A family of isospectral fourth order Sturm Liouville problems and equivalent beam equations

Abstract

In this paper, we consider the class of fourth order Sturm- Liouville equation of the form $y^{(4)}(z)-2(q(z)y^{\prime})^{\prime}+(q^2(z)-q^{\prime\prime}(z))y(z)=\lambda^2y(z),\ 0\leq z \leq L$, with boundary conditions $y(z)=y^{\prime\prime}(z)=0$ at $z=0,L$. We prove that this class is equivalent to a second order Sturm Liouville problem. Using Darboux Lemma we obtain the closed form of fourth order Sturm-Liouville equations that is isospectral to a given one. Also we obtain the Euler Bernoulli beam equation equivalent to this class.