Corrigenda to “The Galerkin method and hinged beam dynamics”

1. Introduction

When [1] was published, it contained errors. Because of this, Theorem One (the sole theorem stated in [1]) was not proven when [1] was published. To remedy this situation the author has made several changes to [1]. Once these changes have been made, Theorem One becomes proven. A list of these changes is as follows. In the first paragraph, the following mix of text and mathematical symbols appears:

“Let \(y=y(x)\) and \(z=z(x)\) be two real-valued functions that are defined on \(\Omega\).” This should read:

“Let \(u_0 = u_0(x)\) and \(u_1=u_1(x)\) be two real-valued functions that are defined on \(\Omega\).”

In the first paragraph, the following mix of text and mathematical symbols appears:

“The initial/boundary value problem \[\begin{split} & u_{tt} + F_1(u_t) + u_{xxxx} + F_2(u) = f \text{ on } \Omega \times (0,T), \\ & u(a,t) = u(b,t) = 0 \text{ for all } t \in (0,T), \\ & u_{xx}(a,t) = u_{xx}(b,t) = 0 \text{ for all } t \in (0,T), \\ & u(x,0) = y(x) \text{ for all } x \in \Omega, \\ & u_t(x,0) = z(x) \text{ for all } x \in \Omega, \end{split}\] occurs naturally in the study of vibrations in beams that are hinged at both ends.”

This should read:

“The initial/boundary value problem \[\begin{split} & u_{tt} + F_1(u_t) + u_{xxxx} + F_2(u) = f \text{ on } \Omega \times (0,T), \\ & u(a,t) = u(b,t) = 0 \text{ for all } t \in (0,T), \\ & u_{xx}(a,t) = u_{xx}(b,t) = 0 \text{ for all } t \in (0,T), \\ & u(x,0) = u_0(x) \text{ for all } x \in \Omega, \\ & u_t(x,0) = u_1(x) \text{ for all } x \in \Omega, \end{split}\] occurs naturally in the study of vibrations in beams that are hinged at both ends.”

\(\mathbf{Page}\) \(\mathbf{237}\)

In the statement of Theorem 1, the following mix of text and mathematical symbols appears:

“Furthermore, let \(y\) be an element of \(H^4_*(\Omega),\) and let \(z\) be an element of \(H^2_*(\Omega)\).”

This should read:

“Furthermore, let \(u_0\) be an element of \(H^4_*(\Omega),\) and let \(u_1\) be an element of \(H^2_*(\Omega)\).”

\(\mathbf{Page}\) \(\mathbf{240}\)

In the second paragraph of step one of the proof of Theorem 1, the following mix of text and mathematical symbols appears:

“The goal of this step is to establish that for any \(k \geq 1\) there exists a unique solution \(u \in C^3([0,T]; W_k)\) to the variational problem \[\begin{split} & ({u}''(t),v)_{L^2} + (u(t),v)_{H_*^2} + (F_1({u}'(t)),v)_{L^2} + (F_2(u(t)),v)_{L^2} = (f(t),v)_{L^2}, \\ & u(0) = u_0^k, u'(0) = u_1^k, \end{split}\] for any \(v \in W_k\) and \(t \in (0,T)\).”

This should read:

“The goal of this step is to establish that for any \(k \geq 1\) there exists a unique solution \(u_k \in C^3([0,T]; W_k)\) to the variational problem \[\begin{split} & ({u}''(t),v)_{L^2} + (u(t),v)_{H_*^2} + (F_1({u}'(t)),v)_{L^2} + (F_2(u(t)),v)_{L^2} = (f(t),v)_{L^2}, \\ & u(0) = u_0^k, u'(0) = u_1^k, \end{split}\] for any \(v \in W_k\) and \(t \in (0,T)\).”

\(\mathbf{Page}\) \(\mathbf{241}\)

In the second paragraph, the following mix of text and mathematical symbols appears:

“It follows that we can write \[\begin{split} & (u_k''(t),e_i)_{L^2} + (u_k(t),e_i)_{H_*^2} + (F_1(u_k'(t)),e_i)_{L^2} + (F_2(u_k(t)),e_i)_{L^2} = (f(t),e_i)_{L^2}, \\ & u(0) = u_0^k, \\ & u'(0) = u_1^k, \end{split}\] for all \(t \in (0,t_k)\) and for any \(i \in \{1,2,3,…,k\}.\)”

This should read:

“It follows that we can write \[\begin{split} & (u_k''(t),e_i)_{L^2} + (u_k(t),e_i)_{H_*^2} + (F_1(u_k'(t)),e_i)_{L^2} + (F_2(u_k(t)),e_i)_{L^2} = (f(t),e_i)_{L^2}, \\ & u_k(0) = u_0^k, \\ & u'_k(0) = u_1^k, \end{split}\] for all \(t \in (0,t_k)\) and for any \(i \in \{1,2,3,…,k\}.\)”

In the second paragraph, the following mix of text and mathematical symbols appears:

“An immediate consequence of the above is that \[\begin{split} & (u_k''(t),v)_{L^2} + (u_k(t),v)_{H_*^2} + (F_1(u_k'(t)),v)_{L^2} + (F_2(u_k(t)),v)_{L^2} = (f(t),v)_{L^2}, \\ & u(0) = u_0^k, \\ & u'(0) = u_1^k, \end{split}\] for all \(v \in W_k\) and \(t \in (0,t_k)\).”

This should read:

“An immediate consequence of the above is that \[\begin{split} & (u_k''(t),v)_{L^2} + (u_k(t),v)_{H_*^2} + (F_1(u_k'(t)),v)_{L^2} + (F_2(u_k(t)),v)_{L^2} = (f(t),v)_{L^2}, \\ & u_k(0) = u_0^k, \\ & u'_k(0) = u_1^k, \end{split}\] for all \(v \in W_k\) and \(t \in (0,t_k)\).”

\(\mathbf{Page}\) \(\mathbf{245}\)

In the first paragraph of Step six, the following mix of text and mathematical symbols appears:

“Proceeding as in section 7.2 of [2], we fix a positive integer \(N\) and choose a function \(v \in C^1([0,T];H^2_*(\Omega)\) of the form \(v(t) = \Sigma_{i=1}^N d_i(t) e_i\), where \(\{d_i\}_{i=1}^N\) are smooth functions.”

This should read:

“Proceeding as in section 7.2 of [2], we fix a positive integer \(N\) and choose a function \(v \in C^1([0,T];H^2_*(\Omega))\) of the form \(v(t) = \Sigma_{i=1}^N d_i(t) e_i\), where \(\{d_i\}_{i=1}^N\) are smooth functions.”

This concludes the corrigenda to “The Galerkin method and hinged beam dynamics.”