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

  • A. S. V. Ravi KanthEmail author
  • Neetu Garg


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.


Time-fractional reaction–diffusion equation Caputo derivative Exponential B-spline method Stability Convergence 

Mathematics Subject Classification

35R11 65M12 65M70 65Y99 



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).


  1. Baeumer B, Kovacs M, Meerschaert MM (2008) Numerical solutions for fractional reaction-diffusion equations. Comput Math Appl 55(10):2212–2226MathSciNetCrossRefGoogle Scholar
  2. Baleanu D (2012) Fractional calculus: models and numerical methods. World Scientific, SingaporeCrossRefGoogle Scholar
  3. Chandra SRS, Kumar M (2008) Exponential B-spline collocation method for self-adjoint singularly perturbed boundary value problems. Appl Numer Math 58(10):1572–1581MathSciNetCrossRefGoogle Scholar
  4. Dag I, Ersoy O (2016) The exponential cubic B-spline algorithm for Fisher equation. Chaos Soliton Fract 86:101–106MathSciNetCrossRefGoogle Scholar
  5. Ersoy O, Dag I (2015) Numerical solutions of the reaction diffusion system by using exponential cubic B-spline collocation algorithms. Open Phys 13(1):414–427CrossRefGoogle Scholar
  6. Gao G, Sun Z (2011) A compact finite difference scheme for the fractional sub-diffusion equations. J Comput Phys 230(3):586–595MathSciNetCrossRefGoogle Scholar
  7. Gong C, Bao WM, Tang G, Jiang YW, Liu J (2014) A domain decomposition method for time fractional reaction-diffusion equation. Sci World J. CrossRefGoogle Scholar
  8. Henry BI, Wearne SL (2000) Fractional reaction-diffusion. Phys A 276:448–455MathSciNetCrossRefGoogle Scholar
  9. Hesameddini E, Asadollahifard E (2016) A new reliable algorithm based on the sinc function for the time fractional diffusion equation. Numer Algor 72(4):893–913MathSciNetCrossRefGoogle Scholar
  10. Hilfer R (2000) Applications of fractional calculus in physics. World Scientific Publishing, New YorkCrossRefGoogle Scholar
  11. 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):892–910zbMATHGoogle Scholar
  12. Karatay I, Kale N (2015) A new difference scheme for time fractional cable equation and stability analysis. Int J Appl Math Res 4(1):52–57CrossRefGoogle Scholar
  13. Kilbas AA, Srivastva HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, AmsterdamGoogle Scholar
  14. Li X (2014) Operational method for solving fractional differential equations using cubic B-spline approximation. Int J Comput Math 91(12):2584–2602MathSciNetCrossRefGoogle Scholar
  15. Liu J, Gong C, Bao W, Tang G, Jiang Y (2014) Solving the Caputo fractional reaction-diffusion equation on GPU. Discrete Dyn Nat Soc. CrossRefzbMATHGoogle Scholar
  16. 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 47:103–117MathSciNetCrossRefGoogle Scholar
  17. McCartin BJ (1991) Theory of exponential splines. J Approx Theory 66(1):1–23MathSciNetCrossRefGoogle Scholar
  18. Mohammadi R (2013) Exponential B-spline solution of convection-diffusion equations. Appl Math 4(6):933–944CrossRefGoogle Scholar
  19. Mohammadi R (2015) Exponential B-spline collocation method for numerical solution of the generalized regularized long wave equation. Chin Phys B 24(5):050206–910CrossRefGoogle Scholar
  20. Oldham KB, Spanier J (1974) The fractional calculus: theory and applications of differentiation and integration to arbitrary order. Academic Press, San DiegozbMATHGoogle Scholar
  21. Podlubny I (1999) Fractional differential equations. Academic press, San DiegozbMATHGoogle Scholar
  22. Povstenko Y (2015) Linear fractional diffusion-wave equation for scientists and engineers. Birkhauser, New YorkCrossRefGoogle Scholar
  23. Rashidinia J, Mohmedi E (2018) Convergence analysis of tau scheme for the fractional reaction-diffusion equation. Eur Phys J Plus. CrossRefGoogle Scholar
  24. Rida SZ, El-sayed AMA, Arafa AAM (2010) On the solutions of time-fractional reaction-diffusion equations. Commun Nonlinear Sci Numer Simulat 15(12):3847–3854MathSciNetCrossRefGoogle Scholar
  25. Turut V, Guzel N (2012) Comparing numerical methods for solving time-fractional reaction-diffusion equations. ISRN Math Anal 2012. CrossRefzbMATHGoogle Scholar
  26. 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 1:207–218MathSciNetCrossRefGoogle Scholar
  27. Zhang J, Yang X (2018) A class of efficient difference method for time fractional reaction-diffusion equation. Comp Appl Math 37(4):4376–4396MathSciNetCrossRefGoogle Scholar
  28. 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. MathSciNetCrossRefzbMATHGoogle Scholar

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© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019

Authors and Affiliations

  1. 1.Department of MathematicsNational Institute of TechnologyKurukshetraIndia

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