Under appropriate restrictions
of material and motion the equation of motion for a vibrating elastic bar is
(∂x4+ ∂t2)u = 0. Because of its mechanical importance, there is a large literature
devoted to the eigenvalue problem for this equation but solutions of boundary value
problems for the equation itself seem to have been ignored. It appears that Pini
was the first to seek a solution in terms of integrals analogous to thermal
potentials. Like Pini, we use a fundamental solution very similar to that of the
heat kernel to obtain potential terms which lead to a system of integral
equations. While Pini uses Laplace transforms to obtain solutions to the
integral equations, we observe that the problem may be reduced to one integral
equation of a complex valued function, f = a + λk ∗f, effecting a significant
simplification.
Along the way, we obtain, by reduction to Abel integral equations, a
general method of solving semi-infinite problems which can solve boundary
value problems not available to Fourier transforms, the technique presently
used.
