Finite difference method nonlinear PDE Problem

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i want to solve a nonlinear PDE with finite difference method , from my academic background , i know how to use finite difference method to solve linear PDE , what's the right way to use FDM to solve nonlinear PDE or could someone provide me with book's titles or articles that can help me solving a nonlinear PDE using FDM
this is my problem and what did i try to do .
\begin{equation}

\begin{equation}
m_{z}\ddot{w}+EIw'''-Tw''-f+c_{1}\dot{w}-EAv''w'-EAv'w''-\dfrac{3}{2}EA(w')^2w''=0
\end{equation}
\begin{equation}
m_{z}\ddot{v}+c_{2}\dot{v}-EAv''-EAw'w''=0
\end{equation}
the boundary conditions of the system :
\begin{equation}
w''(0,t)=w''(L,t)=w(0,t)=v(0,t)=0
\end{equation}
\begin{equation}
-EIw'''(L,t)+Tw'(L,t)+EAv'(L,t)w'(L,t)+\dfrac{1}{2}EA\left[ w'(L,t)\right] ^{3}=u_{T}(t)
\end{equation}
\begin{equation}
\dfrac{1}{2}EA[w'(L,t)]^{2}+EAv'(L,t)=u_{L}(t)
\end{equation}
\begin{equation}
w(x,0)=w'(x,0)=v(x,0)=v'(x,0)=0
\end{equation}
where
\begin{equation}
w'=\dfrac{\partial w(x,t)}{\partial x} \;\; and \;\; \dot{w}=\dfrac{\partial w(x,t)}{\partial t}
\end{equation}
what i've tried to do is:
\begin{equation}
\begin{split}
& m_{z}\left( \dfrac{w_{i}^{j+1}-2w_{i}^{j}+w_{i}^{j-1}}{k^{2}}\right)+ EI\left( \dfrac{w_{i+2}^{j}-2w_{i+1}^{j}+2w_{i-1}^{j}-w_{i-2}^{j}}{2h^{3}}\right)-T\left( \dfrac{w_{i+1}^{j}-2w_{i}^{j}+w_{i-1}^{j}}{h^{2}}\right)+c_{1}\left( \dfrac{w_{i}^{j+1}-w_{i}^{j}}{k}\right)-\\
& EA\left( \dfrac{v_{i+1}^{j}-2v_{i}^{j}+v_{i-1}^{j}}{h^{2}}\right)\left( \dfrac{w_{i+1}^{j}-w_{i}^{j}}{h}\right)-EA \left( \dfrac{v_{i+1}^{j}-v_{i}^{j}}{h}\right)\left( \dfrac{w_{i+1}^{j}-2w_{i}^{j}+w_{i-1}^{j}}{h^{2}}\right)-\\
& \dfrac{3}{2}EA\left( \dfrac{w_{i+1}^{j}-w_{i}^{j}}{h}\right)^{2} \left( \dfrac{w_{i+1}^{j}-2w_{i}^{j}+w_{i-1}^{j}}{h^{2}}\right)=f
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}
& m_{z}\left( \dfrac{v_{i}^{j+1}-2v_{i}^{j}+v_{i}^{j-1}}{k^{2}}\right)+c_{2}\left( \dfrac{v_{i}^{j+1}-v_{i}^{j}}{k}\right)- EA\left( \dfrac{v_{i+1}^{j}-2v_{i}^{j}+v_{i-1}^{j}}{h^{2}}\right)-\\
& EA\left( \dfrac{w_{i+1}^{j}-w_{i}^{j}}{h}\right) \left( \dfrac{w_{i+1}^{j}-2w_{i}^{j}+w_{i-1}^{j}}{h^{2}}\right)=0
\end{split}
\end{equation}
thus
the 1st equation of the system:
\begin{equation}
\begin{split}
&\left(\dfrac{(m_{z}+kc_{1})w_{i}^{j+1}-(2+kc_{1})w_{i}^{j}+w_{i}^{j-1}}{k^{2}}\right)+ \left( \dfrac{EIw_{i+2}^{j}-2(EI+Th)w_{i+1}^{j}+4Thw_{i}^{j}+2(EI-Th)w_{i-1}^{j}-EIw_{i-2}^{j}}{2h^{3}}\right)-\\
& EA\left( \dfrac{v_{i+1}^{j}-2v_{i}^{j}+v_{i-1}^{j}}{h^{2}}\right)\left( \dfrac{w_{i+1}^{j}-w_{i}^{j}}{h}\right)-EA \left( \dfrac{v_{i+1}^{j}-v_{i}^{j}}{h}\right)\left( \dfrac{w_{i+1}^{j}-2w_{i}^{j}+w_{i-1}^{j}}{h^{2}}\right)-\\
& \dfrac{3}{2}EA\left( \dfrac{w_{i+1}^{j}-w_{i}^{j}}{h}\right)^{2} \left( \dfrac{w_{i+1}^{j}-2w_{i}^{j}+w_{i-1}^{j}}{h^{2}}\right)=f
\end{split}
\end{equation}
the 2nd equation of the system:
\begin{equation}
\begin{split}
& \left(\dfrac{(m_{z}+kc_{2})v_{i}^{j+1}-(2+kc_{2})v_{i}^{j}+v_{i}^{j-1}}{k^{2}}\right)- EA\left( \dfrac{v_{i+1}^{j}-2v_{i}^{j}+v_{i-1}^{j}}{h^{2}}\right) -\\
& EA\left( \dfrac{w_{i+1}^{j}-w_{i}^{j}}{h}\right) \left( \dfrac{w_{i+1}^{j}-2w_{i}^{j}+w_{i-1}^{j}}{h^{2}}\right)=0
\end{split}
\end{equation}
where h is delta x and k is delta t
what should i do next ??am i in the right path ?,, is this a good start or there is something else to do before using finite difference method.
thank you.