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A fourth-order accurate extrapolation nonlinear difference method for fourth-order nonlinear PIDEs with a weakly singular kernel

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Abstract

In this work, we propose a fourth-order accurate extrapolation nonlinear difference method for the fourth-order nonlinear partial integrodifferential equations (FON-PIDEs) with a weakly singular kernel. In space direction, we define a operator to handle the nonlinear convection term \(uu_x\), and use the fourth order central difference formula to discretize the fourth-order derivative term. In time direction, we use product-integration rule and Crank–Nicolson method to deal with Riemanm–Liouville fractional order integral (RL-FOI) terms on the graded meshes. Then a difference scheme with second-order accurate in both time and space is obtained. A series of theory analysis are proved, including the existence, stability, convergence and uniqueness. Then we apply the extrapolation method to improve the spatial second-order convergence to fourth-order. Moreover, we provide the theory proof of the extrapolation method. Finally, two numerical examples are given, and the results of examples are in agreement with the theoretical expectations. This proves the effectiveness of our constructed scheme.

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References

  • Chen H, Xu D (2008) A second order fully discrete difference scheme for a nonlinear partial integro-differential equation. J Syst Sci Math Sci 28(1):51

    MathSciNet  Google Scholar 

  • Cui M (2009) Compact finite difference method for the fractional diffusion equation. J Comput Phys 228(20):7792–7804

    Article  MathSciNet  Google Scholar 

  • Ford WF, Sidi A (1987) An algorithm for a generalization of the Richardson extrapolation process. SIAM J Numer Anal 24(5):1212–1232

    Article  MathSciNet  Google Scholar 

  • Gao G, Sun Z (2011) A compact finite difference scheme for the fractional sub-diffusion equations. J Comput Phys 230(3):586–597

    Article  MathSciNet  Google Scholar 

  • Gorguis A (2006) A comparison between Cole-Hopf transformation and the decomposition method for solving Burgers’ equations. Appl Math Comput 173(1):126–136

    MathSciNet  Google Scholar 

  • Hu X, Zhang L (2011) A compact finite difference scheme for the fourth-order fractional diffusion-wave system. Comput Phys Commun 182(8):1645–1650

    Article  MathSciNet  Google Scholar 

  • Kudu M, Amirali I, Amiraliyev GM (2016) A finite-difference method for a singularly perturbed delay integro-differential equation. J Comput Appl Math 308:379–390

    Article  MathSciNet  Google Scholar 

  • Li C, Zhang H, Yang X (2024) A new \(\alpha \)-robust nonlinear numerical algorithm for the time fractional nonlinear KdV equation. Commun Anal Mech 16:147–168

    Article  MathSciNet  Google Scholar 

  • Li C, Zhang H, Yang X (2024) A high-precision Richardson extrapolation method for a class of elliptic Dirichlet boundary value calculation. J Hunan Univ Technol 38(1):91–97

    Google Scholar 

  • Lin C, Gu M, Young D, Chen C (2014) Localized method of approximate particular solutions with Cole-Hopf transformation for multi-dimensional Burgers equations. Eng Anal Boundary Elem 40:78–92

    Article  MathSciNet  Google Scholar 

  • Li C, Zhang H, Yang X (2024) A new nonlinear compact difference scheme for a fourth-order nonlinear Burgers type equation with a weakly singular kernel. J Appl Math Comput 1–33. https://doi.org/10.1007/s12190-024-02039-x.

  • Lopez-Marcos J (1990) A difference scheme for a nonlinear partial integrodifferential equation. SIAM J Numer Anal 27(1):20–31

    Article  MathSciNet  Google Scholar 

  • Luo Z, Zhang X, Wang S, Yao L (2022) Numerical approximation of time fractional partial integro-differential equation based on compact finite difference scheme. Chaos Solitons Fractals 161:112395

    Article  MathSciNet  Google Scholar 

  • McLean W, Mustapha K (2007) A second-order accurate numerical method for a fractional wave equation. Numer Math 105:481–510

    Article  MathSciNet  Google Scholar 

  • Qiu W, Xiao X, Li K (2023) Second-order accurate numerical scheme with graded meshes for the nonlinear partial integrodifferential equation arising from viscoelasticity. Commun Nonlinear Sci Numer Simul 116:106804

    Article  MathSciNet  Google Scholar 

  • Richards SA (1997) Completed Richardson extrapolation in space and time. Commun Numer Methods Eng 13(7):573–582

    Article  MathSciNet  Google Scholar 

  • Safdari H, Mesgarani H, Javidi M, Aghdam YE (2020) Convergence analysis of the space fractional order diffusion equation based on the compact finite difference scheme. Comput Appl Math 39:1–15

    Article  MathSciNet  Google Scholar 

  • Shi Y, Yang X (2024) Pointwise error estimate of conservative difference scheme for supergeneralized viscous Burgers’ equation. Electron Research Arch 32(3):1471–1497

    Article  MathSciNet  Google Scholar 

  • Shi Y, Yang X (2024) A time two-grid difference method for nonlinear generalized viscous Burgers’ equation. J Math Chem 1-28. https://doi.org/10.1007/s10910-024-01592-x.

  • Tang T (1993) A finite difference scheme for partial integro-differential equations with a weakly singular kernel. Appl Numer Math 11(4):309–319

    Article  MathSciNet  Google Scholar 

  • Tian Q, Zhang H, Yang X, Jiang X (2022) An implicit difference scheme for the fourth-order nonlinear non-local PIDEs with a weakly singular kernel. Comput Appl Math 41(7):328

    Article  MathSciNet  Google Scholar 

  • Tian Q, Yang X, Zhang H, Xu D (2023) An implicit robust numerical scheme with graded meshes for the modified Burgers model with nonlocal dynamic properties. Comput Appl Math 42(6):246

    Article  MathSciNet  Google Scholar 

  • Verma P, Kumar M (2020) An analytical solution with existence and uniqueness conditions for fractional integro-differential equations. Int J Model Simul Sci Comput 11(5):2050045

    Article  Google Scholar 

  • Verma P, Kumar M (2020) An analytical solution of linear/nonlinear fractional-order partial differential equations and with new existence and uniqueness conditions. In: Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 1–9

  • Vlasov VV, Rautian NA (2016) Well-posed solvability of Volterra integro-differential equations in Hilbert space. Differ Equ 52:1123–1132

    Article  MathSciNet  Google Scholar 

  • Wang M, Zhang J, Li E, Xin X (2020) The generalized Cole-Hopf transformation to a general variable coefficient burgers equation with linear damping term. Appl Math Lett 105:106299

    Article  MathSciNet  Google Scholar 

  • Wang J, Jiang X, Zhang H (2024) A BDF3 and new nonlinear fourth-order difference scheme for the generalized viscous Burgers’ equation. Appl Math Lett 151:109002

    Article  MathSciNet  Google Scholar 

  • Wang J, Jiang X, Yang X, Zhang H (2024) A nonlinear compact method based on double reduction order scheme for the nonlocal fourth-order PDEs with Burgers’ type nonlinearity. J Appl Math Comput 70(1):489–511

    Article  MathSciNet  Google Scholar 

  • Wang W, Zhang H, Zhou Z, Yang X (2024) A fast compact finite difference scheme for the fourth-order diffusion-wave equation. Int J Comput Math 101(2):170–193

    Article  MathSciNet  Google Scholar 

  • Wu L, Zhang H, Yang X, Wang F (2022) A second-order finite difference method for the multi-term fourth-order integral-differential equations on graded meshes. Comput Appl Math 41(7):313

    Article  MathSciNet  Google Scholar 

  • Wu L, Zhang H, Yang X (2023) The finite difference method for the fourth-order partial integro differential equations with the multi-term weakly singular kernel. Math Methods Appl Sci 46(2):2517–2537

    Article  MathSciNet  Google Scholar 

  • Xiao W, Yang X, Zhou Z (2024) Pointwise-in-time \(\alpha \)-robust error estimate of the ADI difference scheme for three-dimensional fractional subdiffusion equations with variable coefficients. Commun Anal Mech 16(1):53–70

    Article  MathSciNet  Google Scholar 

  • Xu D, Qiu W, Guo J (2020) A compact finite difference scheme for the fourth-order time-fractional integro-differential equation with a weakly singular kernel. Numerl Methods Partial Differ Equ 36(2):439–458

    Article  MathSciNet  Google Scholar 

  • Yan T (2023) The numerical solutions for the nonhomogeneous Burgers’ equation with the generalized Hopf-Cole transformation. Netw Heterog Media 18(1):359–379

    Article  MathSciNet  Google Scholar 

  • Zhang Q, Sun C, Fang Z, Sun H (2022) Pointwise error estimate and stability analysis of fourth-order compact difference scheme for time-fractional Burgers’ equation. Appl Math Comput 418:126824

    MathSciNet  Google Scholar 

  • Zhang H, Jiang X, Wang F, Yang X (2024) The time two-grid algorithm combined with difference scheme for 2D nonlocal nonlinear wave equation. J Appl Math Comput, 1–25, https://doi.org/10.1007/s12190-024-02000-y.

  • Zhou Z, Zhang H, Yang X, Tang J (2023) An efficient ADI difference scheme for the nonlocal evolution equation with multi-term weakly singular kernels in three dimensions. Int J Comput Math 100(8):1719–1736

    Article  MathSciNet  Google Scholar 

  • Zhou Z, Zhang H, Yang X (2023) The compact difference scheme for the fourth-order nonlocal evolution equation with a weakly singular kernel. Math Methods Appl Sci 46(5):5422–5447

    Article  MathSciNet  Google Scholar 

  • Zhou Z, Zhang H, Yang X (2024) CN ADI fast algorithm on non-uniform meshes for the three-dimensional nonlocal evolution equation with multi-memory kernels in viscoelastic dynamics. Appl Math Comput 474:128680

    MathSciNet  Google Scholar 

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Acknowledgements

The author acknowledge Dr. João G. S. Monteiro for helpful discussions on the subject of this paper.

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  1. School of Science, Hunan University of Technology, Zhuzhou, 412007, China

    Caojie Li, Haixiang Zhang & Xuehua Yang

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  1. Caojie Li
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  2. Haixiang Zhang
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The work was supported by National Natural Science Foundation of China Mathematics Tianyuan Foundation (12226337, 12226340), Scientific Research Fund of Hunan Provincial Education Department (21B0550), Hunan Provincial Natural Science Foundation of China (2024JJ7146).

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Li, C., Zhang, H. & Yang, X. A fourth-order accurate extrapolation nonlinear difference method for fourth-order nonlinear PIDEs with a weakly singular kernel. Comp. Appl. Math. 43, 288 (2024). https://doi.org/10.1007/s40314-024-02812-5

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  • DOI: https://doi.org/10.1007/s40314-024-02812-5

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