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Research Article

An efficient ADI difference scheme for the nonlocal evolution equation with multi-term weakly singular kernels in three dimensions

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Pages 1719-1736 | Received 13 Feb 2023, Accepted 06 May 2023, Published online: 16 May 2023

References

  • C. Chen, V. Thomee, and L. Wahlbin, Finite element approximation of a parabolic integro-differential equation with a weakly singular kernel, Math. Comput. 58 (1992), pp. 587–602.
  • H. Chen, D. Xu, J. Cao, and J. Zhou, A formally second order BDF ADI difference scheme for the three-dimensional time-fractional heat equation, Int. J. Comput. Math. 97(5) (2020), pp. 1100–1117.
  • Y. Huang, Time discretization scheme for an integro-differential equation of parabolic type, J. Comput. Math. 3 (1994), pp. 259–264.
  • J.C. Lopez-Marcos, A difference scheme for a nonlinear partial integrodifferential equation, SIAM. J. Numer. Anal. 27(1) (1990), pp. 20–31.
  • J.M. Morel, F. Takens, and B. Teissier, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, Springer, Berlin, 2004.
  • K. Mustapha and H. Mustapha, A second-order accurate numerical method for a semilinear integro-differential equation with a weakly singular kernel, IMA J. Numerical Anal. 30(2) (2009), pp. 555–578.
  • D.W. Peaceman and H.H. Rachford, The numerical solution of parabolic and elliptic differential equations, J. Soc. Ind. Appl. Math. 3(1) (1955), pp. 28–41.
  • A. Pedas and E. Tamme, A discrete collocation method for Fredholm integro-differential equations with weakly singular kernels, Appl. Numer. Math. 61(6) (2011), pp. 738–751.
  • I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  • L. Qiao, W. Qiu, and D. Xu, A second-order ADI difference scheme based on non-uniform meshes for the three-dimensional nonlocal evolution problem, Computers Math. Appl. 102 (2021), pp. 137–145.
  • L. Qiao, D. Xu, B. Tang, and J. Zhou, Fast ADI difference/compact difference schemes for the nonlocal evolution equation with weakly singular kernels in three dimensions, Math. Comput. Simul. 194 (2022), pp. 329–347.
  • W. Qiu, D. Xu, and J. Guo, A formally second-order backward differentiation formula Sinc-Collocation method for the Volterra integro-differential equation with a weakly singular kernel based on the double exponential transformation, Numer. Methods. Partial. Differ. Equ. 38(4) (2022), pp. 830–847.
  • J.M. Sanz-Serna, A numerical method for a partial integro-differential equation, SIAM. J. Numer. Anal. 25(2) (1988), pp. 319–327.
  • Z. Sun, Numerical Methods for Partial Differential Equation (in Chinese), Science Press, Beijing, 2005.
  • Z. Sun, The Method of Order Reduction and Its Application to the Numerical Solutions of Partial Differential Equations, Science Press, Beijing, 2009.
  • T. Tang, A finite difference scheme for partial integro-differential equations with a weakly singular kernel, Appl. Numer. Math. 11(4) (1993), pp. 309–319.
  • Z. Wang, D. Cen, and Y. Mo, Sharp error estimate of a compact L1-ADI scheme for the two-dimensional time-fractional integro-differential equation with singular kernels, Appl. Numer. Math. 159 (2021), pp. 190–203.
  • Z. Wang, Y. Guo, and L. Yi, An hp-version Legendre-Jacobi spectral collocation method for Volterra integro-differential equations with smooth and weakly singular kernels, Math. Comput. 86(307) (2017), pp. 2285–2324.
  • F. Wang, X. Yang, H. Zhang, and L. Wu, A time two-grid algorithm for the two dimensional nonlinear fractional PIDE with a weakly singular kernel, Math. Comput. Simul. 199 (2022), pp. 38–59.
  • Y. Wang and L. Zhu, SCW method for solving the fractional integro-differential equations with a weakly singular kernel, Appl. Math. Comput. 275 (2016), pp. 72–80.
  • D. Xu, On the discretization in time for a parabolic integro-differential equation with a weakly singular kernel I: smooth initial data, Appl. Math. Comput.58(1) (1993), pp. 1–27.
  • D. Xu, On the discretization in time for a parabolic integro-differential equation with a weakly singular kernel II: nonsmooth initial data, Appl. Math. Comput. 57(1) (1993), pp. 29–60.
  • D. Xu, The global behavior of time discretization for an abstract Volterra equation in Hilbert space, Calcolo 34(1) (1997), pp. 71–104.
  • D. Xu, The time discretization in classes of integro-differential equations with completely monotonic kernels: weighted asymptotic stability, Sci. China Math. 56 (2013), pp. 395–424.
  • D. Xu, The time discretization in classes of integro-differential equations with completely monotonic kernels: weighted asymptotic convergence, Numer. Methods. Partial. Differ. Equations 32(3) (2016), pp. 896–935.
  • D. Xu, Numerical asymptotic stability for the integro-differentical equations with the multi-term kernels, Appl. Math. Comput. 309 (2017), pp. 107–132.
  • D. Xu, Numerical solution of partial integro-differential equation with a weakly singular kernel based on sinc methods, Math. Comput. Simul. 190 (2021), pp. 140–158.
  • S. Zaeri, H. Saeedi, and M. Izadi, Fractional integration operator for numerical solution of the integro-partial time fractional diffusion heat equation with weakly singular kernel, Asian-European J. Math. 10(4) (2017), p. 1750071.
  • H. Zhang, X. Han, and X. Yang, Quintic B-spline collocation method for fourth order partial integro-differential equations with a weakly singular kernel, Appl. Math. Comput. 219(12) (2013), pp. 6565–6575.
  • H. Zhang, Y. Liu, and X. Yang, An efficient ADI difference scheme for the nonlocal evolution problem in three-dimensional space, J. Appl. Math. Computing. 69 (2023), pp. 651–674.
  • Y. Zhang, Z. Sun, and H. Wu, Error estimates of Crank-Nicolson type difference schemes for the subdiffusion equation, SIAM. J. Numer. Anal. 49(6) (2011), pp. 2302–2322.
  • J. Zhao, J. Xiao, and N.J. Ford, Collocation methods for fractional integro-differential equations with weakly singular kernels, Numer. Algorithms. 65(4) (2014), pp. 723–743.

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