June  2019, 12(3): 567-590. doi: 10.3934/dcdss.2019037

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

Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein 9300, South Africa

* Corresponding author: mkowolax@yahoo.com (K.M. Owolabi)

Received  January 2017 Revised  September 2017 Published  September 2018

Fund Project: The research contained in this report is supported by South African National Research Foundation.

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.

Citation: Kolade M. Owolabi, Abdon Atangana. High-order solvers for space-fractional differential equations with Riesz derivative. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 567-590. doi: 10.3934/dcdss.2019037
References:
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show all references

References:
[1]

F. B. Adda, The differentiability in the fractional calculus, Nonlinear Analysis, 47 (2001), 5423-5428.  doi: 10.1016/S0362-546X(01)00646-0.

[2]

G. AkrivisM. Crouzeix and C. Makridakis, Implicit xplicit multistep methods for quasilinear parabolic equations, Numerische Mathematik, 82 (1999), 521-541.  doi: 10.1007/s002110050429.

[3]

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.

[4]

B. S. T. Alkahtani, Chua's circuit model with Atangana-Baleanu derivative with fractional order, Chaos, Solitons and Fractals, 89 (2016), 547-551. 

[5]

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.

[6]

L. J. S. Allen, An Introduction to Mathematical Biology, Pearson Education, Inc., New Jersey, 2007.

[7]

E. O. Asante-AsamaniA. 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.

[8]

U. M. AscherS. 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.

[9]

U. M. AscherS. 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.

[10]

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.

[11]

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.

[12]

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. 

[13]

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.

[14]

D. BaleanuR. 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.

[15]

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.

[16]

D. A. BensonS. Wheatcraft and M. M. Meerschaert, pplication of a fractional advection-dispersion equation, Water Resources Research, 36 (2000), 1403-1412. 

[17]

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.

[18]

A. H. BhrawyM. 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.

[19]

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.

[20]

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.

[21]

N. F. Britton, Reaction-diffusion Equations and their Applications to Biology, Academic Press, London, 1986.

[22]

A. Bueno-OrovioD. 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.

[23]

M. P. CalvoJ. 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.

[24]

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. 

[25]

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.

[26]

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. 

[27]

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.

[28]

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. 

[29]

W. Feller, An Introduction to Probability Theory and Its Applications, New York-London-Sydney, 1968.

[30]

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.

[31]

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.

[32]

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.

[33]

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.

[34]

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.

[35]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.

[36]

M. Kot, Elements of Mathematical Ecology, Cambridge University Press, United Kingdom, 2001. doi: 10.1017/CBO9780511608520.

[37]

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.

[38]

C. Li and F. Zeng, Numerical Methods for Fractional Calculus, CRC Press, Taylor and Francis Group, London, 2015.

[39]

D. LiC. ZhangW. 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.

[40]

Y. F. LuchkoH. Matinez and J. J. Trujillo, Fractional Fourier transform and some of its applications, Fractional Calculus and Applied Analysis, 11 (2008), 457-470. 

[41]

R. L. Magin, Fractional Calculus in Bioengineering, Begell House, Connecticut, 2006.

[42]

R. MaginM. D. OrtigueiraI. Podlubny and J. Trujillo, On the fractional signals and systems, Signal Processing, 91 (2011), 350-371.  doi: 10.1016/j.sigpro.2010.08.003.

[43]

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.

[44]

F. MainardiG. 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.

[45]

M. M. MeerschaertD. A. Benson and S. W. Wheatcraft, Subordinated advection-dispersion equation for contaminant transport, Water Resource Research, 37 (2001), 1543-1550. 

[46]

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.

[47]

M. M. MeerschaertH. 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.

[48]

F. C. MeralT. 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.

[49]

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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 $N = 200$
Method$\alpha=0.25$$\alpha=0.50$$\alpha=0.75$$\alpha=1.25$$\alpha=1.50$$\alpha=1.75$
FDM9.2570e-061.8864e-052.8615e-054.8107e-055.7776e-056.7399e-05
0.1674s0.1682s0.1693s0.1718s0.1673s0.1685s
FSM2.7055e-096.2174e-091.0710e-082.4231e-083.4382e-084.7695e-08
0.1664s0.1663s0.1665s0.1677s0.1663s0.1659s
Method$\alpha=0.25$$\alpha=0.50$$\alpha=0.75$$\alpha=1.25$$\alpha=1.50$$\alpha=1.75$
FDM9.2570e-061.8864e-052.8615e-054.8107e-055.7776e-056.7399e-05
0.1674s0.1682s0.1693s0.1718s0.1673s0.1685s
FSM2.7055e-096.2174e-091.0710e-082.4231e-083.4382e-084.7695e-08
0.1664s0.1663s0.1665s0.1677s0.1663s0.1659s
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$$0<\alpha<1$ $1<\alpha< 2$
$\alpha=0.15$CPU(s)$\alpha=0.63$CPU(s)$\alpha=1.37$CPU(s)$\alpha=1.89$CPU(s)
IMEX3RK$100$9.15E-060.214.57E-050.271.33E-040.272.26E-040.27
$200$7.17E-060.273.58E-050.271.04E-040.271.77E-040.27
$300$2.86E-080.261.43E-050.284.14E-050.227.06E-080.26
$400$1.34E-060.266.71E-060.271.93E-050.273.29E-050.27
IMEX3PC$100$4.49E-060.262.43E-050.277.30E-050.271.24E-040.26
$200$3.51E-060.271.90E-050.275.72E-050.279.75E-050.27
$300$1.39E-060.277.54E-060.282.29E-050.293.90E-050.28
$400$6.43E-070.273.48E-060.271.07E-050.271.83E-050.28
ETD3RK$100$1.87E-070.261.01E-060.273.04E-060.265.18E-060.26
$200$1.46E-070.277.93E-070.282.38E-060.274.06E-060.27
$300$5.79E-080.273.14E-070.299.54E-070.271.62E-060.28
$400$2.68E-080.271.45E-070.284.48E-070.277.64E-070.27
Method$N$$0<\alpha<1$ $1<\alpha< 2$
$\alpha=0.15$CPU(s)$\alpha=0.63$CPU(s)$\alpha=1.37$CPU(s)$\alpha=1.89$CPU(s)
IMEX3RK$100$9.15E-060.214.57E-050.271.33E-040.272.26E-040.27
$200$7.17E-060.273.58E-050.271.04E-040.271.77E-040.27
$300$2.86E-080.261.43E-050.284.14E-050.227.06E-080.26
$400$1.34E-060.266.71E-060.271.93E-050.273.29E-050.27
IMEX3PC$100$4.49E-060.262.43E-050.277.30E-050.271.24E-040.26
$200$3.51E-060.271.90E-050.275.72E-050.279.75E-050.27
$300$1.39E-060.277.54E-060.282.29E-050.293.90E-050.28
$400$6.43E-070.273.48E-060.271.07E-050.271.83E-050.28
ETD3RK$100$1.87E-070.261.01E-060.273.04E-060.265.18E-060.26
$200$1.46E-070.277.93E-070.282.38E-060.274.06E-060.27
$300$5.79E-080.273.14E-070.299.54E-070.271.62E-060.28
$400$2.68E-080.271.45E-070.284.48E-070.277.64E-070.27
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