$$uj$/$t $=6$uj$($uj$+1$ uj$1)/(2$x$)($uj$+22$uj$+1+2$uj$1$ uj$2)/(2$x$^3).(4)$

To ensure the solution is stable, the stability condition must be satisfied

$<=1\mathrm{.}089$x$^3.$

Write a MATLAB program that solves the equation in the discretization form using RK$4.$ Solve it in the region $10<=$ x $<=10$ with a grid size $$x $=0\mathrm{.}1,$ and use periodic boundary conditions: x$(10)=$ x$(10)(6)$

Integrate from t $=0$ to t $=2,$ using an appropriate time$-$step that satisfies the stability condition. For each of the initial conditions below, plot the solution at t $=2$ and comment on the results.