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A fourth-order compact difference method for the nonlinear time-fractional fourth-order reaction–diffusion equation

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Abstract

In this paper, a high-order compact scheme is proposed for solving two-dimensional nonlinear time-fractional fourth-order reaction-diffusion equations. The fractional derivative is the Caputo fractional derivative. A scheme with the second-order accuracy is applied to deal with the time derivative, while the spatial derivatives are discretized by the fourth-order compact numerical differential formulas. The unique solvability of the numerical method is proved in detail. Then by using the energy method, it is proved that the proposed algorithm is convergent with order \(O({\tau ^2} + h_1^4 + h_2^4)\), where \(\tau\) is the temporal step size and \(h_1\), \(h_2\) are the spatial step sizes. Finally, some numerical examples are given to verify the theoretical analysis and efficiency of the developed scheme.

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References

  1. Abbaszadeh M (2019) Error estimate of second-order finite difference scheme for solving the Riesz space distributed-order diffusion equation. Appl Math Lett 88:179–185

    MathSciNet  MATH  Google Scholar 

  2. Abbaszadeh M, Dehghan M (2019) Analysis of mixed finite element method (MFEM) for solving the generalized fractional reaction-diffusion equation on nonrectangular domains. Comput Math Appl 78(5):1531–1547

    MathSciNet  MATH  Google Scholar 

  3. Abbaszadeh M, Dehghan M (2020) Direct meshless local Petrov-Galerkin (DMLPG) method for time-fractional fourth-order reaction–diffusion problem on complex domains. Comput Math Appl 79(3):876–888

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  5. Baleanu D, Darzi R, Agheli B (2017) New study of weakly singular kernel fractional fourth-order partial integro-differential equations based on the optimum q-homotopic analysis method. J Comput Appl Math 320:193–201

    MathSciNet  MATH  Google Scholar 

  6. Chen M, Deng W (2014) A second-order numerical method for two-dimensional two-sided space fractional convection diffusion equation. Appl Math Model 38(13):3244–3259

    MathSciNet  MATH  Google Scholar 

  7. Cheng A, Wang H, Wang K (2015) A Eulerian-Lagrangian control volume method for solute transport with anomalous diffusion. Numer Methods Partial Differ Equ 31(1):253–267

    MathSciNet  MATH  Google Scholar 

  8. Cui M (2014) A high-order compact exponential scheme for the fractional convection–diffusion equation. J Comput Appl Math 255:404–416

    MathSciNet  MATH  Google Scholar 

  9. Du Y, Liu Y, Li H, Fang Z, He S (2017) Local discontinuous Galerkin method for a nonlinear time-fractional fourth-order partial differential equation. J Comput Phys 344:108–126

    MathSciNet  MATH  Google Scholar 

  10. Feng R, Liu Y, Hou Y, Li H, Fang Z (2020) Mixed element algorithm based on a second-order time approximation scheme for a two-dimensional nonlinear time fractional coupled sub-diffusion model. Eng Comput 1–18

  11. Gao G-H, Sun H-W, Sun Z-Z (2015) Stability and convergence of finite difference schemes for a class of time-fractional sub-diffusion equations based on certain superconvergence. J Comput Phys 280:510–528

    MathSciNet  MATH  Google Scholar 

  12. Gao G-H, Sun Z-Z, Zhang H-W (2014) A new fractional numerical differentiation formula to approximate the caputo fractional derivative and its applications. J Comput Phys 259:33–50

    MathSciNet  MATH  Google Scholar 

  13. Hendy AS, Zaky MA (2020) Graded mesh discretization for coupled system of nonlinear multi-term time-space fractional diffusion equations. Eng Comput 1–13

  14. Hendy AS, Zaky MA (2021) Combined Galerkin spectral/finite difference method over graded meshes for the generalized nonlinear fractional Schrödinger equation. Nonlinear Dyn 103(3):2493–2507

    Google Scholar 

  15. Hendy AS, Zaky MA, Hafez RM, De Staelen RH (2021) The impact of memory effect on space fractional strong quantum couplers with tunable decay behavior and its numerical simulation. Sci Rep 11(1):1–15

    Google Scholar 

  16. Horn RA, Johnson CR (2012) Matrix analysis. Cambridge University Press, Cambridge

    Google Scholar 

  17. Ilati M (2020) Analysis and application of the interpolating element-free Galerkin method for extended Fisher-Kolmogorov equation which arises in brain tumor dynamics modeling. Numer Algor 85(2):485–502

    MathSciNet  MATH  Google Scholar 

  18. Ilati M, Dehghan M (2018) Direct local boundary integral equation method for numerical solution of extended Fisher–Kolmogorov equation. Eng Comput 34(1):203–213

    Google Scholar 

  19. Ji C-C, Sun Z-Z (2015) The high-order compact numerical algorithms for the two-dimensional fractional sub-diffusion equation. Appl Math Comput 269:775–791

    MathSciNet  MATH  Google Scholar 

  20. Li D, Wu C, Zhang Z (2019) Linearized Galerkin FEMs for nonlinear time fractional parabolic problems with non-smooth solutions in time direction. J Sci Comput 80(1):403–419

    MathSciNet  MATH  Google Scholar 

  21. Li J, Liu F, Feng L, Turner I (2017) A novel finite volume method for the Riesz space distributed-order advection-diffusion equation. Appl Math Model 46:536–553

    MathSciNet  MATH  Google Scholar 

  22. Liu Y, Du Y, Li H, He S, Gao W (2015) Finite difference/finite element method for a nonlinear time-fractional fourth-order reaction–diffusion problem. Comput Math Appl 70(4):573–591

    MathSciNet  MATH  Google Scholar 

  23. Liu Y, Du Y, Li H, Li J, He S (2015) A two-grid mixed finite element method for a nonlinear fourth-order reaction–diffusion problem with time-fractional derivative. Comput Math Appl 70(10):2474–2492

    MathSciNet  MATH  Google Scholar 

  24. Liu Y, Fang Z, Li H, He S (2014) A mixed finite element method for a time-fractional fourth-order partial differential equation. Appl Math Comput 243:703–717

    MathSciNet  MATH  Google Scholar 

  25. Lv C, Xu C (2016) Error analysis of a high order method for time-fractional diffusion equations. SIAM J Sci Comput 38(5):A2699–A2724

    MathSciNet  MATH  Google Scholar 

  26. Meyer CD (2000) Matrix analysis and applied linear algebra, vol 71. Siam, Philadelphia

    Google Scholar 

  27. Oldham KB, Spanier J (1974) The fractional calculus, vol 111 of mathematics in science and engineering

  28. Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier, Amsterdam

    MATH  Google Scholar 

  29. Quarteroni A, Valli A (2008) Numerical approximation of partial differential equations, vol 23. Springer, New York

    MATH  Google Scholar 

  30. Simmons A, Yang Q, Moroney T (2017) A finite volume method for two-sided fractional diffusion equations on non-uniform meshes. J Comput Phys 335:747–759

    MathSciNet  MATH  Google Scholar 

  31. Sousa E (2014) An explicit high order method for fractional advection diffusion equations. J Comput Phys 278:257–274

    MathSciNet  MATH  Google Scholar 

  32. Tian W, Zhou H, Deng W (2015) A class of second order difference approximations for solving space fractional diffusion equations. Math Comput 84(294):1703–1727

    MathSciNet  MATH  Google Scholar 

  33. Vong S, Lyu P, Chen X, Lei S-L (2016) High order finite difference method for time-space fractional differential equations with Caputo and Riemann-Liouville derivatives. Numer Algor 72(1):195–210

    MathSciNet  MATH  Google Scholar 

  34. Wang Z, Vong S (2014) Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation. J Comput Phys 277:1–15

    MathSciNet  MATH  Google Scholar 

  35. Wei L, He Y (2014) Analysis of a fully discrete local discontinuous Galerkin method for time-fractional fourth-order problems. Appl Math Model 38(4):1511–1522

    MathSciNet  MATH  Google Scholar 

  36. Zaky MA, Hendy AS, De Staelen RH (2021) Alikhanov Legendre-Galerkin spectral method for the coupled nonlinear time-space fractional Ginzburg-Landau complex system. Mathematics 9(2):183

    Google Scholar 

  37. Zaky MA, Hendy AS, Macías-Díaz JE (2020) Semi-implicit Galerkin-Legendre spectral schemes for nonlinear time-space fractional diffusion-reaction equations with smooth and nonsmooth solutions. J Sci Comput 82(1):1–27

    MathSciNet  MATH  Google Scholar 

  38. Zeng F, Li C, Liu F, Turner I (2015) Numerical algorithms for time-fractional subdiffusion equation with second-order accuracy. SIAM J Sci Comput 37(1):A55–A78

    MathSciNet  MATH  Google Scholar 

  39. Zeng F, Liu F, Li C, Burrage K, Turner I, Anh V (2014) A Crank-Nicolson ADI spectral method for a two-dimensional Riesz space fractional nonlinear reaction–diffusion equation. SIAM J Numer Anal 52(6):2599–2622

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  41. Zhang Y-N, Sun Z-Z, Wu H-W (2011) Error estimates of Crank-Nicolson-type difference schemes for the subdiffusion equation. SIAM J Numer Anal 49(6):2302–2322

    MathSciNet  MATH  Google Scholar 

  42. Zhao J, Fang Z, Li H, Liu Y (2020) Finite volume element method with the WSGD formula for nonlinear fractional mobile/immobile transport equations. Adv Differ Equ 2020(1):1–20

    MathSciNet  MATH  Google Scholar 

  43. Zheng M, Liu F, Turner I, Anh V (2015) A novel high order space-time spectral method for the time fractional Fokker-Planck equation. SIAM J Sci Comput 37(2):A701–A724

    MathSciNet  MATH  Google Scholar 

  44. Zheng R, Liu F, Jiang X (2020) A Legendre spectral method on graded meshes for the two-dimensional multi-term time-fractional diffusion equation with non-smooth solutions. Appl Math Lett 104:106247

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors thank the reviewers for their useful comments and suggestions that improved the paper.

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Authors and Affiliations

  1. School of Mathematics, Shahid Rajaee Teacher Training University, Lavizan, 16788, Tehran, Iran

    Majid Haghi

  2. Department of Applied Mathematics, Faculty of Basic Sciences, Sahand University of Technology, Tabriz, Iran

    Mohammad Ilati

  3. Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Ave., Tehran, Iran

    Mehdi Dehghan

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  1. Majid Haghi
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  2. Mohammad Ilati
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  3. Mehdi Dehghan
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Correspondence to Mohammad Ilati.

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Haghi, M., Ilati, M. & Dehghan, M. A fourth-order compact difference method for the nonlinear time-fractional fourth-order reaction–diffusion equation. Engineering with Computers 39, 1329–1340 (2023). https://doi.org/10.1007/s00366-021-01524-2

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  • DOI: https://doi.org/10.1007/s00366-021-01524-2

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