boston. doi:10.1007/978-1-4419-9096-9

  18. Buckley JJ, Feuring T (2000) Fuzzy differential equations. Fuzzy Sets Syst 110:43–54. doi:10.1016/S0165-0114(98)00141-9

  19. Chalco-Cano Y, Román-Flores H (2008) On new solutions of fuzzy differential equations. Chaos, Solitons and Fractals 38:112–119. doi:10.1016/j.chaos.2006.10.043

  20. Cui M, Lin Y (2009) Nonlinear numercial analysis in the reproducing kernel space. Nova Science Publisher, New York

  21. Daniel A (2003) Reproducing kernel spaces and applications. Springer, Basel. doi:10.1007/978-3-0348-8077-0

  22. Diamond P (1999) Time-dependent differential inclusions, cocycle attractors and fuzzy differential equations. IEEE Trans Fuzzy Syst 7:734–740. doi:10.1109/91.811243

  23. Effati S, Pakdaman M (2010) Artificial neural network approach for solving fuzzy differential equations. Inf Sci 180:1434–1457. doi:10.1016/j.ins.2009.12.016

  24. Gasilov N, Amrahov ŞE, Fatullayev AG (2014) Solution of linear differential equations with fuzzy boundary values. Fuzzy Sets Syst 257:169–183. doi:10.1016/j.fss.2013.08.008

  25. Geng F (2009) Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method. Appl Math Comput 215:2095–2102. doi:10.1016/j.amc.2009.08.002

  26. Geng F, Cui M (2012) A reproducing kernel method for solving nonlocal fractional boundary value problems. Appl Math Lett 25:818–823. doi:10.1016/j.aml.2011.10.025

  27. Geng F, Qian SP (2013) Reproducing kernel method for singularly perturbed turning point problems having twin boundary layers. Appl Math Lett 26:998–1004. doi:10.1016/j.aml.2013.05.006

  28. Geng F, Qian SP, Li S (2014) A numerical method for singularly perturbed turning point problems with an interior layer. J Comput Appl Math 255:97–105. doi:10.1016/j.cam.2013.04.040

  29. Goetschel R, Voxman W (1986) Elementary fuzzy calculus. Fuzzy Sets Syst 18:31–43. doi:10.1016/0165-0114(86)90026-6

  30. Hüllermeier E (1997) An approach to modelling and simulation of uncertain systems. Int J Uncertain Fuzziness Knowl-Based Syst 5:117–137. doi:10.1142/S0218488597000117

  31. Jiang W, Chen Z (2013) Solving a system of linear Volterra integral equations using the new reproducing kernel method. Appl Math Comput 219:10225–10230. doi:10.1016/j.amc.2013.03.123

  32. Jiang W, Chen Z (2014) A collocation method based on reproducing kernel for a modified anomalous subdiffusion equation. Numer Methods Partial Differ Equ 30:289–300. doi:10.1002/num.21809

  33. Kaleva O (1987) Fuzzy differential equations. Fuzzy Sets Syst 24:301–317. doi:10.1016/0165-0114(87)90029-7

  34. Kaleva O (2006) A note on fuzzy differential equations. Nonlinear Anal: Theory, Methods Appl 64:895–900. doi:10.1016/j.na.2005.01.003

  35. Khastan A, Ivaz K (2009) Numerical solution of fuzzy differential equations by Nystrom method. Chaos, Solitons Fractals 41:859–868. doi:10.1016/j.chaos.2008.04.012

  36. Khastan A, Nieto JJ, López RR (2011) Variation of constant formula for first order fuzzy differential equations. Fuzzy Sets Syst 177:20–33. doi:10.1016/j.fss.2011.02.020

  37. Li C, Cui M (2003) The exact solution for solving a class nonlinear operator equations in the reproducing kernel space. Appl Math Comput 143:393–399. doi:10.1016/S0096-3003(02)00370-3

  38. Ma M, Friedman M, Kandel A (1999) Numerical solution of fuzzy differential equations. Fuzzy Sets Syst 105:133–138. doi:10.1016/S0165-0114(97)00233-9

  39. Maayah B, Bushnaq S, Momani S, Abu Arqub O (2014) Iterative multistep reproducing kernel Hilbert space method for solving strongly nonlinear oscillators. Adv Math Phys. doi:10.1155/2014/758195

  40. Mizukoshi MT, Barros LC, Chalco-Cano Y, Román-Flores H, Bassanezi RC (2007) Fuzzy differential equations and the extension principle. Inf Sci 177:3627–3635. doi:10.1016/j.ins.2007.02.039

  41. Momani S, Abu Arqub O, Hayat T, Al-Sulami H (2014) A computational method for solving periodic boundary value problems for integro-differential equations of Fredholm–Voltera type. Appl Math Comput 240:229–239. doi:10.1016/j.amc.2014.04.057

  42. Mosleh M (2013) Fuzzy neural network for solving a system of fuzzy differential equations. Appl Soft Comput 13:3597–3607. doi:10.1016/j.asoc.2013.04.013

  43. Nguyen HT (1978) A note on the extension principle for fuzzy set. J Math Anal Appl 64:369–380. doi:10.1016/0022-247X(78)90045-8

  44. Nieto JJ, Khastan A, Ivaz K (2009) Numerical solution of fuzzy differential equations under generalized differentiability. Nonlinear Anal Hybrid Syst 3:700–707. doi:10.1016/j.nahs.2009.06.013

  45. Oberguggenberger M, Pittschmann S (1999) Differential equations with fuzzy parameters. Math Comput Model Dyn Syst 5:181–202. doi:10.1076/mcmd.

  46. Palligkinis SCh, Papageorgiou G, Famelis ITh (2009) Runge–Kutta methods for fuzzy differential equations. Appl Math Comput 209:97–105. doi:10.1016/j.amc.2008.06.017

  47. Puri ML, Ralescu DA (1983) Differentials of fuzzy functions. J Math Anal Appl 91:552–558. doi:10.1016/0022-247X(83)90169-5

  48. Puri ML (1986) Fuzzy random variables. J Math Anal Appl 114:409–422. doi:10.1016/0022-247X(86)90093-4

  49. Seikkala S (1987) On the fuzzy initial value problem. Fuzzy Sets Syst 24:319–330. doi:10.1016/0165-0114(87)90030-3

  50. Shawagfeh N, Abu Arqub O, Momani S (2014) Analytical solution of nonlinear second-order periodic boundary value problem using reproducing kernel method. J Comput Anal Appl 16:750–762

  51. Song S, Wu C (2000) Existence and uniqueness of solutions to the Cauchy problem of fuzzy differential equations. Fuzzy Set Syst 110:55–67. doi:10.1016/S0165-0114(97)00399-0

  52. Weinert HL (1982) Reproducing kernel Hilbert spaces: applications in statistical signal processing. Hutchinson Ross Publishing Company, Stroudsburg

Download references


The authors express their thanks to unknown referees for the careful reading and helpful comments.

Author information

Correspondence to Omar Abu Arqub.

Additional information

Communicated by V. Loia.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Abu Arqub, O., AL-Smadi, M., Momani, S. et al. Numerical solutions of fuzzy differential equations using reproducing kernel Hilbert space method. Soft Comput 20, 3283–3302 (2016). https://doi.org/10.1007/s00500-015-1707-4

Download citation


  • Fuzzy differential equations
  • Strongly generalized differentiability
  • Reproducing kernel Hilbert space method
  • Gram–Schmidt process