Problem: Find an approximate analytical solution $u(x, t)$ to the following nonlinear partial differential equation $u_t +0.1(u_x)^2+0.05uu_{xx} = u_{xxxx}-16u+0.2e^{-3x}, \quad u(0,t)=1, u(x,0)=0 $ The solution is sought for $x \ge 0$ and $t \ge 0$. Derive an approximate analytical solution $u(x, t)$ that captures the dominant behavior in the limit $t \to \infty$. Solve fully, such that the final solution contains only known functions (no undefined functions). Please place your final solution in a \boxed{} LaTeX Environment. If there are any free constants, approximate them numerically. There should be no variable (non-evaluated) constants or free constraints, however. Do not box more than one equation! The approximation should include the zeroth-order term and the first-order correction term accounting for the nonlinearity.
Solution: $\boxed{$u(x,t)=0.9994170441764800e^{-2x}+0.003076923076923077e^{-3x}-0.002484639053254438e^{-4x}-0.000009318214941557116e^{-5x}-0.000000009985207100591716e^{-6x}}$