Iterative Solution of Highly Nonlinear Differential Equations Using Mathematica

Frank Kampas
Lockheed Martin

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These are just a few of the many interesting and important problems that were previously excluded from the teaching curriculum as being too difficult but that can now be solved with Mathematica in a simple and lucid manner accessible to students. The solutions generated by NDSolve, Mathematica's function for numerical solution of ordinary and partial differential equations, are (interpolating) functions. This unique feature of Mathematica enables the implementation of iterative solution methods for nonlinear boundary value differential equations in a straightforward fashion. For example, quasi linearization, a form of Newton's method, can be used to interactively solve nonlinear boundary value problems that cannot be handled directly by NDSolve or by the shooting method. In this approach, a linearized form of the differential equation is solved by the shooting method or by superposition, with each successive solution used as input for the next iteration until the solution has the desired accuracy. Variational methods can be used to generate good initial functions for the iterative procedure. Application of this approach to coupled nonlinear differential equations such as those encountered in semiconductor transport physics will be discussed.