---
# System prepended metadata

title: Attempt to solve the analytical solution of Phi

---


# Attempt to solve the analytical solution of Phi
\begin{aligned}		
\frac{\partial^2 }{\partial x^2}(\frac{2Dc}{\Phi_T}\phi)&=a_0\frac{\partial^2}{\partial x^2}\xi -b_0(4\xi^3-6\xi^2+2\xi)+d_0(30\xi^4 - 60\xi^3 + 30\xi^2)i\\
&=a_0\frac{\partial^2}{\partial x^2}\xi -b_0(4\xi^3-6\xi^2+2\xi)+d_0(30\xi^4 - 60\xi^3 + 30\xi^2)(f_0c\times e^{0.5\phi}-p_0\times e^{-0.5\phi})]\\
\stackrel{(Taylor\ apprx.)}{}&\approx
a_0\frac{\partial^2}{\partial x^2}\xi -b_0(4\xi^3-6\xi^2+2\xi)+d_0(30\xi^4 - 60\xi^3 + 30\xi^2)[(f_0c-p_0)+0.5(f_0c+p_0)\phi]\\
\Rightarrow q_0\frac{\partial^2 }{\partial x^2}\phi&=a_0\frac{\partial^2 }{\partial x^2}\xi-b_0P_3(\xi)+d_0f_0P_4(\xi)c-d_0p_0P_4(\xi)+0.5d_0f_0P_4(\xi)c\phi+0.5d_0p_0P_4(\xi)\phi\\
\Rightarrow \frac{\partial^2 }{\partial x^2}\phi &= a_0\frac{\partial^2 }{\partial x^2}\xi+P_4(\xi)+P_4(\xi)c+P_4(\xi)(c+\frac{p_0}{f_0})\phi\\
\end{aligned}
$where\ P_n(\xi)\ represents\ an\ nth-order\ polynomial\ of\ \xi.$
