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Abstract
We propose a generalized theory to construct higher order Grünwald type approximations for fractional derivatives. We use this generalization to simplify the proofs of orders for existing approximation forms for the fractional derivative. We also construct a set of higher order Grünwald type approximations for fractional derivatives in terms of a general real sequence and its generating function. From this, a second order approximation with shift is shown to be useful in approximating steady state problems and time dependent fractional diffusion problems. Stability and convergence for a Crank-Nicolson type scheme for this second order approximation are analyzed and are supported by numerical results.
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References
- Leibniz, G.W. "Letter from Hanover, Germany to G.F.A. L'Hospital, September 30, 1695",
- Mathematische Schriften, reprinted 1962, Hildesheim, Germany (Olns Verlag) 2, 301-302.
- Euler, L. “On Transcendental Progressions. That is, Those Whose General Terms Cannot be Given Algebraically”, Commentarii academiae scientiarum Petropolitanae, 1738, 5, 36-57.
- English Translation by S. Langton at:http://home.sandiego.edu/~langton/eg.pdf
- Ross, B. The development of fractional calculus 1695–1900. Historia Mathematica, 1977, 4(1), 75–89.
- Mainardi, F. Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos, Solitons and Fractals, 1996, 7(9), 1461–1477.
- Bagley, R.L., and Torvik, P. A theoretical basis for the application of fractional calculus to viscoelasticity. Journal of Rheology, 1983, 27(3), 201–210.
- Vinagre, B., Podlubny, I., Hernandez, A., and Feliu, V. Some approximations of fractional order operators used in control theory and applications. Fractional Calculus and Applied Analysis, 2000, 3, 231–248.
- Metzler, R., and Klafter, J. The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. Journal of Physics A: Mathematical and General, 2004, 37(31), R161-R204.
- Barkai, E., Metzler, R., and Klafter, J. From continuous time random walks to the fractional Fokker-Planck equation. Physical Review E, 61, 2000, 1, 132.
- Meerschaert, M.M., and Tadjeran, C. Finite difference approximations for fractional advection–dispersion flow equations. Journal of Computational and Applied Mathematics, 2004, 172(1), 65–77.
- Meerschaert, M.M., and Tadjeran, C. Finite difference approximations for two-sided space-fractional partial differential equations. Applied Numerical Mathematics, 2006, 56(1), 80–90.
- Tadjeran, C., Meerschaert, M.M., and Scheffler, H.-P. A second-order accurate numerical approximation for the fractional diffusion equation. Journal of Computational Physics, 2006, 213(1), 205–213.
- Nasir, H.M., Gunawardana, B.L.K., and Aberathna, H.M.N.P. A second order finite difference approximation for the fractional diffusion equation. International Journal of Applied Physics and Mathematics, 2013, 3(4), 237–243.
- Tian, W., Zhou, H., and Deng, W. A class of second order difference approximations for solving space fractional diffusion equations. Mathematics of Computation, 2015, 84(294), 1703–1727.
- Hao, Z.P., Sun, Z.Z., and Cao, W.R. A fourth-order approximation of fractional derivatives with its applications. Journal of Computational Physics, 2015, 281, 787–805.
- Zhou, H., Tian, W., and Deng, W. Quasi-compact finite difference schemes for space fractional diffusion equations. Journal of Scientific Computing, 2013, 56(1) 45–66.
- Li, C., and Deng, W. A new family of difference schemes for space fractional advection diffusion equation. Advances in Applied Mathematics and Mechanics, 2017, 9(2), 282–306.
- Yu, Y., Deng, W., Wu, Y., and Wu, J. Third order difference schemes (without using points outside of the domain) for one sided space tempered fractional partial differential equations. Applied Numerical Mathematics, 2017, 112, 126–145.
- Zhao, L., and Deng, W. A series of high-order quasi-compact schemes for space fractional diffusion equations based on the superconvergent approximations for fractional derivatives. Numerical Methods for Partial Differential Equations, 2015, 31(5), 1345–1381.
- Lubich, C. Discretized fractional calculus. SIAM Journal on Mathematical Analysis, 1986, 17(3), 704–719.
- Chen, M., and Deng, W. Fourth order difference approximations for space Riemann-Liouville derivatives based on weighted and shifted Lubich difference operators. Communications in Computational Physics 2014,16(02), 516–540.
- Chen, M., and Deng, W. Fourth order accurate scheme for the space fractional diffusion equations. SIAM Journal on Numerical Analysis, 2014, 52(3), 1418–1438.
- Podlubny, I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and some of Their Applications, vol. 198. Academic Press, 1998.
- Henrici, P. Discrete variable methods in ordinary differential equations, SIAM Rev., 1962, 4(3), 261-262.
- Rall, L.B. Automatic differentiation: Techniques and Applications, Lecture Notes in Computer Science, Springer-Verlag, Vol. 120, 1981.
- Weilbeer, M. Efficient Numerical Methods for Fractional Differential Equations and Their Analytical Background. Papierflieger, 2005.
- Walt, S.V.D., Colbert, S.C., and Varoquaux, G. The numpy array: a structure for efficient numerical computation. Computing in Science and Engineering, 2011, 13(2), 22–30.
References
Leibniz, G.W. "Letter from Hanover, Germany to G.F.A. L'Hospital, September 30, 1695",
Mathematische Schriften, reprinted 1962, Hildesheim, Germany (Olns Verlag) 2, 301-302.
Euler, L. “On Transcendental Progressions. That is, Those Whose General Terms Cannot be Given Algebraically”, Commentarii academiae scientiarum Petropolitanae, 1738, 5, 36-57.
English Translation by S. Langton at:http://home.sandiego.edu/~langton/eg.pdf
Ross, B. The development of fractional calculus 1695–1900. Historia Mathematica, 1977, 4(1), 75–89.
Mainardi, F. Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos, Solitons and Fractals, 1996, 7(9), 1461–1477.
Bagley, R.L., and Torvik, P. A theoretical basis for the application of fractional calculus to viscoelasticity. Journal of Rheology, 1983, 27(3), 201–210.
Vinagre, B., Podlubny, I., Hernandez, A., and Feliu, V. Some approximations of fractional order operators used in control theory and applications. Fractional Calculus and Applied Analysis, 2000, 3, 231–248.
Metzler, R., and Klafter, J. The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. Journal of Physics A: Mathematical and General, 2004, 37(31), R161-R204.
Barkai, E., Metzler, R., and Klafter, J. From continuous time random walks to the fractional Fokker-Planck equation. Physical Review E, 61, 2000, 1, 132.
Meerschaert, M.M., and Tadjeran, C. Finite difference approximations for fractional advection–dispersion flow equations. Journal of Computational and Applied Mathematics, 2004, 172(1), 65–77.
Meerschaert, M.M., and Tadjeran, C. Finite difference approximations for two-sided space-fractional partial differential equations. Applied Numerical Mathematics, 2006, 56(1), 80–90.
Tadjeran, C., Meerschaert, M.M., and Scheffler, H.-P. A second-order accurate numerical approximation for the fractional diffusion equation. Journal of Computational Physics, 2006, 213(1), 205–213.
Nasir, H.M., Gunawardana, B.L.K., and Aberathna, H.M.N.P. A second order finite difference approximation for the fractional diffusion equation. International Journal of Applied Physics and Mathematics, 2013, 3(4), 237–243.
Tian, W., Zhou, H., and Deng, W. A class of second order difference approximations for solving space fractional diffusion equations. Mathematics of Computation, 2015, 84(294), 1703–1727.
Hao, Z.P., Sun, Z.Z., and Cao, W.R. A fourth-order approximation of fractional derivatives with its applications. Journal of Computational Physics, 2015, 281, 787–805.
Zhou, H., Tian, W., and Deng, W. Quasi-compact finite difference schemes for space fractional diffusion equations. Journal of Scientific Computing, 2013, 56(1) 45–66.
Li, C., and Deng, W. A new family of difference schemes for space fractional advection diffusion equation. Advances in Applied Mathematics and Mechanics, 2017, 9(2), 282–306.
Yu, Y., Deng, W., Wu, Y., and Wu, J. Third order difference schemes (without using points outside of the domain) for one sided space tempered fractional partial differential equations. Applied Numerical Mathematics, 2017, 112, 126–145.
Zhao, L., and Deng, W. A series of high-order quasi-compact schemes for space fractional diffusion equations based on the superconvergent approximations for fractional derivatives. Numerical Methods for Partial Differential Equations, 2015, 31(5), 1345–1381.
Lubich, C. Discretized fractional calculus. SIAM Journal on Mathematical Analysis, 1986, 17(3), 704–719.
Chen, M., and Deng, W. Fourth order difference approximations for space Riemann-Liouville derivatives based on weighted and shifted Lubich difference operators. Communications in Computational Physics 2014,16(02), 516–540.
Chen, M., and Deng, W. Fourth order accurate scheme for the space fractional diffusion equations. SIAM Journal on Numerical Analysis, 2014, 52(3), 1418–1438.
Podlubny, I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and some of Their Applications, vol. 198. Academic Press, 1998.
Henrici, P. Discrete variable methods in ordinary differential equations, SIAM Rev., 1962, 4(3), 261-262.
Rall, L.B. Automatic differentiation: Techniques and Applications, Lecture Notes in Computer Science, Springer-Verlag, Vol. 120, 1981.
Weilbeer, M. Efficient Numerical Methods for Fractional Differential Equations and Their Analytical Background. Papierflieger, 2005.
Walt, S.V.D., Colbert, S.C., and Varoquaux, G. The numpy array: a structure for efficient numerical computation. Computing in Science and Engineering, 2011, 13(2), 22–30.