(a) Write programs for the explicit and the Crank€”Nicolson methods.

(b) Apply the programs to the heat problem of a laterally insulated bar of length 1 with u(x, 0) = sin Ï€x and u(0, t) = u(1, t) = 0 for all t, using h = 0.2, k = 0.01 for the explicit method (20 steps), h = 0.2 and (9) for the Crank€“Nicolson method (5 steps). Obtain exact 6D-values from a suitable series and compare.

(c) Graph temperature curves in (b) in two figures similar to Fig. 299

(d) Experiment with smaller h (0.1, 0.05, etc.) for both methods to find out to what extent accuracy increases under systematic changes of h and k.

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и(х, t) 100 t = 0 t = 1 t = 2 t = 8 -3 -2 -1 3. H100