This thesis presents a greedy method for the solution of differential equations that accelerates convergence versus standard Finite Element Methods. The Sequentially Optimized Meshfree Approximation method unites meshfree methods, sequential optimization processes, and radial basis functions to solve the strong form of governing equations. The ability to solve the strong form eliminates the need to develop expensive and time consuming variational and/or weak forms of the governing equations currently employed in many numerical methods. The first section introduces and explains the procedures for using this method and then uses increasingly complex examples to detail the finer points of the method across a range of algebraic and differential equations. The second section explains why this method fails for equations and systems that involve discontinuities and explores future avenues through which these shortcomings might be remedied.