TY - JOUR
T1 - Discrete monotone method for space-fractional nonlinear reaction–diffusion equations
AU - Flores, Salvador
AU - Macías-Díaz, Jorge E.
AU - Hendy, Ahmed S.
N1 - Publisher Copyright:
© 2019, The Author(s).
PY - 2019/12/1
Y1 - 2019/12/1
N2 - A discrete monotone iterative method is reported here to solve a space-fractional nonlinear diffusion–reaction equation. More precisely, we propose a Crank–Nicolson discretization of a reaction–diffusion system with fractional spatial derivative of the Riesz type. The finite-difference scheme is based on the use of fractional-order centered differences, and it is solved using a monotone iterative technique. The existence and uniqueness of solutions of the numerical model are analyzed using this approach, along with the technique of upper and lower solutions. This methodology is employed also to prove the main numerical properties of the technique, namely, the consistency, stability, and convergence. As an application, the particular case of the space-fractional Fisher’s equation is theoretically analyzed in full detail. In that case, the monotone iterative method guarantees the preservation of the positivity and the boundedness of the numerical approximations. Various numerical examples are provided to illustrate the validity of the numerical approximations. More precisely, we provide an extensive series of comparisons against other numerical methods available in the literature, we show detailed numerical analyses of convergence in time and in space against fractional and integer-order models, and we provide studies on the robustness and the numerical performance of the discrete monotone method.
AB - A discrete monotone iterative method is reported here to solve a space-fractional nonlinear diffusion–reaction equation. More precisely, we propose a Crank–Nicolson discretization of a reaction–diffusion system with fractional spatial derivative of the Riesz type. The finite-difference scheme is based on the use of fractional-order centered differences, and it is solved using a monotone iterative technique. The existence and uniqueness of solutions of the numerical model are analyzed using this approach, along with the technique of upper and lower solutions. This methodology is employed also to prove the main numerical properties of the technique, namely, the consistency, stability, and convergence. As an application, the particular case of the space-fractional Fisher’s equation is theoretically analyzed in full detail. In that case, the monotone iterative method guarantees the preservation of the positivity and the boundedness of the numerical approximations. Various numerical examples are provided to illustrate the validity of the numerical approximations. More precisely, we provide an extensive series of comparisons against other numerical methods available in the literature, we show detailed numerical analyses of convergence in time and in space against fractional and integer-order models, and we provide studies on the robustness and the numerical performance of the discrete monotone method.
KW - Crank–Nicolson finite-difference scheme
KW - Discrete monotone iterative method
KW - Existence and uniqueness of solutions
KW - Numerical efficiency analysis
KW - Space-fractional diffusion–reaction equations
UR - http://www.scopus.com/inward/record.url?scp=85070193052&partnerID=8YFLogxK
U2 - 10.1186/s13662-019-2267-1
DO - 10.1186/s13662-019-2267-1
M3 - Artículo
AN - SCOPUS:85070193052
SN - 1687-1839
VL - 2019
JO - Advances in Difference Equations
JF - Advances in Difference Equations
IS - 1
M1 - 317
ER -