Problem 9 - Advanced Engineering Mathematics

Question

Use an appropriate Fourier integral transform to solve the given boundary- value problem. Make assumptions about boundedness where necessary. (a) \(a^{2} \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial u}{\partial t},-\infty0\) \(u(x, 0)=f(x),\left.\frac{\partial u}{\partial t}\right|_{t=0}=g(x),-\infty

Step-by-Step Solution

Answer

The solution is \( u(x, t) = \frac{1}{2}[f(x + at) + f(x - at)] \) if \( g(x) = 0 \).

Step 1: Problem Identification

We need to solve the wave equation given by \( a^{2} \frac{\partial^{2} u}{\partial x^{2}} = \frac{\partial u}{\partial t} \) subject to initial conditions \( u(x, 0) = f(x) \) and \( \left.\frac{\partial u}{\partial t}\right|_{t=0} = g(x) \). We will apply the Fourier transform method to solve this boundary-value problem.

Step 2: Apply Fourier Transform

Apply the Fourier transform with respect to \( x \) to the equation. Let \( \hat{u}(k,t) \) be the Fourier transform of \( u(x, t) \). The transform of the wave equation becomes \( a^{2} (-k^{2}) \hat{u}(k,t) = \frac{d\hat{u}}{dt} \). This simplifies to \( \frac{d\hat{u}}{dt} + a^{2} k^{2} \hat{u} = 0 \).

Step 3: Solve the Transformed Ordinary Differential Equation

The transformed equation is an ordinary differential equation (ODE) in \( t \) with a separable variable. Its general solution is \( \hat{u}(k, t) = C(k) e^{-a^{2} k^{2} t} \), where \( C(k) \) is a constant determined by the initial conditions.

Step 4: Apply Initial Conditions

Substitute the initial conditions into the Fourier transform domain:- For \( u(x, 0) = f(x) \), we have \( \hat{u}(k, 0) = \hat{f}(k) \), so \( C(k) = \hat{f}(k) \).- For \( \left.\frac{\partial u}{\partial t}\right|_{t=0} = g(x) \), the equation becomes \(-a^{2} k^{2} \hat{f}(k) = \hat{g}(k)\) to find \( g(x) = 0 \), implying \( \hat{g}(k) = 0 \).

Step 5: Inverse Fourier Transform

The solution in the Fourier domain is \( \hat{u}(k, t) = \hat{f}(k) e^{-a^{2} k^{2} t} \). Take the inverse Fourier transform to return to the \( x \)-domain, giving:\[ u(x, t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \hat{f}(k) e^{-a^{2} k^{2} t + ikx} \, dk. \]

Step 6: If \( g(x) = 0 \), Solve for \( u(x, t) \)

Assuming \( g(x) = 0 \) and applying the inverse Fourier transform results in a symmetrically propagated wave expression. With the method of characteristics and assuming the form of propagating waves, we find:\[ u(x, t) = \frac{1}{2} \left[ f(x + at) + f(x - at) \right], \]which directly solves the given wave equation under the specified condition.

Key Concepts

Wave Equation

The wave equation is a fundamental partial differential equation that expresses how waves, such as sound or light waves, propagate through a medium. The mathematical form of the wave equation given in this problem is:\[ a^{2} \frac{\partial^{2} u}{\partial x^{2}} = \frac{\partial u}{\partial t} \]This equation describes how the second spatial derivative of the wave function \( u(x,t) \) relates to its time derivative. Here, \( a \) is a constant representing the wave's speed.In simple terms, this equation captures the essence of how a point on the wave moves over time due to its surrounding points' movements. Waves move through space, and their properties like amplitude can change depending on the equation's parameters. This representation is crucial for modeling physical phenomena like sound waves, seismic waves, or even electromagnetic waves in physics. Understanding the wave equation helps us predict how such waves will behave under varying conditions.

Boundary-Value Problem

Boundary-value problems involve finding a solution to a differential equation that must satisfy certain conditions at the boundaries of the domain. In the context of the wave equation's domain being infinite, from \(-\infty < x < \infty\), we must satisfy initial conditions as the boundary conditions.For the given wave equation, the initial conditions are:

• Initial displacement: \( u(x, 0) = f(x) \)
• Initial velocity: \( \left.\frac{\partial u}{\partial t}\right|_{t=0} = g(x) \)

These conditions specify how the wave starts at time \( t=0 \) across all space \( x \). Solving boundary-value problems typically involves techniques like separation of variables, Fourier transforms, or considering symmetry properties to find functions \( u(x,t) \) that meet the criteria. Boundary-value problems are fundamental in understanding how waves or heat diffuses across a medium from a given start point.

Partial Differential Equations

Partial Differential Equations (PDEs) like the wave equation describe relations involving unknown multivariable functions and their partial derivatives. They are used extensively in mathematical modeling of physical systems. PDEs are categorized based on their characteristics and behaviors, and the wave equation is a classic example of a second-order linear PDE. The significance of PDEs in this context lies in their ability to model complex systems dynamically. Solutions to PDEs help in predicting future states of these systems. Solving PDEs might involve techniques like the method of characteristics, separation of variables, or transforming them into simpler forms through techniques like the Fourier transform. The transformation simplifies the computations by converting differential operators into algebraic ones, as done in this exercise.

Initial Conditions

In solving differential equations, initial conditions (or boundary conditions) are crucial as they determine the unique solution out of the infinite possibilities for the PDE. In this exercise:

• \( u(x, 0) = f(x) \) represents the initial shape or displacement of the wave at time \( t=0 \).
• \( \left.\frac{\partial u}{\partial t}\right|_{t=0} = g(x) \) represents the initial velocity of each point on the wave.

These conditions fill the role of 'boundary' criteria across the infinite domain in \( x \). When solving using Fourier transforms, these initial conditions help determine constants and particular solutions after the PDE has been transformed and solved in the frequency domain. Initial conditions essentially "set the stage" for how the wave propagates and evolves over time.