TY - JOUR
T1 - Error analysis of interpolating element free Galerkin method to solve non-linear extended Fisher–Kolmogorov equation
AU - Abbaszadeh, Mostafa
AU - Dehghan, Mehdi
AU - Khodadadian, Amirreza
AU - Heitzinger, Clemens
N1 - Funding Information:
The authors are grateful to the two reviewers for carefully reading this paper and for their comments and suggestions which have highly improved the paper. Amirreza Khodadadian and Clemens Heitzinger acknowledge support by FWF (Austrian Science Fund) START Project No. Y660 PDE Models for Nanotechnology.
PY - 2020/7/1
Y1 - 2020/7/1
N2 - Nonlinear partial differential equations (PDEs) play an important role in the modeling of the natural phenomena as they have great significance in real-world applications. This investigation proposes a new algorithm to find the numerical solution of the non-linear extended Fisher–Kolmogorov equation. Firstly, the time variable is discretized by a second-order finite difference scheme. The rate of convergence and stability of the semi-discrete formulation are studied by the energy method. The existence and uniqueness of the solution of the weak form based on the proposed technique have been proved in detail. Furthermore, the interpolating element free Galerkin approach based on the interpolation moving least-squares approximation is employed to derive a fully discrete scheme. Finally, the error estimate of the full-discrete plan is proposed and its convergence order is O(τ2+δm+1) in which τ, δ and m denote the time step, the radius of the weight function and smoothness of the exact solution of the main problem, respectively.
AB - Nonlinear partial differential equations (PDEs) play an important role in the modeling of the natural phenomena as they have great significance in real-world applications. This investigation proposes a new algorithm to find the numerical solution of the non-linear extended Fisher–Kolmogorov equation. Firstly, the time variable is discretized by a second-order finite difference scheme. The rate of convergence and stability of the semi-discrete formulation are studied by the energy method. The existence and uniqueness of the solution of the weak form based on the proposed technique have been proved in detail. Furthermore, the interpolating element free Galerkin approach based on the interpolation moving least-squares approximation is employed to derive a fully discrete scheme. Finally, the error estimate of the full-discrete plan is proposed and its convergence order is O(τ2+δm+1) in which τ, δ and m denote the time step, the radius of the weight function and smoothness of the exact solution of the main problem, respectively.
KW - Convergence analysis
KW - Element free Galerkin method
KW - Error analysis
KW - Interpolating moving least-squares approximation
KW - Non-linear extended Fisher–Kolmogorov equation
UR - http://www.scopus.com/inward/record.url?scp=85083289711&partnerID=8YFLogxK
U2 - 10.1016/j.camwa.2020.03.014
DO - 10.1016/j.camwa.2020.03.014
M3 - Article
AN - SCOPUS:85083289711
VL - 80
SP - 247
EP - 262
JO - Computers and Mathematics with Applications
JF - Computers and Mathematics with Applications
SN - 0898-1221
IS - 1
ER -