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Iterative Solution of Highly Nonlinear Boundary Value Problems

Frank Kampas
Organization: Physicist at Large Consulting
Revision date


Nonlinear boundary value differential equations are usually solved with the "shooting method". In this technique, the initial conditions are adjusted until the boundary conditions at the other boundary are satisfied. In situations in which the shooting method fails, the iterative "relaxation" method can be used. Initial guesses at the solution are improved repeatedly. The relaxation method can be implemented using "quasi-linearization". The differential equation is linearized about the guessed solution and the linearized boundary value problem is solved. This technique can be implemented in a very straight-forward fashion in Mathematica because the numerical solutions to differential equations are interpolating functions.

Quasi-linearization is first demonstrated with a single differential equation and then with the 5 coupled differential equations which describe a p-n junction.

This submission is an update to a previous submission (see URL below). I have updated the code, improved the plots, added references and added some extra discussion and another test case for the p-n junction section.

*Applied Mathematics > Numerical Methods > Approximation Theory
*Engineering > Electrical Engineering
*Mathematics > Calculus and Analysis > Differential Equations
*Science > Physics > Solid State Physics
*Wolfram Technology > Kernel > Numerics

Differential equations, boundary value problems, p-n junction

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nonlinear_boundary_value.nb (358 KB) - Mathematica Notebook