The cable equation ut-γ uzz-at, with γ, α > 0, also known as the lossy heat equa- tion, was derived by the nineteenth-century Scottish physicist William Thomson to model propagation of signals in a transatlantic cable. Later, in honor of his work on thermodynamics, including determining the value of absolute zero temperature, he was named Lord Kelvin by Queen Victoria. The cable equation was later used to model the electrical activity of neurons. (a) Show that the general solution to the cable equation is given by u(t, x)-e-atu(t, z), where u(t, z) solves the heat equation tr γ'm (b) Find a Fourier series solution to the Dirichlet initial-boundary value problem vt = γ uzz _ αν, υ(0, z) = f(x), u(t,0) = 0=r(t, 1), 0 1, t>0 x Does your solution approach an equlibrium value? If so, how fast? (c) Answer part (b) for the Neumann problem