Approximate Solutions to the Allen-Cahn Equation Using the Finite Difference Method
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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.