Euler-Bernoulli beam equation ($u_{tt}+c_b^2{u}_{xxxx}=0$)

The Euler-Bernoulli beam equation models transverse linear bending waves in a slender beam. The normalized equation has $c_b=1$ and may be reduced by the Schrodinger factorization of the elastic beam equation. Prescribing displacement and curvature at an endpoint supplies two boundary traces for its fourth spatial derivative.

Writing $q=u+iv$ in the free Schrodinger equation gives $u_t=-v_{xx}$ and $v_t=u_{xx}$, hence $u_{tt}+u_{xxxx}=0$. To encode initial velocity $u_1$, choose the decaying primitive $v_0(x)=-{\int }_x^{\infty }(s-x)u_1(s)ds$. A prescribed endpoint curvature becomes the time derivative of the imaginary boundary trace. This turns the normalized Euler-Bernoulli beam equation into a complex second-order boundary problem.