TY - JOUR
N2 - In the paper, the numerical method of solving the one-dimensional subdiffusion equation with the source term is presented. In the approach used, the key role is played by transforming of the partial differential equation into an equivalent integro-differential equation. As a result of the discretization of the integro-differential equation obtained an implicit numerical scheme which is the generalized Crank-Nicolson method. The implicit numerical schemes based on the finite difference method, such as the Carnk-Nicolson method or the Laasonen method, as a rule are unconditionally stable, which is their undoubted advantage. The discretization of the integro-differential equation is performed in two stages. First, the left-sided Riemann-Liouville integrals are approximated in such a way that the integrands are linear functions between successive grid nodes with respect to the time variable. This allows us to find the discrete values of the integral kernel of the left-sided Riemann-Liouville integral and assign them to the appropriate nodes. In the second step, second order derivative with respect to the spatial variable is approximated by the difference quotient. The obtained numerical scheme is verified on three examples for which closed analytical solutions are known.
L1 - http://journals.pan.pl/Content/120484/PDF-MASTER/Z_20_02172_Bpast.No.69(6)_OK.pdf
L2 - http://journals.pan.pl/Content/120484
PY - 2021
IS - 6
EP - e138240
DO - 10.24425/bpasts.2021.138240
KW - fractional derivatives and integrals
KW - integro-differential equations
KW - numerical methods
KW - finite difference methods
A1 - Błasik, Marek
VL - 69
DA - 18.08.2021
T1 - The implicit numerical method for the one-dimensional anomalous subdiffusion equation with a nonlinear source term
SP - e138240
UR - http://journals.pan.pl/dlibra/publication/edition/120484
T2 - Bulletin of the Polish Academy of Sciences Technical Sciences
ER -