I would like help taking the inverse Fourier transform to find U(x,t), U(x,t) is supposed to resemble to D'Alembert's solution. The displayed equations appear to be related to Fourier transforms commonly used in the analysis of partial differential equations.1. **Equation for \( \hat{U}(k,t) \):** \[ \hat{U}(k,t) = \frac{\hat{g}(k)}{ik} e^{ikt} + \left( \hat{f}(k) - \frac{\hat{g}(k)}{ik} \right) = \hat{f}(k) + \left(\frac{e^{ikt} - 1}{ik}\right) \hat{g}(k) \] - The term \(\frac{\hat{g}(k)}{ik} e^{ikt}\) involves the exponential function \(e^{ikt}\), suggesting a relationship to time-evolution in a Fourier space. - It simplifies to demonstrate how different components of the Fourier transform interact.2. **Equation for \( U(x,t) \):** \[ U(x,t) = \mathcal{F}^{-1} \{ \hat{f}(k) \} + \mathcal{F}^{-1} \left\{ \frac{e^{ikt} - 1}{ik} \hat{g}(k) \right\} \] - The expression \(\mathcal{F}^{-1} \{ \hat{f}(k) \}\) refers to the inverse Fourier transform, indicating the transformation back to spatial domain. - The second term includes \(\frac{e^{ikt} - 1}{ik}\), showing the time-dependent evolution impacting the function \(\hat{g}(k)\).These equations are essential for understanding how functions evolve over time within the framework of Fourier analysis, often applicable to solving differential equations. The separate terms represent distinct components contributing to the solution, which can be aligned with initial or boundary conditions.

So we have to solve the inverse Fourier transform to get the expression for U(x,t)

Answered: ikt 三 ik ik Ikt. ik こ

ikt 三 ik ik Ikt. ik こ

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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