jedy Consider the following problem Given a positive constant c and function B, find u such that The aim is to du u(0, t) u(x, 0) Suppose the eigenfunctions are denoted by X. Suppose also that is Rie- mann integrable and the Fourier coefficients CA are defined by cou in (0,1) for t> 0, u(1, t) = 0, B(x) for x = (0, 1). ca (X(2)) dx = [" B(x)X (2) dz. S S Ck prove that c exp(-cAkt) X(T) converges uniformly on [0, 1] [to, ti], k=1 1. Provide an example or examples such that where to > 0. (The notation can be simplified if one uses the fact that X(0) = X(1) = 0.) Prove the following steps where to > 0). |ca| 2 / 18 (2)| da 2||18||- 2. Prove that for any positive constant d, 1 C exp(-d) converges. 3. Prove that (1) (2) (3) C exp(-ct) X(T) converges uniformly on [0.1] [to, ti]. k=1 F