Solve the 2-dimensional steady-state heat equation problem

Question

Q1. Solve the 1-dimensional heat equation problem.

∂u/∂t = 2(∂²u/∂x²)

u(0, t) = u(5, t) = 0, for t > 0

u(x, 0) = f(x) = -4sin(πx) + 3sin(2πx), for 0 ≤ x ≤ 5.

Q2. Solve the 1-dimensional zero ends heat equation problem with c = 1, L = 2, and initial temperature

Q3. Solve the 1-dimensional heat equation problem.

∂u/∂t = ∂²u/∂x²

∂u/∂x(0, t) = ∂u/∂x (1, t) = 0, for t > 0

u(x, 0) = f(x) = 5 - 4cos(2πx) + 3cos(6πx), for 0 ≤ x ≤ 1

Q4. Solve the 2-dimensional steady-state heat equation problem.

∂²u/∂x² + ∂²u/∂y² = 0

u(0, y) = u(1, y) = u(x, 0) = 0, for 0 ≤ x, y ≤ 1

u(x, 1) = 1 - x, for 0 ≤ x ≤ 1