Category: Laplace Equation

A New Calculus of Variations Solution of the Schrödinger Equation for the Lithium Like Atom’s Ground State Energy

This computation took a lot longer time to reach a much better solution than my previously published result.

new-variational-solution-for-the-lithiumDownload

A Calculus of Variations Solution to the Quantum Mechanical Schrödinger Wave Equation for the Lithium Like Atom (Atomic Number Z = 3) by James Pate Williams, Jr.

variational-solution-for-the-lithiumDownload

Solving Four Elliptic Partial Differential Equations Using the Finite Element Method by James Pate Williams, Jr.

The equations I solve in this blog entry are from a thesis that uses artificial neural networks to solve partial differential equations:

https://dukespace.lib.duke.edu/dspace/handle/10161/8197

Here are the equations snipped from the thesis:

Alpha is defined as 0.5 * n. In our case n = 1, 2, 3, 4.

The solutions for the four cases above are graphed using Microsoft Mathematics:

Elliptic Partial Differential Equation for n = 1
Elliptic Partial Differential Equation for n = 2
Elliptic Partial Differential Equation for n = 3
Elliptic Partial Differential Equation for n = 4

The Microsoft Mathematics Application Window is illustrated below:

Microsoft Mathematics Application
Finite Element Method Elliptic PDF Solver Main Form 1
Finite Element Method Elliptic PDF Solver Main Form 2
Finite Element Method Elliptic PDF Solver Main Form 3
Finite Element Method Elliptic PDF Solver Equation Class 1
Finite Element Method Elliptic PDF Solver Equation Class 2
pde-elliptic-finite-element-methodDownload

The Laplace Equation in Various Orthogonal Rectilinear Coordinate Systems James Pate Williams, Jr. BA, BS, MSwE, PhD

Solve the Laplace equation in the following orthogonal rectilinear coordinate systems:

  1. Cartesian coordinates
  2. Cylindrical coordinates
  3. Spherical polar coordinates
  4. Parabolic cylindrical coordinates

Solution PDF:

Laplace Equation