Alaterally insulated bar of length 10 cm and constant cross-sectional area 1 cm2, of density 10.6 gm/cm3, thermal conductivity 1.04 cal/cm sec °c, and specific heat 0.056 cal/gm °c has an initial temperature distribution of f(x) = 50 sin 0.1πx and is kept at 0 °c at the ends. (a) using finite difference method and semi-implicit (crank-nicholson) scheme, derive the difference equation for the bar using h = 1 cm and dt = 1 s. (b) solve the temperature at all unknown nodes from t = 0 to 10 s in steps of dt = 4 s. (d) determine the temperature values at x = 3 and 5 cm at t = 20 s using analytical solution. (e) compare the results of analytical and numerical solutions at x = 3 and 5 cm at t = 20 s.