System of Lane-Emden equations as IVPs BVPs and Four Point BVPs & Computation with Haar Wavelets

In this work we present Haar wavelet collocation method and solve the following class of system of Lane-Emden equation defined as \begin{eqnarray*} -(t^{k_1} y'(t))'=t^{-\omega_1} f_1(t,y(t),z(t)),\\ -(t^{k_2} z'(t))'=t^{-\omega_2} f_2(t,y(t),z(t)), \end{eqnarray*} where $t>0$, subject to initial values, boundary values and four point boundary values: \begin{eqnarray*} \mbox{Initial Condition:}&&y(0)=\gamma_1,~y'(0)=0,~z(0)=\gamma_2,~z'(0)=0,\\ \mbox{Boundary Condition:}&&y'(0)=0,~y(1)=\delta_1,~z'(0)=0,~z(1)=\delta_2,\\ \mbox{Four~point~Boundary~Condition:}&&y(0)=0,~y(1)=n_1z(v_1),~z(0)=0,~z(1)=n_2y(v_2), \end{eqnarray*} where $n_1$, $n_2$, $v_1$, $v_2$ $\in (0,1)$ and $k_1\geq 0$, $k_2\geq0$, $\omega_1