Theory and Modern Applications
Table 2 The conditions for linearization in each case
From: Conditional linearization of the quintic nonlinear beam equation
Group | Nonlinear equations | Conditions for linearization |
|---|---|---|
1 | \(\begin{array}[t]{l}EIU'''(1-3U^{2}+\frac{9}{4}U^{4})+EIU^{\prime 3}(\frac {27}{2}U^{2}-3)\\\quad{}+U'(m+P_{0}+\frac{3}{2}P_{0}U^{2})+cU=0 \end{array} \) | \(U=\sqrt{\frac{18}{81}}\), \(P_{0}=0\), c = 0, m = 0 |
\(\begin{array}[t]{l}EIU'''(1-3U^{2}+\frac{9}{4}U^{4})+EIU^{\prime 3}(\frac {27}{2}U^{2}-3)\\\quad{}+U'(m+P_{0}+\frac{3}{2}P_{0}U^{2})+c(U+1)=0 \end{array} \) | \(U=\sqrt{\frac{18}{81}}\), \(P_{0}=0\), c = 0, m = 0 | |
2 | \(\begin{array}[t]{l}EIU'''(1-3U^{2}+\frac{9}{4}U^{4})+EIU^{\prime 3}(\frac {27}{2}U^{2}-3)\\\quad{}+P_{0}U'(1+\frac{3}{2}U^{2})+c=0 \end{array} \) | \(U=\sqrt{\frac{18}{81}}\), \(P_{0}=0\), c = 0 |
\(\begin{array}[t]{l}EIU'''(1-3U^{2}+\frac{9}{4}U^{4})+EIU^{\prime 3}(\frac {27}{2}U^{2}-3)\\\quad{}+P_{0}U'(1+\frac{3}{2}U^{2})=0 \end{array} \) | \(U=\sqrt{\frac{18}{81}}\), \(P_{0}=0\) |