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A numerical approach for a class of time-fractional reaction–diffusion equation through exponential B-spline method

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Abstract

A numerical approach for a class of time-fractional reaction–diffusion equation through exponential B-spline method is presented in this paper. The proposed scheme is a combination of Crank–Nicolson method for the Caputo time derivative and exponential B-spline method for space derivative. The unconditional stability and convergence of the proposed scheme are presented. Several numerical examples are presented to illustrate the feasibility and efficiency of the proposed scheme.

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References

  • Baeumer B, Kovacs M, Meerschaert MM (2008) Numerical solutions for fractional reaction-diffusion equations. Comput Math Appl 55(10):2212–2226

    Article  MathSciNet  Google Scholar 

  • Baleanu D (2012) Fractional calculus: models and numerical methods. World Scientific, Singapore

    Book  Google Scholar 

  • Chandra SRS, Kumar M (2008) Exponential B-spline collocation method for self-adjoint singularly perturbed boundary value problems. Appl Numer Math 58(10):1572–1581

    Article  MathSciNet  Google Scholar 

  • Dag I, Ersoy O (2016) The exponential cubic B-spline algorithm for Fisher equation. Chaos Soliton Fract 86:101–106

    Article  MathSciNet  Google Scholar 

  • Ersoy O, Dag I (2015) Numerical solutions of the reaction diffusion system by using exponential cubic B-spline collocation algorithms. Open Phys 13(1):414–427

    Article  Google Scholar 

  • Gao G, Sun Z (2011) A compact finite difference scheme for the fractional sub-diffusion equations. J Comput Phys 230(3):586–595

    Article  MathSciNet  Google Scholar 

  • Gong C, Bao WM, Tang G, Jiang YW, Liu J (2014) A domain decomposition method for time fractional reaction-diffusion equation. Sci World J. https://doi.org/10.1155/2014/681707

    Article  Google Scholar 

  • Henry BI, Wearne SL (2000) Fractional reaction-diffusion. Phys A 276:448–455

    Article  MathSciNet  Google Scholar 

  • Hesameddini E, Asadollahifard E (2016) A new reliable algorithm based on the sinc function for the time fractional diffusion equation. Numer Algor 72(4):893–913

    Article  MathSciNet  Google Scholar 

  • Hilfer R (2000) Applications of fractional calculus in physics. World Scientific Publishing, New York

    Book  Google Scholar 

  • Karatay I, Kale N, Bayramoglu SR (2013) A new difference scheme for time fractional heat equations based on the Crank-Nicolson method. Frac Calc Appl Anal 16(4):892–910

    MATH  Google Scholar 

  • Karatay I, Kale N (2015) A new difference scheme for time fractional cable equation and stability analysis. Int J Appl Math Res 4(1):52–57

    Article  Google Scholar 

  • Kilbas AA, Srivastva HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, Amsterdam

    Google Scholar 

  • Li X (2014) Operational method for solving fractional differential equations using cubic B-spline approximation. Int J Comput Math 91(12):2584–2602

    Article  MathSciNet  Google Scholar 

  • Liu J, Gong C, Bao W, Tang G, Jiang Y (2014) Solving the Caputo fractional reaction-diffusion equation on GPU. Discrete Dyn Nat Soc. https://doi.org/10.1155/2014/820162

    Article  MATH  Google Scholar 

  • Liu Y, Du Y, Li H, Wang J (2015) An \(H^1\)-Galerkin mixed finite element method for time fractional reaction-diffusion equation. J Appl Math Comput 47:103–117

    Article  MathSciNet  Google Scholar 

  • McCartin BJ (1991) Theory of exponential splines. J Approx Theory 66(1):1–23

    Article  MathSciNet  Google Scholar 

  • Mohammadi R (2013) Exponential B-spline solution of convection-diffusion equations. Appl Math 4(6):933–944

    Article  Google Scholar 

  • Mohammadi R (2015) Exponential B-spline collocation method for numerical solution of the generalized regularized long wave equation. Chin Phys B 24(5):050206–910

    Article  Google Scholar 

  • Oldham KB, Spanier J (1974) The fractional calculus: theory and applications of differentiation and integration to arbitrary order. Academic Press, San Diego

    MATH  Google Scholar 

  • Podlubny I (1999) Fractional differential equations. Academic press, San Diego

    MATH  Google Scholar 

  • Povstenko Y (2015) Linear fractional diffusion-wave equation for scientists and engineers. Birkhauser, New York

    Book  Google Scholar 

  • Rashidinia J, Mohmedi E (2018) Convergence analysis of tau scheme for the fractional reaction-diffusion equation. Eur Phys J Plus. https://doi.org/10.1140/epjp/i2018-12200-2

    Article  Google Scholar 

  • Rida SZ, El-sayed AMA, Arafa AAM (2010) On the solutions of time-fractional reaction-diffusion equations. Commun Nonlinear Sci Numer Simulat 15(12):3847–3854

    Article  MathSciNet  Google Scholar 

  • Turut V, Guzel N (2012) Comparing numerical methods for solving time-fractional reaction-diffusion equations. ISRN Math Anal 2012. https://doi.org/10.5402/2012/737206

    Article  MATH  Google Scholar 

  • Wang QL, Liu J, Gong CY, Tang XT, Fu GT, Xing ZC (2016) An efficient parallel algorithm for Caputo fractional reaction-diffusion equation with implicit finite-difference method. Adv Differ Equ 1:207–218

    Article  MathSciNet  Google Scholar 

  • Zhang J, Yang X (2018) A class of efficient difference method for time fractional reaction-diffusion equation. Comp Appl Math 37(4):4376–4396

    Article  MathSciNet  Google Scholar 

  • Zhu X, Nie Y, Yuan Z, Wang J, Yang Z (2017) An exponential B-spline collocation method for the fractional sub-diffusion equation. Adv Differ Equ. https://doi.org/10.1186/s13662-017-1328-6

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

Authors are grateful to the anonymous reviewers for their insightful comments leading to the improved manuscript. The second author is thankful to the University Grants Commission of India for support under SRF scheme (Sr.No. 2061440951, reference no.22/06/14(i)EU-V).

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Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology, Kurukshetra, 136119, India

    A. S. V. Ravi Kanth & Neetu Garg

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  1. A. S. V. Ravi Kanth
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  2. Neetu Garg
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Correspondence to A. S. V. Ravi Kanth.

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Communicated by José Tenreiro Machado.

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Kanth, A.S.V.R., Garg, N. A numerical approach for a class of time-fractional reaction–diffusion equation through exponential B-spline method. Comp. Appl. Math. 39, 37 (2020). https://doi.org/10.1007/s40314-019-1009-z

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  • DOI: https://doi.org/10.1007/s40314-019-1009-z

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