Problem 8 - Advanced Engineering Mathematics

Question

The initial temperature in a rod of unit length is \(f(x)\) throughout. There is heat transfer from both ends, \(x=0\) and \(x=1\), into a surrounding medium kept at a constant temperature zero. Show that $$ u(x, t)=\sum_{n=1}^{\infty} A_{n} e^{-k \alpha_{n}^{2} t}\left(\alpha_{n} \cos \alpha_{n} x+h \sin \alpha_{n} x\right) $$ where $$ A_{n}=\frac{2}{\left(\alpha_{n}^{2}+2 h+h^{2}\right)} \int_{0}^{1} f(x)\left(\alpha_{n} \cos \alpha_{n} x+h \sin \alpha_{n} x\right) d x $$ The eigenvalues are \(\lambda_{n}=\alpha_{n}^{2}, n=1,2,3, \ldots\), where the \(\alpha_{n}\) are the consecutive positive roots of \(\tan \alpha=2 \alpha h /\left(\alpha^{2}-h^{2}\right)\)

Step-by-Step Solution

Answer

The solution involves expressing temperature as a series using eigenfunctions and decay factors.

Step 1: Understanding the Problem

We need to find the temperature distribution in a 1D rod of unit length with initial temperature function \( f(x) \) and boundary conditions that result in heat transfer at both ends of the rod into a medium at zero temperature. We aim to show the derivation of the expression for \( u(x, t) \) involving an infinite series.

Step 2: Setting Up the Heat Equation

The heat equation for the rod is \( u_t = k u_{xx} \). We apply Fourier series methods to solve this problem under the assumption that the solution can be expressed as a superposition of modes that decay exponentially over time due to the heat loss at the rod ends.

Step 3: Applying Boundary Conditions

For heat transfer at the ends \( x=0 \) and \( x=1 \), Neumann type boundary conditions are used. This results in eigenvalue problem: \( \tan \alpha = \frac{2 \alpha h}{\alpha^2 - h^2} \). The roots \( \alpha_n \) are eigenvalues that describe the system's standing wave modes.

Step 4: Constructing the General Solution

The solution to the heat equation can be expressed as an infinite series: \[ u(x, t) = \sum_{n=1}^{\infty} \left( B_n \cos \alpha_n x + C_n \sin \alpha_n x \right) e^{-k \alpha_n^2 t} \] Due to orthogonality, the coefficients \( B_n \) and \( C_n \) represent the initial temperature distribution \( f(x) \).

Step 5: Finding the Coefficients

The coefficients \( A_n \) are given by:\[ A_n = \int_0^1 f(x) \left( \alpha_n \cos \alpha_n x + h \sin \alpha_n x \right) \, dx \]For normalization, \( A_n \) is divided by the length \( \alpha_n^2 + 2h + h^2 \) since this captures contributions from the boundary conditions and the profile of \( f(x) \).

Step 6: Assembling the Solution

The final expression for the temperature distribution is:\[ u(x, t) = \sum_{n=1}^{\infty} A_n e^{-k \alpha_n^2 t} \left( \alpha_n \cos \alpha_n x + h \sin \alpha_n x \right) \]This represents a series of damped oscillatory modes contributing to the temperature at any point on the rod over time.

Key Concepts

Boundary Conditions

Boundary conditions are crucial when solving differential equations like the heat equation. In this context, they describe how the rod interacts with its environment at each end, which significantly affects the temperature distribution. For our rod, placed in a medium where the temperature is held constant at zero, Neumann boundary conditions apply. This type of condition considers the heat flux, rather than temperature itself, at the boundaries.

Essentially, these conditions allow us to set up an equation, known as the eigenvalue problem, to find roots. Here, at both ends of the rod (\(x=0\) and \(x=1\)), heat will flow out until they reach the surrounding medium's temperature. Because of this interaction, described by a function \( an \alpha = \frac{2 \alpha h}{\alpha^2 - h^2}\), solving for \(\alpha_n\) gives us the various wave numbers that affect how heat dissipates in the rod.

These boundary conditions fundamentally affect how we calculate the Fourier series to accurately reflect how the initial temperature of the rod, \(f(x)\), changes over time.

Fourier Series

The Fourier series plays an essential role in solving the heat equation for the rod. It allows us to express complex functions, like our initial temperature \(f(x)\), as sums of simpler trigonometric functions. This method is beneficial because it breaks down the problem into modes or harmonics, each of which can be handled separately.

For our problem, the solution to the heat equation is constructed using an infinite series of sine and cosine functions, reflecting possible modes of the rod. Each of these modes is weighted by coefficients (\(A_n\)) that arise from projecting the initial temperature over the mode shapes.

The infinite sum involved in the Fourier series expresses the entire heat distribution by superimposing these scaled modes. Hence, it becomes practical to express the temperature distribution as:

• A decay term (\(e^{-k \alpha_n^2 t}\)) reflecting how quickly the mode loses heat over time
• A combination of cosine and sine terms hence captures how the initial heat distribution shapes the future temperature profile

Through Fourier series, we transform a complex temporal heat distribution problem into a manageable series calculation.

Eigenvalues

The concept of eigenvalues is integral in dealing with the heat equation. Specifically, eigenvalues (\(\lambda_n\)) arise when solving boundary value problems associated with differential equations like ours. In our context, the eigenvalues are closely tied to the roots (\(\alpha_n\)) of the equation \(\tan \alpha = \frac{2 \alpha h}{\alpha^2 - h^2}\).

Each \(\alpha_n\) serves as an eigenvalue after it has been squared. These correspond to different modes of heat transfer, otherwise known as standing wave solutions. In simpler terms, eigenvalues show how specific frequencies in the Fourier series expand and interact inside the rod.

The eigenvalue \(\lambda_n = \alpha_n^2\) links to the decay rate of each mode, dictating how fast or slow heat dissipates into the surroundings.

• The larger the eigenvalue, the faster that mode's amplitude will diminish over time.
• These values fundamentally determine the temporal evolution of the temperature as the rod equilibrates with its environment.

In summary, by grasping the role of eigenvalues, one can understand the different ways heat disperses through our rod, governed by both temporal and spatial variables.