To solve this partial differential equation (PDE) using the Fourier transform, let’s break down the problem.

Problem Statement

We have the PDE: $u_{tt} + 2u_t + u = u_{xx}$ for $x \in \mathbb{R}$ and $t > 0$, with initial conditions: $u(x, 0) = \text{sinc}(x)$ $u_t(x, 0) = -\text{sinc}(x)$ where $\text{sinc}(x) = \frac{\sin(x)}{x}$ (with $\text{sinc}(0) = 1$ by convention).

Steps to Solve

1. Apply the Fourier Transform in $x$: The Fourier transform of $u(x, t)$ with respect to $x$ is defined as: $\hat{u}(k, t) = \int_{-\infty}^{\infty} u(x, t) e^{-ikx} \, dx$ Applying this to both sides of the equation, we transform $u_{xx}$ to $-k^2 \hat{u}(k, t)$, and we leave $u_{tt}$ and $u_t$ in terms of $t$.

   This gives us the transformed equation: $\hat{u}_{tt}(k, t) + 2\hat{u}_t(k, t) + \hat{u}(k, t) = -k^2 \hat{u}(k, t)$ which can be simplified to: $\hat{u}_{tt}(k, t) + 2\hat{u}_t(k, t) + (1 + k^2) \hat{u}(k, t) = 0$

2. Solve the Resulting ODE in $t$: The equation above is a second-order linear ordinary differential equation (ODE) in $t$ for each $k$. This can be written as: $\hat{u}_{tt}(k, t) + 2\hat{u}_t(k, t) + (1 + k^2) \hat{u}(k, t) = 0$ To solve this, assume a solution of the form $\hat{u}(k, t) = e^{\lambda t}$ and find the characteristic equation.

3. Determine Initial Conditions in Fourier Space: Using the initial conditions:

   • $u(x, 0) = \text{sinc}(x)$
   • $u_t(x, 0) = -\text{sinc}(x)$

   We compute the Fourier transform of these initial conditions to obtain $\hat{u}(k, 0)$ and $\hat{u}_t(k, 0)$.

4. Inverse Fourier Transform: Once we solve for $\hat{u}(k, t)$, we can use the inverse Fourier transform to find $u(x, t)$.

Would you like a detailed walk-through of each step, including the solution to the ODE and computation of the initial conditions in Fourier space?

Related Questions

1. How does the Fourier transform handle initial conditions in the context of PDEs?
2. What is the significance of the sinc function as an initial condition in wave equations?
3. How would the solution change if we had different initial conditions?
4. What properties of the Fourier transform make it suitable for solving this PDE?
5. Could this method be used for other types of PDEs, such as heat or Laplace equations?

Tip

When solving PDEs with Fourier transforms, always ensure to carefully manage initial and boundary conditions in the transformed space—they play a crucial role in determining the solution's behavior.