(Texas A&M International University, 2016-06-13) Villarreal, Jamil Malik; Lin, Runchang

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Seeking a deeper understanding of the world has been a driving factor in Applied Mathematics.
From counting and measuring physical objects to developing equations and ratios that
resemble patterns in nature, mathematics is used to interpret and explain the intricate structures
that we observe everyday. The field of Applied Mathematics almost always involves setting up and
then solving, or approximating solutions to, at least one partial differential equation that takes the
physical and mathematical properties into consideration. This is the process of creating mathematical
models.
For this thesis, we will investigate approximate solutions to the Allen-Cahn equation whose
analytic solution is still unknown due to the nonlinearities of the problem as well as its sensitivity
to certain constants as we shall see. The numerical schemes involved in these approximations are
obtained from the finite difference method.

(Texas A&M International University, 2015-12) Garcia, Gabriel; Lin, Runchang

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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.