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High-order solvers for space-fractional differential equations with Riesz derivative

The research contained in this report is supported by South African National Research Foundation

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  • This paper proposes the computational approach for fractional-in-space reaction-diffusion equation, which is obtained by replacing the space second-order derivative in classical reaction-diffusion equation with the Riesz fractional derivative of order $ α $ in $ (0, 2] $. The proposed numerical scheme for space fractional reaction-diffusion equations is based on the finite difference and Fourier spectral approximation methods. The paper utilizes a range of higher-order time stepping solvers which exhibit third-order accuracy in the time domain and spectral accuracy in the spatial domain to solve some fractional-in-space reaction-diffusion equations. The numerical experiment shows that the third-order ETD3RK scheme outshines its third-order counterparts, taking into account the computational time and accuracy. Applicability of the proposed methods is further tested with a higher dimensional system. Numerical simulation results show that pattern formation process in the classical sense is the same as in fractional scenarios.

    Mathematics Subject Classification: Primary: 34A34, 35A05, 35K57; Secondary: 65L05, 65M06, 93C10.


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  • Figure 1.  Stability regions of (a) ETD3RK, (b) IMEX3PC with choice $(\mu, \psi, \eta) = (1, 0, 0)$

    Figure 2.  Convergence results of different schemes for one-dimensional problem (1) at (a) $t = 0.1$ and (b) $t = 2.0$ for $\alpha = 1.45$, $d = 8$. Simulation runs for $N = 200$

    Figure 3.  Solution of the fractional chemical system (42) in two-dimensions for subdiffusive (upper-row) and supperdiffusive (lower-row) scenarios. The parameters used are: $D = 0.39, d = 4, \varpi = 0.79, \beta = -0.91, \tau_2 = 0.278$ and $\tau_3 = 0.1$ at $t = 2$ for $N = 200$

    Figure 4.  Superdiffusive distribution of chemical system (42) mitotic patterns in two dimensions at some instances of $\alpha$ with initial conditions: $u_0 = 1-\exp(-10(x-0.5)^2+(y-0.5)^2), \;\;v_0 = \exp(-10(x-0.5)^2+2(y-0.5)^2)$. Other parameters are given in Figure 3 caption

    Figure 5.  Three dimensional results of system (42) showing the species evolution at subdiffusive ($\alpha = 0.35$) and superdiffusive ($\alpha = 1.91$) cases for $\tau_3 = 0.21$, $N = 50$ and final time $t = 5$. Other parameters are given in Figure 3 caption

    Figure 6.  Three dimensional results for system (42) at different instances of fractional power $\alpha$, with $\tau_3 = 0.26$ and final time $t = 5$. The first and second columns correspond to subdiffusive and superdiffusive cases. Other parameters are given in Figure 3 caption

    Table 1.  The maximum norm error and timing results for solving equation (1) in one-dimensional space with the exact solution and source term (40) using the FDM and FSM in conjunction with the IMEX3RK scheme at some instances of fractional power $\alpha$ in sub- and supper-diffusive scenarios for $t = 1$, $d = 0.5$ and $N = 200$

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    Table 2.  The maximum norm errors for two dimensional problem (1) with exact solution and local source term (41) obtained with different scheme at some instances of fractional power $\alpha$ and $N$ at final time $t = 1.5$ and $d = 10$

    Method$N$$0<\alpha<1$ $1<\alpha< 2$
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  •   F. B. Adda , The differentiability in the fractional calculus, Nonlinear Analysis, 47 (2001) , 5423-5428.  doi: 10.1016/S0362-546X(01)00646-0.
      G. Akrivis , M. Crouzeix  and  C. Makridakis , Implicit xplicit multistep methods for quasilinear parabolic equations, Numerische Mathematik, 82 (1999) , 521-541.  doi: 10.1007/s002110050429.
      O. J. J. Algahtani , Comparing the Atangana-Baleanu and Caputo-Fabrizio derivative with fractional order: Allen Cahn model, Chaos Solitons and Fractals, 89 (2016) , 552-559.  doi: 10.1016/j.chaos.2016.03.026.
      B. S. T. Alkahtani , Chua's circuit model with Atangana-Baleanu derivative with fractional order, Chaos, Solitons and Fractals, 89 (2016) , 547-551. 
      B. S. T. Alkahtani  and  A. Atangana , Controlling the wave movement on the surface of shallow water with the Caputo-Fabrizio derivative with fractional order, Chaos Soliton and Fractals, 89 (2016) , 539-546.  doi: 10.1016/j.chaos.2016.03.012.
      L. J. S. Allen, An Introduction to Mathematical Biology, Pearson Education, Inc., New Jersey, 2007.
      E. O. Asante-Asamani , A. Q. M. Khaliq  and  B. A. Wade , A real distinct poles Exponential Time Differencing scheme for reaction diffusion systems, Journal of Computational and Applied Mathematics, 299 (2016) , 24-34.  doi: 10.1016/j.cam.2015.09.017.
      U. M. Ascher , S. J. Ruth  and  R. J. Spiteri , Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations, Applied Numerical Mathematics, 25 (1997) , 151-167.  doi: 10.1016/S0168-9274(97)00056-1.
      U. M. Ascher , S. J. Ruth  and  B. T. R. Wetton , Implicit-explicit methods for time-dependent partial differential equations, SIAM Journal on Numerical Analysis, 32 (1995) , 797-823.  doi: 10.1137/0732037.
      A. Atangana  and  R. T. Alqahtani , Numerical approximation of the space-time Caputo-Fabrizio fractional derivative and application to groundwater pollution equation, Advances in Difference Equations, 2016 (2016) , 1-13.  doi: 10.1186/s13662-016-0871-x.
      A. Atangana  and  D. Baleanu , New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016) , 763-769.  doi: 10.2298/TSCI160111018A.
      A. Atangana  and  B. S. T. Alkahtani , New model of groundwater owing within a confine aquifer: Application of Caputo-Fabrizio derivative, Arabian Journal of Geosciences, 9 (2016) , 3647-3654. 
      A. Atangana  and  I. Koca , Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos, Solitons and Fractals, 89 (2016) , 447-454.  doi: 10.1016/j.chaos.2016.02.012.
      D. Baleanu , R. Caponetto  and  J. T. Machado , Challenges in fractional dynamics and control theory, Journal of Vibration and Control, 22 (2016) , 2151-2152.  doi: 10.1177/1077546315609262.
      D. Baleanu, K. Diethelm and E. Scalas, Fractional Calculus: Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos, World Scientific, 2012. doi: 10.1142/9789814355216.
      D. A. Benson , S. Wheatcraft  and  M. M. Meerschaert , pplication of a fractional advection-dispersion equation, Water Resources Research, 36 (2000) , 1403-1412. 
      H. P. Bhatt  and  A. Q. M. Khaliq , The locally extrapolated exponential time differencing LOD scheme for multidimensional reaction-diffusion systems, Journal of Computational and Applied Mathematics, 285 (2015) , 256-278.  doi: 10.1016/j.cam.2015.02.017.
      A. H. Bhrawy , M. A. Zaky  and  R. A. Van Gorder , A space-time Legendre spectral tau method for the two-sided space-time Caputo fractional diffusion-wave equation, Numerical Algorithms, 71 (2016) , 151-180.  doi: 10.1007/s11075-015-9990-9.
      A. H. Bhrawy  and  M. A. Abdelkawy , A fully spectral collocation approximation for multi-dimensional fractional Schrödinger equations, Journal of Computational Physics, 294 (2015) , 462-483.  doi: 10.1016/j.jcp.2015.03.063.
      A. H. Bhrawy , A Jacobi spectral collocation method for solving multi-dimensional nonlinear fractional sub-diffusion equations, Numerical Algorithms, 73 (2016) , 91-113.  doi: 10.1007/s11075-015-0087-2.
      N. F. Britton, Reaction-diffusion Equations and their Applications to Biology, Academic Press, London, 1986.
      A. Bueno-Orovio , D. Kay  and  K. Burrage , Fourier spectral methods for fractional-in-space reaction-diffusion equations, BIT Numerical mathematics, 54 (2014) , 937-954.  doi: 10.1007/s10543-014-0484-2.
      M. P. Calvo , J. de Frutos  and  J. Novo , Linearly implicit Runge-Kutta methods for advection-reaction-diffusion equations, Applied Numerical Mathematics, 37 (2001) , 535-549.  doi: 10.1016/S0168-9274(00)00061-1.
      M. Caputo  and  M. Fabrizio , Applications of new time and spatial fractional derivatives with exponential kernels, Progress in Fractional Differentiation and Applications, 2 (2016) , 1-11. 
      S. M. Cox  and  P. C. Matthews , Exponential time differencing for stiff systems, Journal of Computational Physics, 176 (2002) , 430-455.  doi: 10.1006/jcph.2002.6995.
      Q. Du  and  W. Zhu , Stability analysis and applications of the exponential time differencing schemes, Journal of Computational and Applied Mathematics, 22 (2004) , 200-209. 
      Q. Du  and  W. Zhu , Analysis and applications of the exponential time differencing schemes and their contour integration modifications, BIT Numerical Mathematics, 45 (2005) , 307-328.  doi: 10.1007/s10543-005-7141-8.
      W. Feller , On a generalization of Marcel Riesz potentials and the semi-groups generated by them, Middlelanden Lunds Universitets Matematiska Seminarium Comm. Sem. Mathm Universit de Lund (Suppl. ddi a M. Riesz), 1952 (1952) , 72-81. 
      W. Feller, An Introduction to Probability Theory and Its Applications, New York-London-Sydney, 1968.
      W. Gear  and  I. Kevrekidis , Projective methods for stiff differential equations: Problems with gaps in their eigenvalue spectrum, SIAM Journal on Scientific Computing, 24 (2003) , 1091-1106.  doi: 10.1137/S1064827501388157.
      I. Grooms  and  K. Julien , Linearly implicit methods for nonlinear PDEs with linear dispersion and dissipation, Journal of Computational Physics, 230 (2011) , 3630-3650.  doi: 10.1016/j.jcp.2011.02.007.
      E. Hairer and G. Wanner, Solving Ordinary Differential Equations Ⅱ: Stiff and Differential Algebraic Problems, Springer-Verlag, New York, 1996. doi: 10.1007/978-3-642-05221-7.
      A. K. Kassam  and  L. N. Trefethen , Fourth-order time-stepping for stiff PDEs, SIAM Journal Scientific Computing, 26 (2005) , 1214-1233.  doi: 10.1137/S1064827502410633.
      C. Kennedy  and  M. Carpenter , Additive Runge-Kutta schemes for covection-diffusion-reaction-diffusion equations, Applied Numerical Mathematics, 44 (2003) , 139-181.  doi: 10.1016/S0168-9274(02)00138-1.
      A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
      M. Kot, Elements of Mathematical Ecology, Cambridge University Press, United Kingdom, 2001. doi: 10.1017/CBO9780511608520.
      T. Koto , IMEX Runge-Kutta schemes for reaction-diffusion equations, Journal of Computational and Applied Mathematics, 215 (2008) , 182-195.  doi: 10.1016/j.cam.2007.04.003.
      C. Li and F. Zeng, Numerical Methods for Fractional Calculus, CRC Press, Taylor and Francis Group, London, 2015.
      D. Li , C. Zhang , W. Wang  and  Y. Zhang , Implicit-explicit predictor-corrector schemes for nonlinear parabolic differential equations, Applied Mathematical Modelling, 35 (2011) , 2711-2722.  doi: 10.1016/j.apm.2010.11.061.
      Y. F. Luchko , H. Matinez  and  J. J. Trujillo , Fractional Fourier transform and some of its applications, Fractional Calculus and Applied Analysis, 11 (2008) , 457-470. 
      R. L. Magin, Fractional Calculus in Bioengineering, Begell House, Connecticut, 2006.
      R. Magin , M. D. Ortigueira , I. Podlubny  and  J. Trujillo , On the fractional signals and systems, Signal Processing, 91 (2011) , 350-371.  doi: 10.1016/j.sigpro.2010.08.003.
      R. L. Magin , Fractional calculus models of complex dynamics in biological tissues, Computers and Mathematics with Applications, 59 (2010) , 1586-1593.  doi: 10.1016/j.camwa.2009.08.039.
      F. Mainardi , G. Pagnini  and  R. K. Saxena , Fox H functions in fractional diffusion, Journal of Computational and Applied Mathematics, 178 (2005) , 321-331.  doi: 10.1016/j.cam.2004.08.006.
      M. M. Meerschaert , D. A. Benson  and  S. W. Wheatcraft , Subordinated advection-dispersion equation for contaminant transport, Water Resource Research, 37 (2001) , 1543-1550. 
      M. M. Meerschaert  and  C. Tadjeran , Finite difference approximations for fractional advectiondispersion flow equations, Journal of Computational and Applied Mathematics, 172 (2004) , 65-77.  doi: 10.1016/j.cam.2004.01.033.
      M. M. Meerschaert , H. P. Scheffler  and  C. Tadjeran , Finite difference methods for twodimensional fractional dispersion equation, Journal of Computational Physics, 211 (2006) , 249-261.  doi: 10.1016/j.jcp.2005.05.017.
      F. C. Meral , T. J. Royston  and  R. Magin , Fractional calculus in viscoelasticity: An experimental study, Communications in Nonlinear Science and Numerical Simulation, 15 (2010) , 939-945.  doi: 10.1016/j.cnsns.2009.05.004.
      R. Metzler  and  J. Klafter , The random walk's guide to anomalous diffusion: A fractional dynamics approach, Physics Reports, 339 (2000) , 1-77.  doi: 10.1016/S0370-1573(00)00070-3.
      R. Metzler  and  J. Klafter , 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, 37 (2004) , R161-R208.  doi: 10.1088/0305-4470/37/31/R01.
      K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
      J. D. Murray, Mathematical Biology Ⅰ: An Introduction, Springer-Verlag, New York, 2002.
      M. D. Ortigueira, Fractional Calculus for Scientists and Engineers, Springer, New York, 2011. doi: 10.1007/978-94-007-0747-4.
      K. M. Owolabi , Mathematical study of two-variable systems with adaptive numerical methods, Numerical Analysis and Applications, 19 (2016) , 218-230.  doi: 10.15372/SJNM20160304.
      K. M. Owolabi , Robust and adaptive techniques for numerical simulation of nonlinear partial differential equations of fractional order, Communications in Nonlinear Science and Numerical Simulations, 44 (2017) , 304-317.  doi: 10.1016/j.cnsns.2016.08.021.
      K. M. Owolabi and A. Atangana, Numerical solution of fractional-in-space nonlinear Schrödinger equation with the Riesz fractional derivative, The European Physical Journal Plus, 131 (2016), 335. doi: 10.1140/epjp/i2016-16335-8.
      K. M. Owolabi , Mathematical analysis and numerical simulation of patterns in fractional and classical reaction-diffusion systems, Chaos Solitons and Fractals, 93 (2016) , 89-98.  doi: 10.1016/j.chaos.2016.10.005.
      K. M. Owolabi , Robust and adaptive techniques for numerical simulation of nonlinear partial differential equations of fractional order, Communications in Nonlinear Science and Numerical Simulation, 44 (2017) , 304-317.  doi: 10.1016/j.cnsns.2016.08.021.
      K. M. Owolabi , Robust IMEX schemes for solving two-dimensional reaction-diffusion models, International Journal of Nonlinear Science and Numerical Simulations, 16 (2015) , 271-284.  doi: 10.1515/ijnsns-2015-0004.
      K. M. Owolabi  and  K. C. Patidar , Higher-order time-stepping methods for time-dependent reaction-diffusion equations arising in biology, Applied Mathematics and Computation, 240 (2014) , 30-50.  doi: 10.1016/j.amc.2014.04.055.
      S. Petrovskii , K. Kawasaki , F. Takasu  and  N. Shigesada , Diffusive waves, dynamic stabilization and spatio-temporal chaos in a community of three competitive species, Japan Journal of Industrial and Applied Mathematics, 18 (2001) , 459-481.  doi: 10.1007/BF03168586.
      E. Pindza  and  K. M. Owolabi , Fourier spectral method for higher order space fractional reaction-diffusion equations, Communications in Nonlinear Science and Numerical Simulation, 40 (2016) , 112-128.  doi: 10.1016/j.cnsns.2016.04.020.
      I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
      J. Sabatier, O. P. Agrawal and J. A. Tenreiro Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Netherlands, 2007.
      S. G. Samko, A. A. Kilbas and O. I. Maritchev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Amsterdam, 1993.
      E. Scalas , R. Gorenflo  and  F. Mainardid , Fractional calculus and continuous-time finance, Physica A: Statistical Mechanics and its Applications, 284 (2000) , 376-384.  doi: 10.1016/S0378-4371(00)00255-7.
      Z. Tomovski , T. Sandev , R. Metzler  and  J. Dubbeldam , Generalized space-time fractional diffusion equation with composite fractional time derivative, Physica A, 391 (2012) , 2527-2542.  doi: 10.1016/j.physa.2011.12.035.
      V. Volpert  and  S. Petrovskii , Reaction-diffusion waves in biology, Physics of Life Reviews, 6 (2009) , 267-310. 
      E. Weinan , Analysis of the heterogeneous multiscale method for ordinary differential equations, Communications in Mathematical Sciences, 3 (2003) , 423-436.  doi: 10.4310/CMS.2003.v1.n3.a3.
      Q. Yang , F. Liu  and  I. Turner , Numerical methods for fractional partial differential equations with Riesz space fractional derivatives, Applied Mathematical Modelling, 34 (2010) , 200-218.  doi: 10.1016/j.apm.2009.04.006.
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