Show that (9) in Sec. 12.5 with coefficients (10) is a

Chapter 15, Problem 15.5

(choose chapter or problem)

Show that (9) in Sec. 12.5 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 Iillln/iltl < An2 Ke-An2to if 1 ~ to and theseries of the expressions on the right converges. by theratio test. Conclude from this. the Weierstrass test, andTheorem 4 that the series (9) can be differentiated termby term with respect to t and the resulting series hasthe sum duliN. Show that (9) can be differentiated twicewith respect to x and the resulting series has the suma2u/ilx2 . Conclude from this and the result to Prob. 19that (9) is a solution of the heat equation for allt ~ to. (The proof that (9) satisfies the given initialcondition can be found in Ref. [CIO] listed in App. 1.)