Heat equation for a rectangular region

Question

Assignment:

Consider the heat equation for a rectangular region,   0 < x  < a, 0 < y < b, t > 0

u_t= k(u_xx + u_yy) , 0 < x  < a, 0 < y < b, t > 0

subject to the initial conditions: u(x,y) = f(x,y)

a) u_x (0, y, t) = 0, u_x(a, y, t) = 0, 0 < y < b, t > 0

u_y (x, 0, t) = 0, u_y (x, b, t) = 0, 0 0

b) u (0, y, t) = 0, u_x (a, y, t) = 0, 0 < y < b, t > 0

u_y (x, 0, t) = 0, u_y (x, b, t) = 0, 0 0

Provide complete and step by step solution for the question and show calculations and use formulas.