Skip to main content
SpringerLink
Log in

An efficient computational approach for nonlinear variable order fuzzy fractional partial differential equations

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

Abstract

This research work employs the Bernstein spectral technique based on Bernstein polynomials to analyze and to obtain the approximate numerical solution of a class of variable order fuzzy partial differential equations (PDEs) and its some particular cases using the basic properties of fuzzy theory. We analyze a variable order mathematical fuzzy model where the coefficients, unknown functions, initial and boundary conditions are some fuzzy numbers and fuzzy valued functions. The variable order fuzzy operational matrix of Bernstein polynomials is derived for fuzzy fractional derivatives with respect to space and time where the fuzzy derivative is taken in Caputo sense. The Bernstein fuzzy operational matrix is applied to concerned non-linear fuzzy space-time fractional variable order reaction–diffusion equations which reduce into a system of non-linear fuzzy algebraic equations and can be deal with using the method given in the literature. To validate the high efficiency and capability of the proposed numerical scheme few test examples are reported with computation of the absolute error for the obtained numerical solution.

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
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

References

  • Allahviranloo T, Kermani MA (2006) Solution of a fuzzy system of linear equation. Appl Math Comput 175(1):519–531

    MathSciNet  MATH  Google Scholar 

  • Allahviranloo T, Taheri N (2009) An analytic approximation to the solution of fuzzy heat equation by a domian decomposition method. Int J Contemp Math Sci 4(3):105–114

    MathSciNet  MATH  Google Scholar 

  • Arikoglu A, Ozkol I (2007) Solution of fractional differential equations by using differential transform method. Chaos Solitons Fract 34(5):1473–1481

    Article  MathSciNet  Google Scholar 

  • Babolian E, Fattahzadeh F (2007) Numerical computation method in solving integral equations by using Chebyshev wavelet operational matrix of integration. Appl Math Comput 188(1):1016–1022

    MathSciNet  MATH  Google Scholar 

  • Bhrawy AH, Doha EH, Ezz-Eldien SS, Abdelkawy MA (2016) A numerical technique based on the shifted Legendre polynomials for solving the time-fractional coupled kdv equations. Calcolo 53(1):1–17

    Article  MathSciNet  Google Scholar 

  • Chen Y, Liu L, Liu D, Boutat D (2018) Numerical study of a class of variable order nonlinear fractional differential equation in terms of Bernstein polynomials. Ain Shams Eng J 9(4):1235–1241

    Article  Google Scholar 

  • Dabiri A, Moghaddam BP, Tenreiro Machado JA (2018) Optimal variable-order fractional pid controllers for dynamical systems. J Comput Appl Math 339:40–48

    Article  MathSciNet  Google Scholar 

  • Das S, Rajeev S (2010) Solution of fractional diffusion equation with a moving boundary condition by variational iteration method and adomian decomposition method. Zeitsch Nat A 65(10):793–799

    Google Scholar 

  • Ganji RM, Jafari H (2019) A numerical approach for multi-variable orders differential equations using Jacobi polynomials. Int J Appl Comput Math 5(2):34

    Article  MathSciNet  Google Scholar 

  • Jameel AF, Nidal Anakira, Alomari AK, Hashim I, Momani S (2016) A new approximation method for solving fuzzy heat equations. J Comput Theoret Nanosci 13(11):7825–7832

    Article  Google Scholar 

  • Johnston SJ, Jafari H, Moshokoa SP, Ariyan VM, Baleanu D (2016) Laplace homotopy perturbation method for burgers equation with space-and time-fractional order. Open Phys 14(1):247–252

    Article  Google Scholar 

  • Kumar S, Pandey P, Das S, Craciun EM (2019) Numerical solution of two dimensional reaction-diffusion equation using operational matrix method based on Genocchi polynomial-part i: Genocchi polynomial and operatorial matrix. Proc Rom Acad Ser A Math Phys Tech Sci Inf Sci 20(4):393–399

    MathSciNet  MATH  Google Scholar 

  • Li Y, Zhao W (2010) Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations. Appl Math Comput 216(8):2276–2285

    MathSciNet  MATH  Google Scholar 

  • Moghaddam BP, Machado JAT (2017a) Time analysis of forced variable-order fractional van der pol oscillator. Eur Phys J Sp Top 226(16):3803–3810

    Article  Google Scholar 

  • Moghaddam BP, Mostaghim ZA (2017b) Modified finite difference method for solving fractional delay differential equations. Bol Soc Parana Mat 35(2):49–58

    Article  MathSciNet  Google Scholar 

  • Moghaddam BP, Dabiri A, Lopes AM, Machado JAT (2019a) Numerical solution of mixed-type fractional functional differential equations using modified Lucas polynomials. Comput Appl Math 38(2):1–12

    Article  MathSciNet  Google Scholar 

  • Moghaddam BP, Dabiri A, Machado JAT (2019b) Application of variable-order fractional calculus in solid mechanics. In: Baleanu D, Lopes AM (eds) Volume 7 applications in engineering, life and social sciences, part A. De Gruyter, pp 207–224

  • Moghaddam BP, Mendes Lopes A, Machado JAT, Mostaghim ZS (2019c) Computational scheme for solving nonlinear fractional stochastic differential equations with delay. Stoch Anal Appl 37(6):893–908

    Article  MathSciNet  Google Scholar 

  • Ortigueira MD, Valério D, Tenreiro Machado J (2019) Variable order fractional systems. Commun Nonlinear Sci Numer Simul 71:231–243

    Article  MathSciNet  Google Scholar 

  • Pandey P, Gómez-Aguilar JF (2021) On solution of a class of nonlinear variable order fractional reaction-diffusion equation with Mittag-Leffler kernel. Numer Methods Partial Differ Equ 37(2):998–1011

    Article  MathSciNet  Google Scholar 

  • Pandey P, Kumar S, Das S (2019) Approximate analytical solution of coupled fractional order reaction-advection-diffusion equations. Eur Phys J Plus 134(7):364

    Article  Google Scholar 

  • Pandey P, Kumar S, Gómez-Aguilar JF (2022) Numerical solution of the time fractional reaction-advection-diffusion equation in porous media. J Appl Comput Mech 8(1):84–96

  • Pandey P, Das S, Craciun EM, Sadowski T (2021) Two-dimensional nonlinear time fractional reaction-diffusion equation in application to sub-diffusion process of the multicomponent fluid in porous media. Meccanica 56(1):99–115

    Article  MathSciNet  Google Scholar 

  • Parsa Moghaddam B, Mostaghim Salamat Z (2014) A novel matrix approach to fractional finite difference for solving models based on nonlinear fractional delay differential equations. Ain Shams Eng J 5(2):585–594

    Article  Google Scholar 

  • Senol M, Atpinar S, Zararsiz Z, Salahshour S, Ahmadian A (2019) Approximate solution of time-fractional fuzzy partial differential equations. Comput Appl Math 38(1):1–18

    Article  MathSciNet  Google Scholar 

  • Stepnicka M, Valasek R (2005) Numerical solution of partial differential equations with help of fuzzy transform. In: The 14th IEEE International Conference on fuzzy systems, 2005. FUZZ’05., pp 1104–1109. IEEE

  • Tavassoli Kajani M, Ghasemi M, Babolian E (2007) Comparison between the homotopy perturbation method and the sine-cosine wavelet method for solving linear integro-differential equations. Comput Math Appl 54(7–8):1162–1168

    Article  MathSciNet  Google Scholar 

  • Valério D, Sá da Costa J (2013) Variable order fractional controllers. Asian J Control 15(3):648–657

    Article  MathSciNet  Google Scholar 

  • Xiaobin Guo X, Shang D, Lu X (2013) Fuzzy approximate solutions of second-order fuzzy linear boundary value problems. Bound Value Probl 1:212

    MathSciNet  MATH  Google Scholar 

  • Yousefi SA, Behroozifar M (2010) Operational matrices of Bernstein polynomials and their applications. Int J Syst Sci 41(6):709–716

    Article  MathSciNet  Google Scholar 

  • Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi, 221005, India

    Prashant Pandey

  2. Department of Mathematics, Government M.G.M. P.G. College, Itarsi, 461111, India

    Prashant Pandey

  3. Department of Mathematics, JECRC University, Jaipur, Rajasthan, 303905, India

    Jagdev Singh

Authors
  1. Prashant Pandey
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jagdev Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

Ethics declarations

Conflict of interest

This work does not have any conflicts of interest.

Additional information

Communicated by Vasily E. Tarasov.

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

Pandey, P., Singh, J. An efficient computational approach for nonlinear variable order fuzzy fractional partial differential equations. Comp. Appl. Math. 41, 38 (2022). https://doi.org/10.1007/s40314-021-01710-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-021-01710-4

Keywords

Mathematics Subject Classification

Search

Navigation