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 -