Theory and Modern Applications
Table 3 The exact solutions to the quintic nonlinear equations of motion
From: Conditional linearization of the quintic nonlinear beam equation
Group | Generators | Exact solutions |
|---|---|---|
1 | \(X_{1}+X_{2}\) | \(w=\frac{2m^{2}}{c^{2}}e^{-\frac {c}{m}t}+\frac{h_{1}c_{1}^{2}}{6b_{1}}(t-x)^{3} +\frac{h_{2}c_{1}}{2b_{1}}(t-x)^{2}+\frac{h_{3}}{b_{1}}(t-x)+r\) |
\(X_{1}+X_{2}+X_{3}\) | \(w=\frac{m^{2}}{c^{2}}(2+e)e^{-\frac{c}{m}t}+\frac {h_{1}c_{1}^{2}}{6b_{1}}(t-x)^{3} +\frac{h_{2}c_{1}}{2b_{1}}(t-x)^{2}+\frac{h_{3}}{b_{1}}(t-x)+r\) | |
\(X_{1}+X_{2}+X_{4}\) | \(w=\frac{3m^{2}}{c^{2}}e^{-\frac{c}{m}t}+t+\frac {h_{1}c_{1}^{2}}{6b_{1}}(t-x)^{3} +\frac{h_{2}c_{1}}{2b_{1}}(t-x)^{2}+\frac{h_{3}}{b_{1}}(t-x)+r\) | |
2 | \(X_{2}+X_{4}\) | \(w=\frac{2m^{2}}{c^{2}}e^{-\frac {c}{m}t}+t+\frac{h_{1}c_{1}^{2}}{6b_{1}}x^{3} +\frac{h_{2}c_{1}}{2b_{1}}x^{2}+\frac{h_{3}}{b_{1}}x+r\) |
\(X_{2}+X_{3}+X_{4}\) | \(w=\frac{m^{2}}{c^{2}}(2+e)e^{-\frac{c}{m}t}+t+\frac {h_{1}c_{1}^{2}}{6b_{1}}x^{3} +\frac{h_{2}c_{1}}{2b_{1}}x^{2}+\frac{h_{3}}{b_{1}}x+r\) | |
\(X_{2}+X_{3}\) | \(w=\frac{m^{2}}{c^{2}}(1+e)e^{-\frac{c}{m}t}+\frac {h_{1}c_{1}^{2}}{6b_{1}}x^{3} +\frac{h_{2}c_{1}}{2b_{1}}x^{2}+\frac{h_{3}}{b_{1}}x+r\) |