Category: C++ Computer Applications

Blog Entry for Early Morning Friday, May 31, 2024, Solutions of a Second Order Self-Adjoint Ordinary Differential Equation by Three Methods

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Text and Exercise from “Boundary Value Problems Second Edition” by David L. Powers in Progress (c) Wednesday, April 17, 2024, James Pate Williams, Jr.

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Solution of the One-Dimensional Heat Equation for a Rod Using Finite Differences by James Pate Williams, Jr. Created on Wednesday April 3, 2024

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Undamped Mass-Spring Eigenvalue – Eigenvector Problem by James Pate Williams, Jr. (c) Monday April 1, 2024

We extend the results of the following website:

https://math.libretexts.org/Bookshelves/Differential_Equations/Differential_Equations_for_Engineers_(Lebl)/3%3A_Systems_of_ODEs/3.6%3A_Second_order_systems_and_applications

The five masses in the problem have a maximum value of 8. The six springs have a maximum value of 4 for their Hooke’s coefficients. The first 5 by 5 matrix is the inverse mass matrix, the second matrix is the Hooke’s coefficient 5 by 5 matrix, the third 5 by 5 matrix is the product of the inverse mass matrix times the Hooke’s coefficient matrix. The final row vector is the eigenvalue vector.

New 1d Integration Results (c) March 24, 2024, by James Pate Williams, Jr.

I tested NUMAL’s integration function versus homegrown trapezoidal rule and Simpson’s rule. The second and third algorithms were closed (included both endpoints). There exist higher order Newton-Cotes integration formulas. I did not test the Gauss-Legendre and/or Gauss-Laguerre integration method(s). The trapezoidal rule can be improved if the derivative of the integrand is known and is easily calculated.

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