An alternating direction implicit fractional trapezoidal rule type difference scheme for the two-dimensional fractional evolution equation

References

• C. Chen, V. Thomee, and L.B. Wahlbin, Finite element approximation of a parabolic integro-differential equation with a weakly singular kernel, Math. Comput. 58 (1992), pp. 587–602. doi: 10.1090/S0025-5718-1992-1122059-2
   Web of Science ®Google Scholar
• H. Chen and D. Xu, A second-order fully discrete difference scheme for a partial integro-differential equation (in Chinese), Math. Numer. Sin. 28 (2006), pp. 141–154.
   Google Scholar
• H. Chen and D. Xu, A compact difference scheme for an evolution equation with a weakly singular kernel, Numer. Math. Theoret. Meth. Appl. 5 (2012), pp. 559–572. doi: 10.4208/nmtma.2012.m11032
   Web of Science ®Google Scholar
• W. Chen, L. Ye, and H. Sun, Fractional diffusion equations by the Kansa method, Computers Math. Appl. 59 (2010), pp. 1614–1620. doi: 10.1016/j.camwa.2009.08.004
   Web of Science ®Google Scholar
• E. Cuesta and C. Palencia, A fractional trapezoidal rule for integro-differential equations of fractional order in Banach spaces, Appl. Numer. Math. 45 (2003), pp. 139–159. doi: 10.1016/S0168-9274(02)00186-1
   Web of Science ®Google Scholar
• Y. Fujita, Integro-differential equation which interpolates the heat equation and the wave equation, Osaka J. Math. 27 (1990), pp. 319–327.
   Google Scholar
• Y. Fujita, Integro-differential equation which interpolates the heat equation and the wave equation (II), Osaka J. Math. 27 (1990), pp. 797–804.
   Web of Science ®Google Scholar
• Z. Fu, W. Chen, and H. Yang, Boundary particle method for Laplace transformed time fractional diffusion equations, J. Comp. Phys. 235 (2013), pp. 52–66. doi: 10.1016/j.jcp.2012.10.018
   Web of Science ®Google Scholar
• L. Li and D. Xu, Alternating direction implicit-Euler method for the two-dimensional fractional evolution equation, J. Comput. Phys. 236 (2013), pp. 157–168. doi: 10.1016/j.jcp.2012.11.005
   Web of Science ®Google Scholar
• H. Liao and Z. Sun, Maximum error estimates of ADI and compact ADI methods for solving parabolic equations, Numer. Methods Partial Differ. Equ. 26 (2010), pp. 37–60. doi: 10.1002/num.20414
   Web of Science ®Google Scholar
• J.C. Lopez-Marcos, A difference scheme for a nonlinear partial integro-differential equation, SIAM. J. Numer. Anal. 27 (1990), pp. 20–31. doi: 10.1137/0727002
   Web of Science ®Google Scholar
• C. Lubich, Convolution quadrature and discretized operational calculus, I, Numer. Math. 52 (1988), pp. 187–199. doi: 10.1007/BF01398689
   Web of Science ®Google Scholar
• C. Lubich, On convolution quadrature and Hille-Philips operational calculus, Appl. Numer. Math. 9 (1992), pp. 704–719. doi: 10.1016/0168-9274(92)90014-5
   Web of Science ®Google Scholar
• W. Mclean and V. Thomee, Numerical solution of an evolution equation with a positive type memory term, J. Austral. Math. Soc. Ser. B35 (1993), pp. 23–70. doi: 10.1017/S0334270000007268
   Web of Science ®Google Scholar
• A.K. Pani, G. Fairweather, and R.I. Fernandes, Alternating direction implicit orthogonal spline collocation methods for an evolution equation with a positive-type memory term, SIAM J. Numer. Anal. 46 (2008), pp. 344–364. doi: 10.1137/050634967
   Web of Science ®Google Scholar
• A.K. Pani, G. Fairweather, and R.I. Fernandes, ADI orthogonal spline collocation methods for parabolic partial integro-differential equations, IMA J. Numer. Anal. 30 (2010), pp. 248–276. doi: 10.1093/imanum/drp024
   Web of Science ®Google Scholar
• I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
   Google Scholar
• G. Pang, W. Chen, and K.Y. Sze, Gauss–Jacobi-type quadrature rules for fractional directional integrals, Comput. Math. Appl. 59 (2010), pp. 1614–1620. doi: 10.1016/j.camwa.2009.08.004
   Web of Science ®Google Scholar
• Z. Sun, Numerical Methods for Partial Differential Equations (in Chinese), Science Press, Beijing, 2005.
   Google Scholar
• Z. Sun, The method of order reduction and its application to the numerical solutions of partial differential equations, Science Press, Beijing, 2009.
   Google Scholar
• T. Tang, A finite difference scheme for partial integro-differential equations with a weakly singular kernel, Appl. Numer. Math. 11 (1993), pp. 309–319. doi: 10.1016/0168-9274(93)90012-G
   Web of Science ®Google Scholar
• D. Xu, The global behavior of time discretization for an abstract Volterra equation in Hilbert space, CALCOLO 34 (1997), pp. 71–104.
   Google Scholar
• Y. Zhang and Z. Sun, Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation, J. Comput. Phys. 230 (2011), pp. 8713–8728. doi: 10.1016/j.jcp.2011.08.020
   Web of Science ®Google Scholar
• Y. Zhang, Z. Sun, and H. Wu, Error estimates of Crank–Nicolson type difference schemes for the sub-diffusion equation, SIAM. J. Numer. Anal. 49 (2011), pp. 2302–2322. doi: 10.1137/100812707
   Web of Science ®Google Scholar
• Y. Zhang, Z. Sun, and X. Zhao, Compact alternating direction implicit scheme for the two-dimensional fractional diffusion-wave equation, SIAM. J. Numer. Anal. 50 (2012), pp. 1535–1555. doi: 10.1137/110840959
   Web of Science ®Google Scholar