Numerical Simulation of the Nagumo Equation by Finite Difference Method
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The Nagumo equation is an important nonlinear reaction-diffusion equation used to model the transmission of nerve impulses. In this thesis, traveling wave solutions to the Nagumo equation are studied. A pseudo-Crank-Nicolson finite difference scheme is developed to find numerical solutions. The exact solution of Kawahara and Tanaka is used to demonstrate the efficiency of the scheme. It is confirmed that the numerical errors, evaluated in the discrete maximum norm, converge in $O(\Delta x^2 + \Delta t)$, where $\Delta x$ and $\Delta t$ are spatial and temporal step sizes, respectively. More simulations with different initial conditions are conducted. In particular, it is observed that the amplitude of the impulse is the major factor in determining if the wave damps down to the resting or rises to the excited state. For values $u(x, 0) \le \alpha$, the wave approaches the 0 resting state and for $u(x, 0) \gt \alpha$ the wave rose to the 1 excited state.