Show that (9) in Sec. 12.6 with coefficients (10) is a solution of the heat equation for t > 0 assuming that f(x) is continuous on the interval 0 ‰¤ x ‰¤ L and has one-sided derivatives at all interior points of that interval. Proceed as follows.

Show that |ˆ‚u_n/ˆ‚t| < λ_n²Ke^-λn2t0 if t ‰¥ t₀ and the series of the expressions on the right converges, by the ratio test. Conclude from this, the Weierstrass test, and Theorem 4 that the series (9) can be differentiated term by term with respect to t and the resulting series has the sum ˆ‚u/ˆ‚t. Show that (9) can be differentiated twice with respect to x and the resulting series has the sum ˆ‚²u/ˆ‚x². Conclude from this and the result to Prob. 19 that (9) is a solution of the heat equation for all t ‰¥ t₀.

Data from Prob. 19

Show that (9) in Sec. 12.6 with coefficients (10) is a solution of the heat equation for t > 0 assuming that f(x) is continuous on the interval 0 ‰¤ x ‰¤ L and has one-sided derivatives at all interior points of that interval. Proceed as follows.

Show that |B_n| is bounded, say |B_n| < K for all n. Conclude that

and, by the Weierstrass test, the series (9) converges uniformly with respect to x and t for t ‰¥ t_0, 0 ‰¤ x ‰¤ L. Using Theorem 2, show that u(x, t) is continuous for t ‰¥ t₀ and thus satisfies the boundary conditions (2) for t ‰¥ t₀.

|un] < Ke-Ato if t2 to > 0