Question

In this problem we will consider the heat equation on [0,1] : ∂u−∂

x
2
​
u=f(x) a) Consider (1) when f(x)=0. Show that, u₁ =e−⁴π²ᵗ
cos(2πx), and u₂ =e−¹⁶π²ᵗ
cos(4πx), solves (1) in this case.

Answer 1

To show that the solutions u₁ = e^(-4π²t) cos(2πx) and u₂ = e^(-16π²t) cos(4πx) satisfy the heat equation when f(x) = 0, we substitute these solutions into the equation and simplify.

Step-by-step explanation:

In this problem, we are considering the heat equation on the interval [0,1]: ∂u/∂t - ∂²u/∂x² = f(x). For part (a) of the question, when f(x) = 0, we need to show that the solutions u₁ = e^(-4π²t) cos(2πx) and u₂ = e^(-16π²t) cos(4πx) satisfy the heat equation.

To verify this, we substitute these solutions into the heat equation and confirm that the equation holds for both u₁ and u₂:

∂u₁/∂t - ∂²u₁/∂x² = e^(-4π²t) cos(2πx)

∂u₂/∂t - ∂²u₂/∂x² = e^(-16π²t) cos(4πx)

By computing the derivatives and simplifying the equations, we can see that these solutions satisfy the heat equation when f(x) = 0.