Skip to main content
SpringerLink
Log in

Fuzzy natural transform method for solving fuzzy differential equations

  • Fuzzy systems and their mathematics
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

This article establishes a new transform method to seek the solution to some fuzzy differential equations. For the first time in the literature, the fuzzy natural transform method (FNTM) is introduced to solve the integer-order differential equations and non-integer-order differential equations in an uncertain environment. To do this, some primary results of FNTM are presented for integer- and non-integer-order derivatives in the Riemann–Liouville and Caputo senses under generalized Hukuhara differentiability. The method is illustrated by providing some examples.

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.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  • Ahmadian A, Salahshour S, Chan CS (2017) Fractional differential systems: a fuzzy solution based on operational matrix of shifted Chebyshev polynomials and its applications. IEEE Transa Fuzzy Syst 25:218–236

    Google Scholar 

  • Ahmadian A, Ismail F, Salahshour S, Baleanu D, Ghaemi F (2017) Uncertain viscoelastic models with fractional order: a new spectral tau method to study the numerical simulations of the solution. Commun Nonlinear Sci Numer Simul 53:44–64

    MathSciNet  MATH  Google Scholar 

  • Alijani Z, Baleanu D, Shiri B, Wu GC (2020) Spline collocation methods for systems of fuzzy fractional differential equations. Chaos Solitons Fractals 131:109510

    MathSciNet  MATH  Google Scholar 

  • Alijani Z, Kangro U (2021) On the smoothness of the solution of fuzzy Volterra integral equations of the second kind with weakly singular kernels. Numer Funct Anal Optim 42:819–833

    MathSciNet  MATH  Google Scholar 

  • Alijani Z, Kangro U (2022) Numerical solution of a linear fuzzy Volterra integral equation of the second kind with weakly singular kernels. Soft Comput 26:12009–12022

    Google Scholar 

  • Allahviranloo T, Gouyandeh Z, Armand A, Hasanoglu A (2015) On fuzzy solutions for heat equation based on generalized Hukuhara differentiability. Fuzzy Sets Syst 265:1–23

    MathSciNet  MATH  Google Scholar 

  • Allahviranloo T, Ahmadi MB (2010) Fuzzy Laplace transforms. Soft Comput 14:235–243

    MATH  Google Scholar 

  • Allahviranloo T (2021) Fuzzy fractional differential operators and equation, Studies in Fuzziness and Soft Computing, 397

  • Asiru MA (2002) Further properties of Sumudu Transform and its applications. Int J Math Educ Sci Technol 33(2):441–449

    MathSciNet  MATH  Google Scholar 

  • Asiru MA (2003) Classroom note: application of the Sumudu transform to discrete dynamic systems. Int J Math Educ Sci Technol 34:944–949

    Google Scholar 

  • Baleanu D, Shiri B (2022) Generalized fractional differential equations for past dynamic. AIMS Math 7:14394–14418

    MathSciNet  Google Scholar 

  • Baleanu D, Shiri B (2022) Nonlinear higher order fractional terminal value problems. AIMS Math 7:7489–7506

    MathSciNet  Google Scholar 

  • Bede B, Rudas I, Bencsik A (2007) First order linear fuzzy differential equations under generalized differentiability. Inf Sci 177:1648–1662

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Belgacem FBM, Karaballi AA, Kalla SL (2003) Analytical investigations of the Sumudu transform and applications to integral production equations. Math Probl Eng 3:103–118

    MathSciNet  MATH  Google Scholar 

  • Belgacem FBM (2010) Sumudu transform applications to Bessel’s Functions and Equations. Appl Math Sci 4:3665–3686

    MathSciNet  MATH  Google Scholar 

  • da Costa TM, Chalco-Cano Y, Lodwick WA, Silva GN (2018) A new approach to linear interval differential equations as a first step toward solving fuzzy differential. Fuzzy Sets Syst 347:129–141

    MathSciNet  MATH  Google Scholar 

  • Debnath L, Bhatta D (2007) Integral transforms and their applications, 2nd edn. C.R.C. Press, London

    MATH  Google Scholar 

  • Gasilov NA, Amrahov ŞE, Fatullayev AG (2016) On a solution of the fuzzy Dirichlet problem for the heat equation. Int J Therm Sci 103:67–76

    Google Scholar 

  • Ghaffari M, Allahviranloo T, Abbasbandy S, Azhini M (2021) On the fuzzy solutions of time-fractional problems. Iranian J Fuzzy Syst 18:51–66

    MathSciNet  MATH  Google Scholar 

  • Guo Y, Ye G, Liu W, Zhao D, Treanţă S (2021) Optimality conditions and duality for a class of generalized convex interval-valued optimization problems. Mathematics 9:2979

    Google Scholar 

  • Guo Y, Ye G, Liu W, Zhao D, Treanţă S (2022) On symmetric gH-derivative: applications to dual interval-valued optimization problems. Chaos Solitons Fractals 158:112068

    MathSciNet  MATH  Google Scholar 

  • Hoa NV, Lupulescu V, O’Regan D (2018) A note on initial value problems for fractional fuzzy differential equations. Fuzzy Sets Syst 347:54–69

    MathSciNet  MATH  Google Scholar 

  • Hoa NV, Vu H, Minh Duc T (2019) Fuzzy fractional differential equations under Caputo Katugampola fractional derivative approach. Fuzzy Sets Syst 375:70–99

    MathSciNet  MATH  Google Scholar 

  • Hoa NV (2015) Fuzzy fractional functional differential equations under Caputo gH- differentiability. Commun Nonlinear Sci Numer Simul 22:1134–1157

    MathSciNet  MATH  Google Scholar 

  • Khan ZH, Khan WA (2008) N-transform properties and applications. NUST J Eng Sci 1(1):127–133

    Google Scholar 

  • Keshavarz M, Allahviranloo T (2021) Fuzzy fractional diffusion processes and drug release, Fuzzy Sets and Systems

  • Kilicman A, Eltayeb, H, Agarwal RP (2010) On Sumudu transform and system of differential equations. Abstr Appl Anal. 2010

  • Long HV, Son NTK, Tam HTT (2017) The solvability of fuzzy fractional partial differential equations under Caputo gH-differentiability. Fuzzy Sets Syst 309:35–63

    MathSciNet  MATH  Google Scholar 

  • Long HV, Son NTK, Hoa NV (2017) Fuzzy fractional partial differential equations in partially ordered metric spaces. Iranian J Fuzzy Syst 14:107–126

    MathSciNet  MATH  Google Scholar 

  • Long HV, Son NTK, Tam HTT, Yao JC (2017) Ulam stability for fractional partial integro-differential equation with uncertainty. Acta Math Vietnam 42:675–700

    MathSciNet  MATH  Google Scholar 

  • Mazandarani M, Kamyad AV (2013) Modified fractional Euler method for solving fuzzy fractional initial value problem. Commun Nonlinear Sci Numer Simul 18:12–21

    MathSciNet  MATH  Google Scholar 

  • Mazandarani M, Pariz N, Kamyad AV (2018) Granular differentiability of fuzzy-number-valued functions. IEEE Trans Fuzzy Syst 26:310–323

    Google Scholar 

  • Mazandarani M, Zhao Y (2020) Z-differential equations. IEEE Trans Fuzzy Syst 28:462–473

    Google Scholar 

  • Najariyan M, Zhao Y (2020) On the stability of fuzzy linear dynamical systems. J Franklin Inst 357:5502–5522

    MathSciNet  MATH  Google Scholar 

  • Najariyan M, Zhao Y (2017) Fuzzy fractional quadratic regulator problem under granular fuzzy fractional derivatives. IEEE Trans Fuzzy Syst 26:2273–2288

    Google Scholar 

  • Pedrycz W, Gomide F (1998) An introduction to fuzzy sets: analysis and design, Massachusetts Institute of Technology

  • Pham VN, Ngo LT, Pedrycz W (2016) Interval-valued fuzzy set approach to fuzzy co-clustering for data classification. Knowl-Based Syst 107:1–13

    Google Scholar 

  • Salahshour S, Allahviranloo T, Abbasbandy S (2012) Solving fuzzy fractional differential equations by fuzzy Laplace transforms. Commun Nonlinear Sci Numer Simul 17:1372–1381

    MathSciNet  MATH  Google Scholar 

  • Salahshour S, Allahviranloo T (2013) Applications of fuzzy Laplace transforms. Soft Comput 17:145–158

    MATH  Google Scholar 

  • Schiff LJ (2005) Laplace transform theory and applications, Auckland New-Zealand,. Springer, Berlin

    Google Scholar 

  • Shiri B (2023) A unified generalization for Hukuhara types differences and derivatives: solid analysis and comparisons. AIMS Math 8:2168–2190

    MathSciNet  Google Scholar 

  • Son NTK, Long HV, Dong NP (2019) Fuzzy delay differential equations under granular differentiability with applications. Comput Appl Math 38:107

    MathSciNet  MATH  Google Scholar 

  • Spiegel MR (1965) Theory and problems of Laplace transforms, Schaums outline series. McGraw Hill, New York

    Google Scholar 

  • Treanţă S (2021) LU-optimality conditions in optimization problems with mechanical work objective functionals. IEEE Trans Neural Netw Learn Syst 33:4971–4978

    MathSciNet  Google Scholar 

  • Treanţă S (2021) On a class of constrained interval-valued optimization problems governed by mechanical work cost functionals. J Optim Theory Appl 188:913–924

    MathSciNet  MATH  Google Scholar 

  • Treanţă S (2021) Efficiency in uncertain variational control problems. Neural Comput Appl 33:5719–5732

    Google Scholar 

  • Treanţă S (2022) Characterization results of solutions in interval-valued optimization problems with mixed constraints. J Global Optim 82:951–964

    MathSciNet  MATH  Google Scholar 

  • Treanţă S (2022) Saddle-point optimality criteria involving (p, b, d)-invexity and (p, b, d)-pseudoinvexity in interval-valued optimisation problems. Int J Control 95:1042–1050

    MathSciNet  MATH  Google Scholar 

  • Watugala GK (1998) Sumudu transform-a new integral transform to solve differential equations and control engineering problems. Math Eng Ind 6:319–329

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to express their deep gratitude to the anonymous referees for their valuable comments and suggestions, which have greatly improved this paper.

Funding

No funding was provided to the completion of this study.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Malakand, Chakdara, Dir Lower, Khyber Pakhunkhwa, Pakistan

    Shabir Ahmad, Aman Ullah & Abd Ullah

  2. Laboratory for Applied and Industrial Mathematics, Institute for Computational Science and Artificial Intelligence, Van Lang University, Ho Chi Minh City, Vietnam

    Ngo Van Hoa

  3. Faculty of Basic Sciences, Van Lang University, Ho Chi Minh City, Vietnam

    Ngo Van Hoa

Authors
  1. Shabir Ahmad
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Aman Ullah
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Abd Ullah
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ngo Van Hoa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ngo Van Hoa.

Ethics declarations

Competing Interests

The authors declare that they has no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Informed consent

Informed consent was obtained from all individual participants included in the study.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ahmad, S., Ullah, A., Ullah, A. et al. Fuzzy natural transform method for solving fuzzy differential equations. Soft Comput 27, 8611–8625 (2023). https://doi.org/10.1007/s00500-023-08194-w

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-023-08194-w

Keywords

Search

Navigation