과제정보
This work was supported by the State Key Laboratory of Synthetical Automation for Process Industries Fundamental Research Funds, No. 2013ZCX02.
참고문헌
- P. L. Butzer and U. Westphal, An introduction to fractional calculus, in Applications of fractional calculus in physics, 1-85, World Sci. Publ., River Edge, NJ, 2000. https://doi.org/10.1142/9789812817747_0001
- L. Debnath, Recent applications of fractional calculus to science and engineering, Int. J. Math. Math. Sci. 2003 (2003), no. 54, 3413-3442. https://doi.org/10.1155/S0161171203301486
- R. E. Ewing, T. Lin, and Y. Lin, On the accuracy of the finite volume element method based on piecewise linear polynomials, SIAM J. Numer. Anal. 39 (2002), no. 6, 1865-1888. https://doi.org/10.1137/S0036142900368873
- G. Gao, Z. Sun, and H. Zhang, A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications, J. Comput. Phys. 259 (2014), 33-50. https://doi.org/10.1016/j.jcp.2013.11.017
- Y. Jiang and J. Ma, High-order finite element methods for time-fractional partial differential equations, J. Comput. Appl. Math. 235 (2011), no. 11, 3285-3290. https://doi.org/10.1016/j.cam.2011.01.011
- B. Jin, R. Lazarov, and Z. Zhou, Error estimates for a semidiscrete finite element method for fractional order parabolic equations, SIAM J. Numer. Anal. 51 (2013), no. 1, 445-466. https://doi.org/10.1137/120873984
- S. Karaa, K. Mustapha, and A. K. Pani, Optimal error analysis of a FEM for fractional diffusion problems by energy arguments, J. Sci. Comput. 74 (2018), no. 1, 519-535. https://doi.org/10.1007/s10915-017-0450-7
- A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006.
- N. Kumar and M. Mehra, Collocation method for solving non-linear fractional optimal con trol problems by using Hermite scaling function with error estimates, Optimal Control, Appl. Meth. (2020). https://doi.org/10.1002/oca.2681
- Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys. 225 (2007), no. 2, 1533-1552. https://doi.org/10.1016/j.jcp.2007.02.001
- J. Lv and Y. Li, L2 error estimates and superconvergence of the finite volume element methods on quadrilateral meshes, Adv. Comput. Math. 37 (2012), no. 3, 393-416. https://doi.org/10.1007/s10444-011-9215-2
- W. McLean and K. Mustapha, A second-order accurate numerical method for a fractional wave equation, Numer. Math. 105 (2007), no. 3, 481-510. https://doi.org/10.1007/s00211-006-0045-y
- V. Mehandiratta and M. Mehra, A difference scheme for the time-fractional diffusion equation on a metric star graph, Appl. Numer. Math. 158 (2020), 152-163. https://doi.org/10.1016/j.apnum.2020.07.022
- V. Mehandiratta, M. Mehra, and G. Leugering, An approach based on Haar wavelet for the approximation of fractional calculus with application to initial and boundary value problems, Math. Methods Appl. Sci. 44 (2021), no. 4, 3195-3213. https://doi.org/10.1002/mma.6800
- K. S. Patel and M. Mehra, Fourth order compact scheme for space fractional advection-diffusion reaction equations with variable coefficients, J. Comput. Appl. Math. 380 (2020), 112963, 15 pp. https://doi.org/10.1016/j.cam.2020.112963
- I. Podlubny, Fractional differential equations, Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, CA, 1999.
- J. Ren, X. Long, S. Mao, and J. Zhang, Superconvergence of finite element approximations for the fractional diffusion-wave equation, J. Sci. Comput. 72 (2017), no. 3, 917-935. https://doi.org/10.1007/s10915-017-0385-z
- A. K. Singh and M. Mehra, Uncertainty quantification in fractional stochastic integro-differential equations using Legendre wavelet collocation method, Lecture Notes in Computer Science 12138 (2020), 58-71.
- M. Stynes, E. O'Riordan, and J. L. Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal. 55 (2017), no. 2, 1057-1079. https://doi.org/10.1137/16M1082329
- Z. Sun and X. Wu, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math. 56 (2006), no. 2, 193-209. https://doi.org/10.1016/j.apnum.2005.03.003
- Z. Wang and S. Vong, Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation, J. Comput. Phys. 277 (2014), 1-15. https://doi.org/10.1016/j.jcp.2014.08.012
- S. B. Yuste, Weighted average finite difference methods for fractional diffusion equations, J. Comput. Phys. 216 (2006), no. 1, 264-274. https://doi.org/10.1016/j.jcp.2005.12.006
- F. Zeng, C. Li, F. Liu, and I. Turner, The use of finite difference/element approaches for solving the time-fractional subdiffusion equation, SIAM J. Sci. Comput. 35 (2013), no. 6, A2976-A3000. https://doi.org/10.1137/130910865
- T. Zhang, Superconvergence of finite volume element method for elliptic problems, Adv. Comput. Math. 40 (2014), no. 2, 399-413. https://doi.org/10.1007/s10444-013-9313-4
- T. Zhang and Q. Guo, The finite difference/finite volume method for solving the fractional diffusion equation, J. Comput. Phys. 375 (2018), 120-134. https://doi.org/10.1016/j.jcp.2018.08.033
- Y. Zhao, Y. Zhang, D. Shi, F. Liu, and I. Turner, Superconvergence analysis of nonconforming finite element method for two-dimensional time fractional diffusion equations, Appl. Math. Lett. 59 (2016), 38-47. https://doi.org/10.1016/j.aml.2016.03.005