Theory and Modern Applications
Table 1 Variable separation solution, inhomogeneity terms, and decay rate of ILLG equation (2)
Solutions I–III | q(r) and decay rate |
|---|---|
Solution I: \(\begin{pmatrix} \cos ( \Theta_{1} ) \sin ( B ) \\ \sin ( \Theta_{1} ) \sin ( B ) \\ \cos ( B ) \end{pmatrix}\), | \(-{\frac{C_{{1}}{r}^{2-2 n} ( C_{{2}}{r}^{n}+C_{{3}}n ) ^{ 2}\cos ( B ) {\beta}^{2}}{\alpha C_{{2}}^{2}{n}^{2} }}\) |
\(\Theta_{1} = -{\frac{\alpha}{\cos ( B ) \beta}\ln ( -{\frac{\cos ( B ) \beta ( C_{{2}}{r}^{n}+C _{{3}}n ) }{\alpha n}} ) } + C_{{1}}t \) | and O(1) |
Solution II: \(\begin{pmatrix} \cos ( \Theta_{2} ) \sin ( B ) \\ \sin ( \Theta_{2} ) \sin ( B ) \\ \cos ( B ) \end{pmatrix}\), | \(-{\frac{C_{{1}}\cos ( B ) {\beta}^{2} ( \overline{r} +K_{{n+1}} ) ^{2}}{\alpha ( \overrightarrow{K} \cdot \overrightarrow{K} ) }}\) |
\(\Theta_{2} = C_{{1}}t -{\frac{\alpha\ln ( \overline{r} +K_{{n+1}} ) }{\cos ( B ) \beta}}+K_{{n+2}} \) | and O(1) |