# Numerical Solutions for Partial Differential Equations

Some Aspects of Numerical Solutions for Partial Differential Equations Austin Andries University of Southern Mississippi May 5, 2011 Dr. Hironori Shimoyama

Hirophysics.com Laplaces Equation Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as:

Laplace's equation are all harmonic functions and are important in many fields of science including electromagnetism, astronomy, and fluid dynamics. Where is the Laplace operator and is a scalar function.

= Hirophysics.com The solution of Laplaces equation under one of the boundary conditions Hirophysics.com

Poissons Equation Electrostatics is the posing and solving of problems that are described by the Poisson equation. Finding for some given is an important practical problem, since this is the usual way to find the electric potential for a given charge distribution. If vanishes to zero then the express becomes Laplaces equation. Hirophysics.com

The case of two unlike point charges Hirophysics.com Multiple point charges Hirophysics.com

Solitons A soliton is a self-reinforcing solitary wave (a wave packet or pulse) that travels with constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium. They arise as the solutions of a widespread class of weakly nonlinear dispersive partial differential equations describing physical systems.

The blue line is the carrier waves, while the red line is the envelope soliton. Hirophysics.com Examples of solitons Hirophysics.com

Interaction of soliton waves Hirophysics.com Future Research Implementation of a higher order of finite difference methods Investigating some of pitfalls with the simple algorithm of finite difference

Hirophysics.com Work Cited: Images: http://en.wikipedia.org/wiki/Laplace%27s_equation (slide 2 ) http://en.wikipedia.org/wiki/Poisson%27s_equation (slide 7) http://en.wikipedia.org/wiki/Poisson%27s_equation (slide 18). Code referred to Computational Physics by Landau and Paez

Hirophysics.com Appendix: Young-Laplace Equation YoungLaplace equation is a nonlinear partial differential equation that describes the capillary pressure difference sustained across the interface between two static fluids, such as

water and air, due to the phenomenon of surface tension. p = pressure difference across the fluid interface = surface tension, is the unit normal pointing out of the surface, H = is the mean curvature R1 , R2 = are the principal radii of curvature. Soap Films

If the pressure difference is zero, as in a soap film without gravity, the interface will assume the shape of a minimal surface. Hirophysics.com Hirophysics.com Hirophysics.com

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