Arnaldo Rodriguez-Gonzalez

@aghostinthefigures.bsky.social

194 Followers 26 Following

Dynamicist, writer, asker of questions. Assistant professor of teaching in the University at Buffalo. Ph.D. in theoretical & applied mechanics from Cornell, B.S. in mechanical engineering from UPR Mayagüez. 🇵🇷 He/him.

A plot of the maximum absolute value of a numerical approximation to the solution of an altered version of the original BVP, as the grid size becomes smaller. Because this altered problem actually has an analytical solution, the value converges rapidly onto a single value as the grid size increases.
A graph showing an increase in the maximum of the absolute value of the numerical "solution" as the grid becomes finer and finer. As a result, the computed solution keeps changing and growing as the grid is made smaller!
A plot of a numerical approximation to the non-existent solution of the original BVP. It is considerably sine-like, and has a larger amplitude than the previous computed result.
A plot of a numerical approximation to the non-existent solution of the original boundary value problem for a grid size of 5. The numerical function appears sine-like, and fits the required boundary conditions.
An alternative initial value problem is shown; the same differential equation, except that now the value of the function and its derivative are both specified at a single value of t. This variant of the problem always has a solution!
When attempting to apply the second boundary condition, there is no value of A that will make x(pi) equal to 1, as A*sin(pi) is always equal to zero for any value of A.
When the first boundary condition is applied to the general solution, the coefficient B in front of the cosine term must equal zero.
A boundary value problem: the second derivative of x as a function of t, plus x as a function of t, equal zero. When t = 0, x must be 0; when t = pi, x = 1.