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Finite difference and spectral collocation methods for the solution of semilinear time fractional convection-reaction-diffusion equations with time delay

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Abstract

In this paper, an efficient numerical method is constructed to solve the nonlinear fractional convection-reaction-diffusion equations with time delay. Firstly, we discretize the time fractional derivative with a second order finite difference scheme. Then, Chebyshev spectral collocation method is utilized to space component and to obtain full discretization of problem. We show that the proposed method is unconditionally stable and convergent. Numerical experiments are carried out to demonstrate the accuracy of the proposed method and to compare the results with analytical solutions and the numerical solutions of other schemes in the literature. The results show that the present method is accurate and efficient. It is illustrated that the numerical results are in good agreement with theoretical ones.

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References

  1. Alikhanov, A.A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424–438 (2015)

    Article  MathSciNet  Google Scholar 

  2. Bellen, A., Zennaro, M.: Numerical Methods for Delay Differential Equations. Oxford University Press, Oxford (2003)

    Book  Google Scholar 

  3. Bhalekar, S., Daftardar-Gejji, V.: A predictor-corrector scheme for solving nonlinear delay differential equations of fractional order. J. Fract. Calc. Appl. 1, 1–8 (2011)

    MATH  Google Scholar 

  4. Bressan, A., Quarteroni, A.: An implicit/explicit spectral method for Burgers equation. Calcolo 23, 265–84 (1986)

    Article  MathSciNet  Google Scholar 

  5. Canuto, C., Quarteroni, A.: Approximation results for orthogonal polynomials in Sobolev spaces. Math. Comput. 38, 67–86 (1982)

    Article  MathSciNet  Google Scholar 

  6. Culshaw, R.V., Ruan, S., Webb, G.: A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay. J. Math. Biol. 46, 425–444 (2003)

    Article  MathSciNet  Google Scholar 

  7. Daftardar-Gejji, V., Sukale, Y., Bhalekar, S.: Solving fractional delay differential equations: a new approach. Fract. Calc. Appl. Anal. 18, 400–418 (2015)

    Article  MathSciNet  Google Scholar 

  8. Davis, L.C.: Modication of the optimal velocity traffic model to include delay due to driver reaction time. Phys. A. 319, 557–567 (2002)

    Article  Google Scholar 

  9. Dehghan, M., Abbaszadeh, M., Mohebbi, A.: Numerical solution of system of n-coupled nonlinear schrodinger equations via two variants of the meshless local Petrov-Galerkin (mlpg) method. Comput. Model. Eng. Sci. 100, 399–444 (2014)

    MathSciNet  MATH  Google Scholar 

  10. Dehghan, M., Abbaszadeh, M., Mohebbi, A.: The use of element free Galerkin method based on moving Kriging and radial point interpolation techniques for solving some types of Turing models. Eng. Anal. Bound. Elem. 62, 93–111 (2016)

    Article  MathSciNet  Google Scholar 

  11. Deng, W., Wu, Y., Li, C.P.: Stability analysis of differential equations with time-dependent delay. Int. J. Bifurc. Chaos 16, 465–472 (2006)

    Article  MathSciNet  Google Scholar 

  12. Deng, W., Li, C.P., Lu, J.: Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dyn. 48, 409–416 (2007)

    Article  MathSciNet  Google Scholar 

  13. Gao, G., Alikhanov, A.A., Sun, Z.Z.: The temporal second order difference schemes based on the interpolation approximation for solving the time multi-term and distributed-order fractional sub-diffusion equations. J. Sci. Comput. 73, 93–121 (2017)

    Article  MathSciNet  Google Scholar 

  14. Hao, Z., Fan, K., Cao, W., Sun, Z.Z.: A finite difference scheme for semilinear space-fractional diffusion equations with time delay. Appl. Math. Comput. 275, 238–254 (2016)

    MathSciNet  MATH  Google Scholar 

  15. Khader, M.M., Hendy, A.S.: The approximate and exact solutions of the fractional-order delay differential equations using Legendre pseudospectral method. Int. J. Pure Appl. Math. 74, 287–297 (2012)

    MATH  Google Scholar 

  16. Kuang, Y.: Delay Differential Equations with Applications in Population Biology. Academic Press, Boston (MA) (1993)

    MATH  Google Scholar 

  17. Li, C.P., Zeng, F.: Numerical Methods for Fractional Calculus. CRC Press, New York (2015)

    Book  Google Scholar 

  18. Miller, K.S., Ross, B.: An Introductionto the Fractional Calculus and Fractional Differential Equations. Academic Press, New York (1974)

    Google Scholar 

  19. Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)

    MATH  Google Scholar 

  20. Ouyang, Z.: Existence and uniqueness of the solutions for a class of nonlinear fractional order partial differential equations with delay. Comput. Math. Appl. 61, 860–870 (2011)

    Article  MathSciNet  Google Scholar 

  21. Podulbny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    Google Scholar 

  22. Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer Series in Computational Mathematics 23. Springer, Berlin (1994)

    Book  Google Scholar 

  23. Rihan, F.A.: Computational methods for delay parabolic and time-fractional partial differential equations. Numer. Method Partial Differ. Equ. 26, 1556–1571 (2010)

    MathSciNet  MATH  Google Scholar 

  24. Sakara, M., Uludag, F., Erdogan, F.: Numerical solution of time-fractional nonlinear PDEs with proportional delays by homotopy perturbation method. Appl. Math. Model. 40, 6639–6649 (2016)

    Article  MathSciNet  Google Scholar 

  25. Sun, Z.Z., Zhang, Z.B.: A linearized compact difference scheme for a class of nonlinear delay partial differential equations. Appl. Math. Model. 37, 742–752 (2013)

    Article  MathSciNet  Google Scholar 

  26. Wang, Z., Huang, X., Zhou, J.: A numerical method for delayed fractional-order differential equations: based on G-L defiition. Appl. Math. Inf. Sci. 7, 525–529 (2013)

    Article  MathSciNet  Google Scholar 

  27. Yang, Z., Cao, J.: Initial value problems for arbitrary order fractional dierential equations with delay. Commun. Nonlinear Sci. Numer. Simul. 18, 2993–3005 (2013)

    Article  MathSciNet  Google Scholar 

  28. Yao, Z., Wang, Z.: A compact difference scheme for fourth-order fractional sub-diffusion equations with Neumann boundary conditions. J. Appl. Anal. Comput. 4, 1159–1169 (2018)

    MathSciNet  Google Scholar 

  29. Zayernouri, M., Cao, W., Zhang, Z., Karniadakis, G.E.: Spectral and discontinuous spectral element methods for fractional delay equations. SIAM J. Sci. Comput. 36, B904–B929 (2014)

    Article  MathSciNet  Google Scholar 

  30. Zhang, Q., Ran, M., Xu, D.: Analysis of the compact difference scheme for the semilinear fractional partial differential equation with time delay. Appl. Anal. 11, 1867–1884 (2017)

    Article  MathSciNet  Google Scholar 

  31. Zhang, Q., Zhang, C.: A new linearized compact multisplitting scheme for the nonlinear convection-reaction-diffusion equations with delay. Commun. Nonlinear Sci. Numer. Simul. 18, 3278–3288 (2013)

    Article  MathSciNet  Google Scholar 

  32. Zhou, Y., Jiao, F., Li, J.: Existence and uniqueness for fractional neutral dierential equations with infinite delay. Nonlinear Anal. 71, 3249–3256 (2009)

    Article  MathSciNet  Google Scholar 

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  1. Department of Applied Mathematics, Faculty of Mathematical Science, University of Kashan, Kashan, Iran

    Akbar Mohebbi

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  1. Akbar Mohebbi
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Mohebbi, A. Finite difference and spectral collocation methods for the solution of semilinear time fractional convection-reaction-diffusion equations with time delay. J. Appl. Math. Comput. 61, 635–656 (2019). https://doi.org/10.1007/s12190-019-01267-w

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  • DOI: https://doi.org/10.1007/s12190-019-01267-w

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