Problem 7 - Applied Partial Differential Equations

Question

Prove that the Fourier series of a continuous function \(u(x, t)\) can be differentiated term by term with respect to the parameter \(t\) if \(\partial u / \partial t\) is piecewise smooth.

Step-by-Step Solution

Answer

The Fourier series of a continuous function \(u(x, t)\) can be differentiated term by term with respect to parameter \(t\) when \(\partial u / \partial t\) is piecewise smooth. This is because \(\partial u / \partial t\) being piecewise smooth guarantees no sudden changes that would prevent term-by-term differentiation, resulting in the Fourier series of \(\partial u / \partial t\) being equivalent to the derived series.

Step 1: Express the Function as a Fourier Series

Represent the function \(u(x, t)\) as a Fourier series. This series is typically in the form of \[u(x, t) = a_0 + \sum_{n=1}^{\infty}{[a_n \cdot \cos(nx) + b_n \cdot \sin(nx)]}\] where the coefficients \(a_n\) and \(b_n\) are dependent on \(t\), but are constants with respect to \(x\).

Step 2: Differentiate the Fourier Series Term by Term

Differentiate the Fourier series with respect to \(t\). Remember to only differentiate \(a_n\) and \(b_n\) as they're the only variables dependent on \(t\). This results in \[\frac{\partial u}{\partial t} = \frac{\partial a_0}{\partial t} + \sum_{n=1}^{\infty}{[\frac{\partial a_n}{\partial t} \cdot \cos(nx) + a_n \cdot \cos(nx) \cdot \sin(nx) + \frac{\partial b_n}{\partial t} \cdot \sin(nx) - b_n \cdot \sin(nx) \cdot \cos(nx)]}\]

Step 3: Justify the Term-By-Term Differentiation

The term-by-term differentiation is valid because \(\partial u / \partial t\) is piecewise smooth, which guarantees that the differentiation operation does not create any discontinuities. In this case, it means that the Fourier series of \(\partial u / \partial t\) is the same as the derived series, as demonstrated in step 2.

Key Concepts

Continuous function

When dealing with Fourier series, the behavior of continuous functions plays a pivotal role. A continuous function is one that has no abrupt changes in value - it can be graphed without lifting your pencil from the paper. In the context of our exercise, the function denoted as u(x, t) is continuous in terms of the variable x. Continuity is essential for Fourier series representation as it allows the function to be expressed as an infinite sum of sine and cosine terms.

The main advantage of Fourier series for continuous functions is that they simplify the analysis and processing of the function by breaking it down into simpler periodic components, which are the sine and cosine terms. With continuity ensured, if we differentiate or integrate the Fourier series, we expect that the resulting function will maintain this smooth behavior, provided certain conditions are met.

Piecewise smooth function

A piecewise smooth function is a function that is smooth on each piece of its domain, possibly excluding a finite number of points. Smoothing refers to the function possessing derivatives of all orders, except perhaps at certain isolated points where the function may be discontinuous or have a 'corner'. In the exercise, we are assured that \( \partial u / \partial t \) is piecewise smooth. This implies that the parameter t leads to a function that may have distinct sections within which the function behaves nicely - is differentiable, but the function may have finite points of non-smooth behavior.

For Fourier series, if a function is piecewise smooth, it can often be represented very accurately because the pieces that are smooth are handled in the same way as continuous functions. The implications for differentiation are vital; it means that we should be careful when differentiating at the points of non-smooth behavior, but otherwise, term-by-term differentiation is legitimate.

Partial derivative

The concept of a partial derivative is at the core of multivariable calculus. It describes how a function changes as only one of the variables is allowed to vary, treating all other variables as constants. For example, the partial derivative \( \partial u / \partial t \) in the exercise represents the rate of change of the function u with respect to the variable t, while x is held constant.

Understanding partial derivatives is essential for asserting the behavior of multivariable functions, such as those found in physics and engineering. When functions are represented by Fourier series, the coefficients of the series themselves may depend on variables that permit partial differentiation. This ability to take partial derivatives of Fourier series term by term under certain conditions simplifies the complexity involved in analyzing the underlying functions.

Term-by-term differentiation

Term-by-term differentiation is a technique where each term of an infinite series is differentiated individually. This method is particularly useful for Fourier series, where functions are represented as sums of sine and cosine functions with coefficients that may vary with respect to another variable, such as time t.

In the exercise, term-by-term differentiation of the Fourier series is justified because the function \( \partial u / \partial t \) is piecewise smooth, meaning it can tolerate this method without introducing inconsistencies or losing integrity of the original function representation. This approach is powerful because it allows us to handle the differentiation of potentially complex functions by focusing on simpler components, simplifying calculations, and helping us understand the dynamics involved in each part of the function.