Skip to main content
SpringerLink
Log in

Novel operational matrix method for the numerical solution of nonlinear reaction–advection–diffusion equation of fractional order

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

Abstract

In this work, a new scheme has been developed for the numerical solution of the fractional order reaction–advection–diffusion equation. To approximate the problem the authors have used Vieta–Fibonacci polynomials as basis functions and derived for the first time the operational matrices with the said polynomials for integer and fractional order Caputo differential operator. Using these operational matrices and collocating the residual together with initial and boundary conditions at certain collocation points, the problem is reduced to a system of algebraic equations. An approximate solution to the problem can be obtained by solving this system of equations. The efficiency and accuracy of the proposed method are validated through error analysis between the obtained numerical results and the existing analytical results for the particular forms of the considered model.

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

Similar content being viewed by others

References

  • Agarwal P, El-Sayed A (2018) Non-standard finite difference and Chebyshev collocation methods for solving fractional diffusion equation. Phys A Stat Mech Appl 500:40–49

    Article  MathSciNet  Google Scholar 

  • Agarwal P, El-Sayed A, Tariboon J (2021) Vieta–Fibonacci operational matrices for spectral solutions of variable-order fractional integro-differential equations. J Comput Appl Math 382:113063

    Article  MathSciNet  Google Scholar 

  • Bazm S, Hosseini A (2020) Bernoulli operational matrix method for the numerical solution of nonlinear two-dimensional Volterra–Fredholm integral equations of Hammerstein type. Comput Appl Math 39(2):49

    Article  MathSciNet  Google Scholar 

  • Bhrawy AH, Alofi A (2013) The operational matrix of fractional integration for shifted Chebyshev polynomials. Appl Math Lett 26(1):25–31

    Article  MathSciNet  Google Scholar 

  • Das S (2009) Analytical solution of a fractional diffusion equation by variational iteration method. Comput Math Appl 57(3):483–487

    Article  MathSciNet  Google Scholar 

  • Das S, Gupta P (2011) Homotopy analysis method for solving fractional hyperbolic partial differential equations. Int J Comput Math 88(3):578–588

    Article  MathSciNet  Google Scholar 

  • Das S, Vishal K, Gupta P (2011) Solution of the nonlinear fractional diffusion equation with absorbent term and external force. Appl Math Modell 35(8):3970–3979

    Article  MathSciNet  Google Scholar 

  • Doha EH, Bhrawy AH, Ezz-Eldien SS (2011) A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order. Comput Math Appl 62(5):2364–2373

    Article  MathSciNet  Google Scholar 

  • El-Sayed AA, Agarwal P (2019) Numerical solution of multiterm variable-order fractional differential equations via shifted Legendre polynomials. Math Methods Appl Sci 42(11):3978–3991

    Article  MathSciNet  Google Scholar 

  • Farooq U, Khan H, Baleanu D, Arif M (2019) Numerical solutions of fractional delay differential equations using Chebyshev wavelet method. Comput Appl Math 38(4):195

    Article  MathSciNet  Google Scholar 

  • Ganji RM, Jafari H, Nemati S (2020) A new approach for solving integro-differential equations of variable order. J Comput Appl Math 379:112946

    Article  MathSciNet  Google Scholar 

  • Ganji R, Jafari H, Kgarose M, Mohammadi A (2021) Numerical solutions of time-fractional Klein–Gordon equations by clique polynomials. Alex Eng J 60(5):4563–4571

    Article  Google Scholar 

  • Golbabai A, Nikan O, Nikazad T (2019) Numerical analysis of time fractional Black–Scholes European option pricing model arising in financial market. Comput Appl Math 38(4):173

    Article  MathSciNet  Google Scholar 

  • Gorenflo R, Mainardi F, Scalas E, Raberto M (2001) Fractional calculus and continuous-time finance III: the diffusion limit. In: Kohlmann M, Tang S (eds) Mathematical finance. Springer, Berlin, pp 171–180

    Chapter  Google Scholar 

  • Hilfer R (2000) Applications of fractional calculus in physics, vol 2000. Word Scientific, Singapore

    Book  Google Scholar 

  • Jafari H, Babaei A, Banihashemi S (2019) A novel approach for solving an inverse reaction–diffusion–convection problem. J Optim Theory Appl 183(2):688–704

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Liu J, Li X, Wu L (2016) An operational matrix technique for solving variable order fractional differential–integral equation based on the second kind of Chebyshev polynomials. Adv Math Phys 2016:6345978

    Article  MathSciNet  Google Scholar 

  • Magin RL (2004) Fractional calculus in bioengineering, part 1. Crit RevTN Biomed Eng 32(1):104

    Google Scholar 

  • Mainardi F, Raberto M, Gorenflo R, Scalas E (2000) Fractional calculus and continuous-time finance II: the waiting-time distribution. Phys A Stat Mech Appl 287(3–4):468–481

    Article  Google Scholar 

  • Oud MAA, Ali A, Alrabaiah H, Ullah S, Khan MA, Islam S (2021) A fractional order mathematical model for Covid-19 dynamics with quarantine, isolation, and environmental viral load. Adv Differ Equ 1:1–19

    MathSciNet  MATH  Google Scholar 

  • Rekhviashvili S, Pskhu A, Agarwal P, Jain S (2019) Application of the fractional oscillator model to describe damped vibrations. Turk J Phys 43(3):236–242

    Article  Google Scholar 

  • Saadatmandi A, Dehghan M (2010) A new operational matrix for solving fractional-order differential equations. Comput Math Appl 59(3):1326–1336

    Article  MathSciNet  Google Scholar 

  • Sabatelli L, Keating S, Dudley J, Richmond P (2002) Waiting time distributions in financial markets. Eur Phys J B Condens Matter Complex Syst 27(2):273–275

    Article  MathSciNet  Google Scholar 

  • Sadeghi Roshan S, Jafari H, Baleanu D (2018) Solving FDEs with Caputo–Fabrizio derivative by operational matrix based on Genocchi polynomials. Math Methods Appl Sci 41(18):9134–9141

    Article  MathSciNet  Google Scholar 

  • Sadri K, Hosseini K, Baleanu D, Salahshour S, Park C (2021) Designing a matrix collocation method for fractional delay integro-differential equations with weakly singular kernels based on Vieta–Fibonacci polynomials. Fractal Fract 6(1):2

    Article  Google Scholar 

  • Salahshour S, Ahmadian A, Senu N, Baleanu D, Agarwal P (2015) On analytical solutions of the fractional differential equation with uncertainty: application to the basset problem. Entropy 17(2):885–902

    Article  MathSciNet  Google Scholar 

  • Saldır O, Sakar MG, Erdogan F (2019) Numerical solution of time-fractional Kawahara equation using reproducing kernel method with error estimate. Comput Appl Math 38(4):198

    Article  MathSciNet  Google Scholar 

  • Scalas E, Gorenflo R, Mainardi F (2000) Fractional calculus and continuous-time finance. Phys A Stat Mech Appl 284(1–4):376–384

    Article  MathSciNet  Google Scholar 

  • Singh M, Das S, Rajeev Craciun EM (2021) Numerical solution of two-dimensional nonlinear fractional order reaction–advection–diffusion equation by using collocation method. Analele Universitatii “Ovidius’’ Constanta-Seria Matematica 29(2):211–230

    Article  MathSciNet  Google Scholar 

  • Sokolov IM, Klafter J, Blumen A (2002) Fractional kinetics. Phys Today 55(11):48–54

    Article  Google Scholar 

  • Wituła R, Słota D (2006) On modified Chebyshev polynomials. J Math Anal Appl 324(1):321–343

    Article  MathSciNet  Google Scholar 

  • Xie S (2015) Positive solutions for a system of higher-order singular nonlinear fractional differential equations with nonlocal boundary conditions. Electron J Qual Theory Differ Equ 2015(18):1–17

    Article  MathSciNet  Google Scholar 

  • Yousefi MHN, Najafabadi SHG, Tohidi E (2019) A fast and efficient numerical approach for solving advection–diffusion equations by using hybrid functions. Comput Appl Math 38(4):171

    Article  MathSciNet  Google Scholar 

  • Zhang Y, Sun Z, Liao H (2014) Finite difference methods for the time fractional diffusion equation on non-uniform meshes. J Comput Phys 265:195–210

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are extending their heartfelt thanks to the revered reviewers for their constructive suggestions towards the improvement of the article. The second author S. Das acknowledges the project grant provided by the BRE, BRNS, BARC, Government of India (Sanction No: 58/14/07/2022-BRNS/37041).

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, IIT (BHU), Varanasi, 221005, India

    Manpal Singh, S. Das &  Rajeev

  2. Institute of Actuarial Science and Data Analytics, UCSI University, Kuala Lumpur, 56000, Malaysia

    S. H. Ong

Authors
  1. Manpal Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Das
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rajeev
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. S. H. Ong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Manpal Singh.

Additional information

Communicated by Kassem Mustapha.

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 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

Singh, M., Das, S., Rajeev et al. Novel operational matrix method for the numerical solution of nonlinear reaction–advection–diffusion equation of fractional order. Comp. Appl. Math. 41, 306 (2022). https://doi.org/10.1007/s40314-022-02017-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-022-02017-8

Keywords

Mathematics Subject Classification

Search

Navigation