Evaluate the space factor and time factor in equation

Question

Differential Equation : Space Factor, Time Factor, Eigenvalues and Sturm-Liouville 

Solve the following problem:

Consider the differential equation u_t =1/2 u_xx + u_x for 0 ≤x ≤ π, t > 0 with boundary conditions u(0,t) = u(π,t) = 0.

(a) Separate variables and write the ordinary differential equations that the space factor X(x) and the time factor T(t) must satisfy.

(b) Show that 0 is not an eigenvalue of the Sturm-Liouville problem for X.

(c) Show that for any integer n≥1, X_n(x) = e^-x sin nx is an eigenfunction of the Sturm-Liouville problem for X and determine the corresponding eigenvalue.

(d) Assuming that these are all the eigenvalues, write down in series form the general solution of the boundary value problem above assuming a general initial condition u(x, 0) = f(x).