Abstract
This work presents two different finite difference methods to compute the numerical solutions for Newell–Whitehead–Segel partial differential equation, which are implicit exponential finite difference method and fully implicit exponential finite difference method. Implicit exponential methods lead to nonlinear systems. Newton method is used to solve the resulting systems. Stability and consistency are discussed. To illustrate the accuracy of the proposed numerical methods, some examples are delivered at the end.
Highlights
Nonlinear partial differential equations play an important role in modeling complicated phenomenon in physics, chemistry, biology, and mechanics
Different finite difference schemes have been developed for solving different differential equations, Bahadir [3] applied exponential finite difference method to KDV equation for small times, Ramos [21] used explicit finite difference methods for the equal width (EW) and regularized long-wave (RLW) equations, Inan and Bahadir [9] used Hopf–Cole transform to linearize Burgers’ equation, they applied an explicit exponential finite difference method to find the numerical solution, they presented an implicit exponential finite difference scheme for solving generalized Burgers–Huxley equation [10], Huang and Abduwali [7] used Crank-Nicolson method to modify the numerical scheme of generalized Burgers–Huxley equation, Celikten et al [4] presented four different explicit exponential finite difference methods to solve modified Burgers’ equation, and Inan [8] applied Crank–Nicolson exponential finite difference scheme to generalized Fitzhugh–Nagumo equation
These tables show that the errors of implicit exponential finite difference scheme (I-EFD) and fully implicit exponential finite difference scheme (FI-EFD) methods are almost equal, and they offer high accurate solutions
Summary
Nonlinear partial differential equations play an important role in modeling complicated phenomenon in physics, chemistry, biology, and mechanics. Different finite difference schemes have been developed for solving different differential equations, Bahadir [3] applied exponential finite difference method to KDV equation for small times, Ramos [21] used explicit finite difference methods for the equal width (EW) and regularized long-wave (RLW) equations, Inan and Bahadir [9] used Hopf–Cole transform to linearize Burgers’ equation, they applied an explicit exponential finite difference method to find the numerical solution, they presented an implicit exponential finite difference scheme for solving generalized Burgers–Huxley equation [10], Huang and Abduwali [7] used Crank-Nicolson method to modify the numerical scheme of generalized Burgers–Huxley equation, Celikten et al [4] presented four different explicit exponential finite difference methods to solve modified Burgers’ equation, and Inan [8] applied Crank–Nicolson exponential finite difference scheme to generalized Fitzhugh–Nagumo equation. Some examples are presented to show the efficiency of these methods to solve the equation
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