Q$\mathrm{.}1.$ Consider the vertically hanging rod with linearly varying cross$-$sectional area in the form of $A(x)=A_0(1+\frac{x}{L}).$ The rod is of length L and is acted by a point load P at $x=L$ in addition to its self$-$weight. Find the vertical deflection of the rod by various methods of weighted residuals $($i$.$e$.,$ least squares, collocation, sub$-$domain, and Galerkin$)$ at points $x=\frac{L}{3}.$ Assume the material to be steel with $L=1\mathrm{.}0m,E=210$GPa,$p=1\mathrm{.}0N$ and $A_0=0\mathrm{.}001m^2$ and P to be twice the self$-$weight of the rod. Compare the results with exact solution. Use a cubic polynomial trial function.