6. [20 marks; (a) 12 marks (b) 4 marks (c) 4 marks]
(a) Solve the homogeneous heat equation with homogeneous boundary condition:
$$w_t(x, t) = w_{xx}(x, t),$$
$$w_x(0,t) = 0,$$
$$w(\pi, t) = 0,$$
$$t > 0, 0 \leq x \leq \pi$$
$$t>0$$
$$t>0$$
$$w(x, 0) = x(\pi - x),$$
$$0 < x < \pi.$$
(b) Find a solution to the following initial/boundary value problem:
$$v_t(x, t) = v_{xx}(x, t) + 2(\pi + x)t,$$
$$v_x(0,t) = t^2,$$
$$v(\pi,t) = 2\pi t^2,$$
$$t > 0, 0 \leq x \leq \pi$$
$$t>0$$
$$t>0$$
$$v(x, 0) = 0,$$
$$0 \leq x \leq \pi.$$
(c) Solve the heat equation for u(x, t) with mixed boundary condition (using principle
of superposition)
$$u_t(x,t) = u_{xx}(x, t) + 2(\pi + x)t,$$
$$u_x(0,t) = t^2,$$
$$u(\pi,t) = 2\pi t^2,$$
$$t > 0, 0 \leq x \leq \pi$$
$$t>0$$
$$t>0$$
$$u(x, 0) = x(\pi - x),$$
$$0 \leq x \leq \pi.$$