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Abstract
The purpose of this paper is to explore the nature of partial differential equation (PDE) models and numerical iterative methods to find solutions. A summary of the historical development is provided as well as a brief introduction to PDEs and their classifications. Classic and weak solutions are covered along with numerical iterative methods to find solutions. Several common PDEs of interest are discussed; these are the Heat, Wave, Laplace, Poisson, Stokes, and Navier-Stokes equations. Methods for discretization techniques that produce large sparse matrices are presented along with the direct methods for solving; these are the LU and Cholesky factorization methods. Four iterative methods are presented in Chapter 4; these are Jacobi, Gauss-Seidel, SOR, and GMRES iteration methods. Numerical experiments were then performed via the software extension IFISS in MATLAB. Results in chapter 4 show the effectiveness and efficiency of the numerical solvers.
