Determine the parabolic equations and heat equation

Question

Green's Functions, Parabolic Equations and Heat Equation

Response to the following question:

Question 1 - Parabolic PDE

Find in 0 < x' 0, the Green's function G(x, t, x', t') satisfying equations 6.18 and 6.20 and the additional conditions G_^x'(x,t,a,t')=G(x,t,0,t')=0.

Equation 6.18 is P'₃G=δ(x'-x)δ(y'-y), where P'₃=u_xx+u_^t is the adjoint of the heat equation.

Equation 6.20 is G=0,t'>t