Rapid development of Mathematica built-in functions and interfaces provides the possibility for solving a large variety of problems arising in mathematics, physics, engineering, and other sciences. Especially, Versions 5, 5.1, and 5.2 possess exciting new features. The goal of the presentation is to show the numerical approximation of nonlinear partial differential equations using variational method and finite element approach based on Mathematica. A semi-discrete algorithm (time t continuous, spatial variable x is discrete) was constructed to solve nonlinear partial differential equations, the main technical points of the scheme are focused on several aspects:
- Functional analysis and variational method is considered to nonlinear partial differential equations.
- Gauss-Legendre quadrature is employed to deal with integration appearing in a nonlinear term of weak form, which was established by variational method.
- Runge-Kutta iteration method is used to find the solutions of converted ordinary differential equations.
- The main point is based on Mathematica to implement a whole solving process (starting from weak form construction, matrix and vectors equations, numerical solution searching, until obtaining a 3D graphical expression).
The results are divided into two parts to show obtained achievement:
- Mathematica-based numerical approach of several nonlinear partial differential equations
- Mathematica-based numerical approximate for optimal control of nonlinear parameter-distributed systems, for example Hofield neural network with diffusion term
As to further work, referring to MathOptimizer, MathOptimizer Professional, and Global Optimization, nonlinear large-scale complex systems, nonlinear design optimization, and other nonlinear phenomena can be considered.