Math 21b The Heat Equation

The Heat Equation1. The temperature f(t, x) in a thermally insulated rod of length π satisfies the partial differential equation ∂f ∂2f = µ ∂t ∂x2 where µ is some constant.

Example. Which of the following functions are solutions to the heat equation subject to the boundary conditions f(t, 0) = f(t, π) = 0? 1. f(t, x) = µt sin x

2. f(t, x) = cos(µt) sin x

3. f(t, x) = e−µt cos x

4. f(t, x) = e−µt sin x

1Despite the name, the heat equation has applications outside of thermodynamics. Schr¨odinger’swave equation (the equation that governs quantum systems) is actually a heat equation; Black–Scholes equation in financial mathematics is a heat equation; many image analysis and machine learning algorithms make use of the heat equation. Example. Suppose f(t, x) defined for x in [0, π] is a function satisfying the boundary conditions f(t, 0) = f(t, π) = 0. 1. At any fixed time t, we can think of f(t, x) as a function of just x. What can we say about the Fourier series for f(t, x) for a fixed t?

2. If f(t, x) satisfies the heat equation, what does that say about the coefficients of the Fourier series for f(t, x)?

3. Use your answer to part (2) to find the solution to the heat equation with boundary conditions f(t, 0) = f(t, π) = 0. Solutions to Heat Equation. The heat equation

∂f ∂2f = µ ∂t ∂x2 with boundary conditions f(t, 0) = f(t, π) = 0 and initial condition f(0, x) has solution

∞ X −µk2t f(t, x) = bke sin(kx) k=1 where bk are the coefficients in a Fourier sine series for the initial condition f(0, x).

Example. Solve the heat equation ft = µfxx, 0 ≤ x ≤ π with boundary conditions f(t, 0) = f(t, π) = 0 and initial ondition f(0, x) = sin(5x) + 3 sin(8x).

Example. A thermally insulated rod of length π is heated to a uniform temperature of 50◦C. The ends are then plunged into an ice bath of temperature 0◦C.

1. Find the temperature f(t, x) of the rod at time t and position x, assuming f satisfies the heat equation.

2. As t → ∞, what happens to the temperature of the rod? Example. (Adjusting the Boundary Conditions) Suppose that in the previous example we constantly heated the ends of the rod to 50◦C instead of plunging them into an ice bath.

1. What would happen to the rod’s temperature? Write down a function that models the temperature of the rod at time t and position x. Verify that this function satisfies the heat equation.

2. Now we want to solve the heat equation with boundary conditions f(t, 0) = f(t, π) = 50. Let fp be the particular solution we found in part (1). Show that if fh is any solution to the heat equation satisfying the boundary conditions fh(t, 0) = fh(t, π) = 0 then f = fh + fp is a solution to the heat equation with boundary conditions f(t, 0) = f(t, π) = 50.

3. Find the general solution to the heat equation satisfying the boundary conditions f(t, 0) = f(t, π) = 0.