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Numerical Simulation Of The Nagumo Equation By Finite Difference Method, Gabriel Garcia
Numerical Simulation Of The Nagumo Equation By Finite Difference Method, Gabriel Garcia
Theses and Dissertations
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 …
Approximate Solutions To The Allen-Cahn Equation Using The Finite Difference Method, Jamil Malik Villarreal
Approximate Solutions To The Allen-Cahn Equation Using The Finite Difference Method, Jamil Malik Villarreal
Theses and Dissertations
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 …