Discretization of Fractional Differential Equations by a Piecewise Constant Approximation

There has recently been considerable interest in using a nonstandard piecewise approximation to formulate fractional order differential equations as difference equations that describe the same dynamical behaviour and are more amenable to a dynamical systems analysis. Unfortunately, due to mistakes in the fundamental papers, the difference equations formulated through this process do not capture the dynamics of the fractional order equations. We show that the correct application of this nonstandard piecewise approximation leads to a one parameter family of fractional order differential equations that converges to the original equation as the parameter tends to zero. A closed formed solution exists for each member of this family and leads to the formulation of a difference equation that is of increasing order as time steps are taken. Whilst this does not lead to a simplified dynamical analysis it does lead to a numerical method for solving the fractional order differential equation. The method is shown to be equivalent to a quadrature based method, despite the fact that it has not been derived from a quadrature. The method can be implemented with non-uniform time steps. An example is provided showing that the difference equation can correctly capture the dynamics of the underlying fractional differential equation

Cauchy integrals for computational solutions of master equations

Cauchy contour integrals are demonstrated to be effective in computationally solving master equations. A fractional generalization of a bimolecular master equation is one interesting application. References A. Andreychenko, L. Mikeev, D. Spieler, and V. Wolf. Approximate maximum likelihood estimation for stochastic chemical kinetics. EURASIP J. Bioinf. Sys. Biol., 2012:9, 2012. doi:10.1186/1687-4153-2012-9 C. N. Angstmann, I. C. Donnelly, B. I. Henry, and J. A. Nichols. A discrete time random walk model for anomalous diffusion. J. Comput. Phys., 293:53–69, 2014. doi:10.1016/j.jcp.2014.08.003 Y. Berkowitz, Y. Edery, H. Scher, and B. Berkowitz. Fickian and non-Fickian diffusion with bimolecular reactions. Phys. Rev. E, 87:032812, 2013. doi:10.1103/PhysRevE.87.032812 J. C. Butcher. On the numerical inversion of Laplace and Mellin transforms. Conference on Data Processing and Automatic Computing Machines, 117:1–8, 1957. D. Ding, D. A. Benson, A. Paster, and D. Bolster. Modeling bimolecular reactions and transport in porous media via particle tracking. Adv. Water Resour., 53:56–65, 2013. doi:10.1016/j.advwatres.2012.11.001 B. Drawert, S. Engblom, and A. Hellander. URDME: a modular framework for stochastic simulation of reaction-transport processes in complex geometries. BMC Syst. Biol., 6:76, 2012. doi:10.1186/1752-0509-6-76 T. A Driscoll, N. Hale, and L. N. Trefethen. Chebfun Guide. Pafnuty Publications, 2014. http://www.chebfun.org/docs/guide/ N. Dunford and J. Schwartz. Linear Operators I, II, III. Wiley New York, 1971. http://au.wiley.com/WileyCDA/WileyTitle/productCd-0471608483.html, http://au.wiley.com/WileyCDA/WileyTitle/productCd-0471608475.html, http://au.wiley.com/WileyCDA/WileyTitle/productCd-0471608467.html D. Fulger, E. Scalas, and G. Germano. Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation. Phys. Rev. E, 77:021122, 2008. doi:10.1103/PhysRevE.77.021122 D. Gillespie. Markov Processes: An Introduction for Physical Scientists. Academic Press, 1991. http://www.elsevier.com/books/markov-processes/gillespie/978-0-12-283955-9 M. Hegland, C. Burden, L. Santoso, S. MacNamara, and H. Booth. A solver for the stochastic master equation applied to gene regulatory networks. J. Comput. Appl. Math., 205(2):708–724, 2006. doi:10.1016/j.cam.2006.02.053 B. I. Henry, T. A. M. Langlands, and S. L. Wearne. Anomalous diffusion with linear reaction dynamics: From continuous time random walks to fractional reaction-diffusion equations. Phys. Rev. E, 74(3):031116, 2006. doi:10.1103/PhysRevE.74.031116 N. J. Higham. Functions of Matrices. SIAM, 2008. doi:10.1137/1.9780898717778 R. Hilfer and L. Anton. Fractional master equations and fractal time random walks. Phys. Rev. E, 51:R848, 1995. doi:10.1103/PhysRevE.51.R848 K. J. in 't Hout and J. A. C. Weideman. A contour integral method for the Black–Scholes and Heston equations. SIAM J. Sci. Comput., 33:763–785, 2011. doi:10.1137/090776081 T. Jahnke and D. Altintan. Efficient simulation of discrete stochastic reaction systems with a splitting method. BIT, 50:797–822, 2010. doi:doi:10.1007/s10543-010-0286-0 T. Kato. Perturbation theory for linear operators. Springer-Verlag, 1976. http://link.springer.com/book/10.1007%2F978-3-642-66282-9 V. M. Kenkre, E. W. Montroll, and M. F. Shlesinger. Generalized master equations for continuous-time random walks. J. Stat. Phys., 9(1):45, 1973. doi:10.1007/BF01016796 M. Lopez-Fernandez and C. Palencia. On the numerical inversion of the Laplace transform in certain holomorphic mappings. Appl. Numer. Math., 51:289–303, 2004. doi:10.1016/j.apnum.2004.06.015 S. MacNamara, K. Burrage, and R. B. Sidje. Multiscale modeling of chemical kinetics via the master equation. SIAM Multiscale Model. Simul., 6(4):1146–1168, 2008. doi:10.1137/060678154 M. Magdziarz, A. Weron, and K. Weron. Fractional Fokker–Planck dynamics: Stochastic representation and computer simulation. Phys. Rev. E, 75:016708, 2007. doi:10.1103/PhysRevE.75.016708 F. Mainardi, R. Gorenflo, and A. Vivoli. Beyond the Poisson renewal process: A tutorial survey. J. Comput. Appl. Math., 2007. doi:10.1016/j.cam.2006.04.060 W. McLean. Regularity of solutions to a time-fractional diffusion equation. ANZIAM J., 52(2):123–138, 2010. doi:10.1017/S1446181111000617 W. McLean and V. Thomee. Time discretization of an evolution equation via Laplace transforms. IMA J. Numer. Anal., 24:439–463, 2004. doi:10.1093/imanum/24.3.439 R. Metzler and J. Klafter. The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep., 339:1–77, 2000. doi:10.1016/S0370-1573(00)00070-3 C. Moler and C. Van Loan. Nineteen Dubious Ways to Compute the Exponential of a Matrix, 25 Years Later. SIAM Rev., 45(1):3–49, 2003. doi:10.1137/S00361445024180 E. W. Montroll and G. H. Weiss. Random Walks on Lattices. II. J. Math. Phys., 6(2):167–181, 1965. doi:10.1063/1.1704269 I. Moret and P. Novati. On the Convergence of Krylov Subspace Methods for Matrix Mittag–Leffler Functions. SIAM J. Numer. Anal., 49(5):2144–2164, 2011. doi:10.1137/080738374 I. Podlubny. Fractional Differential Equations. Academic Press, San Diego, 1999. http://www.elsevier.com/books/fractional-differential-equations/podlubny/978-0-12-558840-9 M. Raberto, F. Rapallo, and E. Scalas. Semi-Markov Graph Dynamics. PLoS ONE, 6(8):e23370, 2011. doi:10.1371/journal.pone.0023370 S. C. Reddy and L. N. Trefethen. Pseudospectra of the convection-diffusion operator. SIAM J. Appl. Math., 54(6):1634–1649, 1994. doi:10.1137/S0036139993246982 E. B. Saff and A. D. Snider. Fundamentals of complex analysis with applications to engineering and science. Pearson Education, 2003. http://www.pearsonhighered.com/educator/product/Fundamentals-of-Complex-Analysis-with-Applications-to-Engineering-Science-and-Mathematics/9780139078743.page E. Scalas, R. Gorenflo, and F. Mainardi. Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation. Phys. Rev. E, 69:011107, 2004. doi:10.1103/PhysRevE.69.011107 D. Sheen, I. H. Sloan, and V. Thomee. A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature. IMA J. Numer. Anal., 23:269–299, 2003. doi:10.1093/imanum/23.2.269 M. J. Shon and A. E. Cohen. Mass action at the single-molecule level. J. Am. Chem. Soc., 134(35):14618–14623, 2012. doi:10.1021/ja3062425 G. Strang and S. MacNamara. Functions of difference matrices are Toeplitz plus Hankel. SIAM Rev., 56(3):525–546, 2014. doi:10.1137/120897572 A. Talbot. The accurate numerical inversion of Laplace transforms. J. Inst. Math. Appl., 23:97–120, 1979. doi:10.1093/imamat/23.1.97 L. N. Trefethen. Approximation Theory and Approximation Practice. SIAM, Philadelphia, 2013. http://bookstore.siam.org/ot128/ L. N. Trefethen and M. Embree. Spectra and pseudospectra: the behavior of nonnormal matrices and operators. Princeton University Press, 2005. http://press.princeton.edu/titles/8113.html L. N. Trefethen and J. A. C. Weideman. The exponentially convergent trapezoidal rule. SIAM Rev., 56(3):385–458, 2014. doi:10.1137/130932132 N. G. van Kampen. Stochastic Processes in Physics and Chemistry. Elsevier Science, 2001. http://www.elsevier.com/books/stochastic-processes-in-physics-and-chemistry/van-kampen/978-0-444-52965-7 J. A. C. Weideman. Improved contour integral methods for parabolic PDEs. IMA J. Numer. Anal., 30:334–350, 2010. doi:10.1093/imanum/drn074 J. A. C. Weideman and L. N. Trefethen. Parabolic and hyperbolic contours for computing the bromwich integral. Math. Comput., 76:1341–1356, 2007. doi:10.1090/S0025-5718-07-01945-X T. G. Wright. Eigtool, 2002. http://www.comlab.ox.ac.uk/pseudospectra/eigtool/. Q. Yang, T. Moroney, K. Burrage, I. Turner, and F. Liu. Novel numerical methods for time-space fractional reaction diffusion equations in two dimensions. ANZIAM J., 52:395–409, 2011. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/3791/146