Skip to main content
SpringerLink
Log in

Combining fractional differential transform method and reproducing kernel Hilbert space method to solve fuzzy impulsive fractional differential equations

  • Original Article
  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

The aim of this paper is to use the combination of Reproducing kernel Hilbert space method (RKHSM) and fractional differential transform method (FDTM) to solve the linear and nonlinear fuzzy impulsive fractional differential equations. Finding the numerical solutions of this class of equations are a difficult topic to analyze. In this study, convergence analysis, estimations error and bounds errors are discussed in detail under some hypotheses which provide the theoretical basis of the proposed algorithm. Some numerical examples indicate that this method is an efficient one to solve the mentioned equations.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2

Similar content being viewed by others

References

  • Allahviranloo T, Gouyandeh Z, Armand A (2015) A full fuzzy method for solving differential equation based on taylor expansion. J Intell Fuzzy Syst 1(4):1–16

    MathSciNet  MATH  Google Scholar 

  • Alvandi A, Paripour M (2016) Combined reproducing kernel method and taylor series for solving nonlinear Volterra–Fredholm integro-differential equations. Int J Math Model 6(4):301–312

    MATH  Google Scholar 

  • Armand A, Allahviranloo T, Abbasbandy S, Gouyandeh Z (2019) The fuzzy generalized taylor’s expansion with application in fractional diferential equations. Iran J Fuzzy Syst 16(2):57–72

    MathSciNet  MATH  Google Scholar 

  • Arqub OA, Maayah B (2018) Solutions of bagley-torvik and painlevé equations of fractional order using iterative reproducing kernel algorithm. Neural Comput 29(4):1465–1479

    Article  Google Scholar 

  • Arqub OA, Al-Smadi M, Momani S (2012) Application of reproducing kernel method for solving nonlinear fredholm-volterra. Hindawi Publ Corp Abstr Appl Anal 29(4):1–16

    MATH  Google Scholar 

  • Bede B, Stefanini (2013) Generalized differentiability of fuzzy-valued function. Fuzzy Sets Syst 230(4):119–141

    Article  MathSciNet  MATH  Google Scholar 

  • Benchohra M, Seba D (2009) Impulsive fractional differential equations. Electron J Qual Theory Differ Eq Spec Ed I 8(4):1–14

    MATH  Google Scholar 

  • Benchohra M, Slimani A (2009) Existence and uniqueness of solutions to impulsive fractional, differential equations. Electron J Differ Eq 4:12–19

    MathSciNet  MATH  Google Scholar 

  • Cui MG, Geng FZ (2007) Soiving singular two-point boundary value problem in reproducing kernel space. J Comput Appl Math 205:6–15

    Article  MathSciNet  MATH  Google Scholar 

  • Du H, Cui M (2008) Approximate solution of the fredhom integral equation of the first kind in a reproducing kernel hilbert space. Appl Math Lett 21(4):617–623

    Article  MathSciNet  MATH  Google Scholar 

  • Du H, JiHong S (2014) Reproducing kernel method for solving singular inregral equation with cosecant kernel. J Comput Appl Math 255(4):308–314

    MATH  Google Scholar 

  • Du H, Zhao C (2014) Reproducing kernel method for solving fredholm integro-differential equations with weakly singularity. J Comput Appl Math 255(4):308–314

    MathSciNet  MATH  Google Scholar 

  • Dwijendra NAC, Pandey (2016) Approximations of solutions for an impulsive fractional differential equation with a deviated argument. Int J Appl Comput Math 2(4):269–289

    MathSciNet  MATH  Google Scholar 

  • FarzanehJavan S, Abbasbandy S, Araghi MA (2017) Reproducing kernel hilbert space method for solving a class of nonlinear integral equations. Math Probl Eng 10:1–17

    Article  MathSciNet  Google Scholar 

  • Friedman M, Ma M, Kandel (1999) Numerical solutions of fuzzy differential and integral equations. Fuzzy Sets Syst 106:35–48

    Article  MathSciNet  MATH  Google Scholar 

  • Genga F, Minggen Cui b, Bo Z (2010) Method for solving nonlinear initial value problems by combining homotopy perturbation and reproducing kernel hilbert space methods. Nonlinear Anal Real World Appl 11:637–644

    Article  MathSciNet  MATH  Google Scholar 

  • Kaleva O (1987) Fuzzy differential equations. Fuzzy Sets Syst 24:301–317

    Article  MathSciNet  MATH  Google Scholar 

  • Mohammadi M, Mokhtari R (2011) Solving the generalized regularized long wave equation on the basis of a reproducing kernel space. J Comput Appl Math 235:4003–4014

    Article  MathSciNet  MATH  Google Scholar 

  • Najafi N (2018a) Computing the fuzzy fractional differential transform method based on the concept of generalized hukuhara differentiability. In: Conference IDS03-IDS03-024

  • Najafi N (2018b) Solving fuzzy impulsive fractional differential equations by homotopy pertourbation method. Int J Math Model Comput (IJM2C) (31:2–3

    Google Scholar 

  • Najafi N, Allahviranloo T (2017) Semi-analytical methods for solving fuzzy impulsive fractional differential equations. J Intell Fuzzy Syst 33:3539–3560

    Article  Google Scholar 

  • Nazari D, Shahmorad S (2018) Application of the fractional differential transform methodto fractional-orderintegro-differential equations withnonlocal boundary conditions. J Comput Appl Math 234:883–891

    Article  MATH  Google Scholar 

  • Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier 198:1–10

    Google Scholar 

  • Sahihi H, Abbasbandy S, Allahviranloo T (2018) Generalized hukuhara differentiability of interval-valued function and interval differential equation. J Comput Appl Math Arch 328:30–43

    Article  MATH  Google Scholar 

  • Stefanini L, Bede B (2009) Generalized hukuhara differentiability of interval-valued function and interval differential equation. Nonlinear Anal 71:1311–1328

    Article  MathSciNet  MATH  Google Scholar 

  • Wang Q, Lu D, Fang Y (2015) Stability analysis of impulsive fractional differential. Appl Math Lett 40:1–6

    Article  MathSciNet  MATH  Google Scholar 

  • Wang W, Long Y (2010) On some impulsive fractional differential equations in banach spaces. Opuscula Math 30:507–525

    Article  MathSciNet  Google Scholar 

  • Xueqin L, Sixing S (1965) The combined rkm and adm forsolving nonlinear weakly singular volterra integrodifferential equations. Inf Control 8:1–10

    Article  MATH  Google Scholar 

  • Zadeh L (1965) Fuzzy sets. Hindawi Publ Corp Abst Appl Anal 2:338–353

    MATH  Google Scholar 

  • Zimmerman HJ (1996) Fuzzy set theory and its applications. Kluwer Academic, NewYork, pp 33–53

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Islamic Azad University, Hamedan Branch, Hamedan, Iran

    Nematallah Najafi

  2. Faculty of engineering and natural sciences, Bahcesehir university, Istanbul, Turkey

    Tofigh Allahviranloo

Authors
  1. Nematallah Najafi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tofigh Allahviranloo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tofigh Allahviranloo.

Additional information

Communicated by Rosana Sueli da Motta Jafelice.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Najafi, N., Allahviranloo, T. Combining fractional differential transform method and reproducing kernel Hilbert space method to solve fuzzy impulsive fractional differential equations. Comp. Appl. Math. 39, 122 (2020). https://doi.org/10.1007/s40314-020-01140-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-020-01140-8

Keywords

Mathematics Subject Classification

Search

Navigation