Category: Laplace Equation
Blog Entry © Tuesday, March 25, 2025, by James Pate Williams, Jr. Hydrogen Radial Wavefunctions and Related Functions
Classification of Two Mathematical Entities by James Pate Williams, Jr. Sunday, May 12, 2024
Happy Mother’s Day
Solution of the Laplace (Potential) Equation on a Two-Dimensional Square via Finite Differences
Some Helium Coulomb Integrals over Six Dimensions by James Pate Williams, Jr. Source Code in C++ Development over December 15 – 16, 2023
Revised Translated Source Code from May 15, 2015, by James Pate Williams, Jr.
New and Corrected Ground State Energy Numerical Computation for the Helium Like Atom (Atomic Number 2) by James Pate Williams, Jr.
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.
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.
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:
The Microsoft Mathematics Application Window is illustrated below:
Numerical Explorations 