Solved Examples: Fourier Transforms for Solving PDEs

📚 Quick Study Guide

• 🔑 Fourier Transforms are used to convert PDEs from the time or space domain to the frequency domain, often simplifying the equation.
• 💡 Key property: The Fourier Transform of a derivative involves multiplication by $i\omega$, where $\omega$ is the frequency variable. This significantly simplifies differentiation in PDEs.
• ➗ Convolution Theorem: The Fourier Transform of a convolution is the product of the Fourier Transforms. Mathematically, $F[f * g] = F[f]F[g]$.
• 📝 Common Fourier Transform Pairs:
  • Box function: $F[rect(t)] = sinc(f)$
  • Gaussian function: $F[e^{-at^2}] = \sqrt{\frac{\pi}{a}} e^{-\pi^2 f^2 / a}$
• 📈 Applications: Heat equation, wave equation, Laplace's equation, and other linear PDEs.
• 🧮 Solution Steps: (1) Apply Fourier Transform to the PDE. (2) Solve the transformed (algebraic) equation. (3) Apply the Inverse Fourier Transform to obtain the solution in the original domain.
• 🌡️ Example: For the heat equation $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$, the Fourier transform leads to $\frac{\partial U}{\partial t} = -\alpha \omega^2 U$, where $U$ is the Fourier transform of $u$.

🧪 Practice Quiz

1. What is the primary purpose of using Fourier Transforms to solve PDEs?
   1. To increase the complexity of the equation.
   2. To convert the PDE into an algebraic equation.
   3. To solve ordinary differential equations (ODEs).
   4. To find particular solutions only.
2. The Fourier Transform of a derivative, $\frac{\partial f(x)}{\partial x}$, is proportional to which of the following?
   1. $f(x)$
   2. $\omega$
   3. $i\omega$
   4. $\omega^2$
3. According to the Convolution Theorem, what is the Fourier Transform of $f(x) * g(x)$?
   1. $F[f(x)] + F[g(x)]$
   2. $F[f(x)] - F[g(x)]$
   3. $F[f(x)] \cdot F[g(x)]$
   4. $\frac{F[f(x)]}{F[g(x)]}$
4. Which of the following PDEs is most commonly solved using Fourier Transforms?
   1. Burgers' equation
   2. The Heat equation
   3. The Logistic equation
   4. The Black-Scholes equation
5. If $F[f(x)] = F(\omega)$, what is $F[f(x-a)]$?
   1. $e^{-ia\omega}F(\omega)$
   2. $e^{ia\omega}F(\omega)$
   3. $F(\omega - a)$
   4. $F(\omega + a)$
6. For the heat equation $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$, what does its Fourier transform become?
   1. $\frac{\partial U}{\partial t} = \alpha \omega U$
   2. $\frac{\partial U}{\partial t} = -\alpha \omega^2 U$
   3. $\frac{\partial U}{\partial t} = \alpha \omega^2 U$
   4. $\frac{\partial U}{\partial t} = -\alpha \omega U$
7. Which property of the Fourier Transform is crucial for solving PDEs with boundary conditions on an infinite domain?
   1. Linearity
   2. Time shifting
   3. Differentiation
   4. Duality