Soft Computing

, Volume 21, Issue 23, pp 7191–7206 | Cite as

Application of reproducing kernel algorithm for solving second-order, two-point fuzzy boundary value problems

  • Omar Abu Arqub
  • Mohammed Al-Smadi
  • Shaher Momani
  • Tasawar Hayat
Methodologies and Application


In this paper, we investigate the analytic and approximate solutions of second-order, two-point fuzzy boundary value problems based on the reproducing kernel theory under the assumption of strongly generalized differentiability. The solution methodology is based on generating the orthogonal basis from the obtained kernel functions, while the orthonormal basis is constructing in order to formulate and utilize the solutions with series form in terms of their r-cut representation in the space \(\oplus _{j=1}^2 W_2^3 \left[ {a,b}\right] \). An efficient computational algorithm is provided to guarantee the procedure and to confirm the performance of the proposed method. Results of numerical experiments are provided to illustrate the theoretical statements in order to show potentiality, generality, and superiority of our algorithm for solving such fuzzy equations. Graphical results, tabulated data, and numerical comparisons are presented and discussed quantitatively to illustrate the possible fuzzy solutions.


Fuzzy boundary value problems Reproducing kernel theory Strongly generalized differentiability Gram–Schmidt process 


  1. Abu Arqub O (2015) Adaptation of reproducing Kernel algorithm for solving fuzzy Fredholm-Volterra integrodifferential equations. Neural Comput Appl. doi:10.1007/s00521-015-2110-x Google Scholar
  2. Abu Arqub O (2016) The reproducing kernel algorithm for handling differential algebraic systems of ordinary differential equations. Math Method Appl Sci. doi:10.1002/mma.3884 MATHGoogle Scholar
  3. 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 MATHMathSciNetGoogle Scholar
  4. Abu Arqub O, Al-Smadi M, Shawagfeh N (2013) 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 MATHMathSciNetGoogle Scholar
  5. 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 CrossRefMathSciNetGoogle Scholar
  6. Abu Arqub O, Momani S, Al-Mezel S, Kutbi M (2015a) Numerical solutions of fuzzy differential equations using reporducing kernel Hilbert space mehtod. Soft Comput. doi:10.1007/s00500-015-1707-4 MATHGoogle Scholar
  7. 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 MATHMathSciNetGoogle Scholar
  8. Al-Smadi M, Abu Arqub O, Shawagfeh N, Momani S (2016) Numerical investigations for systems of second-order periodic boundary value problems using reproducing kernel method. Appl Math Comput. In PressGoogle Scholar
  9. Bede B (2006) A note on “two-point boundary value problems associated with non-linear fuzzy differential equations”. Fuzzy Sets Syst 157:986–989. doi:10.1016/j.fss.2005.09.006 CrossRefMATHMathSciNetGoogle Scholar
  10. 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 CrossRefMATHMathSciNetGoogle Scholar
  11. 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 CrossRefMATHGoogle Scholar
  12. 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 MATHGoogle Scholar
  13. Chalco-Cano Y, Román-Flores H (2008) On new solutions of fuzzy differential equations. Chaos Solitons Fractals 38:112–119. doi:10.1016/j.chaos.2006.10.043 CrossRefMATHMathSciNetGoogle Scholar
  14. Chen M, Fu Y, Xue X, Wu C (2008) Two-point boundary value problems of undamped uncertain dynamical systems. Fuzzy Sets Syst 159:2077–2089. doi:10.1016/j.fss.2008.03.006 CrossRefMATHMathSciNetGoogle Scholar
  15. Chen M, Wu C, Xue X, Liu G (2008b) On fuzzy boundary value problems. Inf Sci 178:1877–1892. doi:10.1016/j.ins.2007.11.017 CrossRefMATHMathSciNetGoogle Scholar
  16. Cherki A, Plessis G, Lallemand B, Tison T, Level P (2000) Fuzzy behavior of mechanical systems with uncertain boundary conditions. Comput Methods Appl Mech Eng 189:863–873. doi:10.1016/S0045-7825(99)00401-6 CrossRefMATHGoogle Scholar
  17. Cui M, Lin Y (2009) Nonlinear numerical analysis in the reproducing kernel space. Nova Science, New YorkMATHGoogle Scholar
  18. Daniel A (2003) Reproducing kernel spaces and applications. Springer, Basel. doi:10.1007/978-3-0348-8077-0 MATHGoogle Scholar
  19. Gasilov N, Amrahov E, 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 CrossRefMATHMathSciNetGoogle Scholar
  20. Gasilov N, Amrahov E, Fatullayev AG, Hashimoglu IF (2015) Solution method for a boundary value problem with fuzzy forcing function. Inf Sci 317:349–368. doi:10.1016/j.ins.2015.05.002
  21. 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/
  22. 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 CrossRefMATHMathSciNetGoogle Scholar
  23. 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 CrossRefMATHMathSciNetGoogle Scholar
  24. Goetschel R, Voxman W (1986) Elementary fuzzy calculus. Fuzzy Sets Syst 18:31–43. doi:10.1016/0165-0114(86)90026-6
  25. Guo X, Shang D, Lu X (2013) Fuzzy approximate solutions of second-order fuzzy linear boundary value problems. Bound Value Probl. doi:10.1186/1687-2770-2013-212 MATHMathSciNetGoogle Scholar
  26. 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 CrossRefMATHMathSciNetGoogle Scholar
  27. 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 MATHMathSciNetGoogle Scholar
  28. 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 CrossRefMATHMathSciNetGoogle Scholar
  29. Kaleva O (1987) Fuzzy differential equations. Fuzzy Sets Syst 24:301–317. doi:10.1016/0165-0114(87)90029-7 CrossRefMATHMathSciNetGoogle Scholar
  30. Khastan A, Bahrami F, Ivaz K (2009) New results on multiple solutions for Nth-order fuzzy differential equations under generalized differentiability. Bound Value Probl. doi:10.1155/2009/395714 MATHGoogle Scholar
  31. Khastan A, Nieto JJ (2010) A boundary value problem for second order fuzzy differential equations. Nonlinear Anal 72:3583–3593. doi:10.1016/ CrossRefMATHMathSciNetGoogle Scholar
  32. Lakshmikantham V, Murty KN, Turner J (2001) Two-point boundary value problems associated with non-linear fuzzy differential equations. Math Inequal Appl 4:527–533. doi:10.7153/mia-04-47 MATHMathSciNetGoogle Scholar
  33. 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 MATHMathSciNetGoogle Scholar
  34. Li D, Chen M, Xue X (2011) Two-point boundary value problems of uncertain dynamical systems. Fuzzy Sets Syst 179:50–61. doi:10.1016/j.fss.2011.05.012 CrossRefMATHMathSciNetGoogle Scholar
  35. Mohammadi M, Mokhtari R (2011) Solving the generalized regularized long wave equation on the basis of a reproducing kernel space. J Comput Appl Math 235:4003–4014CrossRefMATHMathSciNetGoogle Scholar
  36. 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 MATHMathSciNetGoogle Scholar
  37. Mosleh M, Otadi M (2015) Approximate solution of fuzzy differential equations under generalized differentiability. Appl Math Model 39:3003–3015. doi:10.1016/j.apm.2014.11.035 CrossRefMATHMathSciNetGoogle Scholar
  38. 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 CrossRefMATHMathSciNetGoogle Scholar
  39. O’Regana D, Lakshmikantham V, Nieto JJ (2003) Initial and boundary value problems for fuzzy differential equations. Nonlinear Anal 54:405–415. doi:10.1016/S0362-546X(03)00097-X CrossRefMATHMathSciNetGoogle Scholar
  40. Wang W, Cui M, Han B (2008) A new method for solving a class of singular two-point boundary value problems. Appl Math Comput 206:721–727. doi:10.1016/j.amc.2008.09.019 MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Omar Abu Arqub
    • 1
  • Mohammed Al-Smadi
    • 2
  • Shaher Momani
    • 3
  • Tasawar Hayat
    • 4
  1. 1.Department of Mathematics, Faculty of ScienceAl Balqa Applied UniversitySaltJordan
  2. 2.Department of Applied Science, Ajloun CollegeAl Balqa Applied UniversityAjlounJordan
  3. 3.Department of Mathematics, Faculty of ScienceUniversity of JordanAmmanJordan
  4. 4.Department of Mathematics, Faculty of ScienceQuaid-I-Azam UniversityIslamabadPakistan

Personalised recommendations