Question

Consider the conduction of heat in a rod \(40 \mathrm{cm}\) in length whose ends are maintained at \(0^{\circ} \mathrm{C}\) for all \(t>0 .\) In each of Problems 9 through 12 find an expression for the temperature \(u(x, t)\) if the initial temperature distribution in the rod is the given function. Suppose that \(\alpha^{2}=1\) $$ u(x, 0)=x, \quad 0

Short answer

The expression for the temperature distribution \(u(x, t)\) of the rod with length 40 cm is given by: $$ u(x, t) = \sum_{n=1}^{\infty}\frac{2}{n\pi}(-1)^{n+1}\sin\left(\frac{n\pi x}{40}\right)e^{-\frac{n^2\pi^2 t}{40^2}} $$

Step-by-step solution

Separation of variables

Consider the function \(u(x, t) = X(x)T(t)\). First, plug this function into the heat conduction equation to separate the variables. $$ X(x)T'(t) = \alpha^2 X''(x)T(t) $$ Divide both sides by \(X(x)T(t)\): $$ \frac{T'(t)}{\alpha^2 T(t)} = \frac{X''(x)}{X(x)} $$ Now, both sides of the equation only depend on one variable, so they must be equal to a constant, let's call it \(-k^2\): $$ \frac{T'(t)}{\alpha^2 T(t)} = \frac{X''(x)}{X(x)} = -k^2 $$

Solve the spatial problem

We will solve the spatial problem given by $$ X''(x) + k^2 X(x) = 0 $$ This is an ordinary differential equation with constant coefficients, and we have the following boundary conditions: $$ X(0) = X(40) = 0 $$ The general solution of the spatial ODE is $$ X(x) = A\sin(kx) + B\cos(kx) $$ Using the boundary conditions, we have $$ X(0) = B\cos(0) = 0 \implies B = 0 $$ So, the spatial solution simplifies to $$ X(x) = A\sin(kx) $$ Now, using the second boundary condition \(X(40) = 0\), we can see that \(k\) must take on the value of \(\frac{n\pi}{40}\) for \(n\in\mathbb{Z}\). Thus, the spatial solution becomes: $$ X_n(x) = A_n\sin\left(\frac{n\pi x}{40}\right) $$

Solve the temporal problem

Now, we will solve the temporal problem, given by $$ T'(t) + \alpha^2 k^2 T(t) = 0 $$ Plugging in \(k^2 = \left(\frac{n\pi}{40}\right)^2\), this becomes $$ T'(t) +\left(\frac{n\pi}{40}\right)^2 T(t) = 0 $$ This is a first-order linear ODE. The general solution is given by $$ T_n(t) = C_n e^{-\frac{n^2\pi^2 t}{40^2}} $$

Combine the solutions and apply the initial condition

Now, we combine the spatial and temporal solutions into an expression for \(u(x, t)\): $$ u(x, t) = \sum_{n=1}^{\infty}A_n \sin\left(\frac{n\pi x}{40}\right)e^{-\frac{n^2\pi^2 t}{40^2}} $$ To determine the coefficients \(A_n\), we apply the initial condition \(u(x, 0) = x\) for \(0 < x < 40\): $$ x = \sum_{n=1}^{\infty}A_n \sin\left(\frac{n\pi x}{40}\right) $$ To find the coefficients \(A_n\), we use the orthogonality property of sine functions over the interval \(0<x<40\): $$ A_n = \frac{2}{40}\int_{0}^{40} x \sin\left(\frac{n\pi x}{40}\right) dx $$ To evaluate this integral, we can use integration by parts, with \(u=x\) and \(dv=\sin\left(\frac{n\pi x}{40}\right)dx\). This leads to: $$ A_n = \frac{2}{n\pi}(-1)^{n+1} $$

Write the expression for u(x, t)

Combine the \(A_n\) values with the spatial and temporal solutions from above, to obtain the expression for the desired temperature distribution, \(u(x, t)\): $$ u(x, t) = \sum_{n=1}^{\infty}\frac{2}{n\pi}(-1)^{n+1}\sin\left(\frac{n\pi x}{40}\right)e^{-\frac{n^2\pi^2 t}{40^2}} $$ This expression represents the temperature distribution in the rod as a function of position \(x\) and time \(t\).

Key concepts

Separation of Variables

The separation of variables is a powerful mathematical method used to solve partial differential equations (PDEs), including the heat conduction equation. This technique simplifies a complex PDE by expressing the unknown function as a product of functions, each depending on a single independent variable.

In the context of heat conduction in a rod, we express the temperature as a product, say, \(u(x, t) = X(x)T(t)\), where \(X\) depends only on the spatial coordinate \(x\) and \(T\) only on the time \(t\). We then substitute this product into the heat conduction equation, leading to two separate ordinary differential equations (ODEs) for \(X\) and \(T\) when we equate them to a separation constant.

In the provided exercise, this technique allows the complex problem to be divided into simpler, more manageable parts: one concerning the spatial distribution \(X(x)\) and the other the temporal evolution \(T(t)\). This method also employs the given boundary conditions to further refine the solution by determining permissible values of separation constants.

Boundary Value Problem

A boundary value problem arises when we look for solutions to differential equations that also satisfy certain specified conditions at the boundaries of the domain. In heat conduction problems, these conditions often represent physical constraints, such as maintaining certain temperatures at the ends of a rod.

The problem presented requires solving for the temperature distribution inside a rod with specific temperature conditions at the ends. Using the separation of variables method, we derived an equation for \(X(x)\), the spatial component of the temperature. With the boundary conditions \(X(0) = X(40) = 0\), only certain 'modes' or frequencies—determined by the value of \(k\)—provide non-trivial solutions that satisfy both conditions. In essence, these 'modes' correspond to permissible vibration frequencies of the rod if it were to be imagined as a stretched string. The boundary conditions form an essential part of the problem, influencing the final shape of the solution.

Fourier Series

The Fourier series is a powerful mathematical tool for expressing a function as an infinite sum of sine and cosine functions. It comes into play after using the separation of variables and solving the boundary value problem.

When the initial condition of a heat conduction problem is provided, the Fourier series enables us to develop a representation of that initial condition using a base set of sinusoidal functions that respect the boundary conditions. These sine (and sometimes cosine) functions are orthogonal to each other over the problem's domain, ensuring that each term in the series uniquely corresponds to a specific 'frequency' of the solution.

In the case of the rod, the initial temperature distribution is given by a simple function \(u(x, 0) = x\). By using the orthogonality of sine functions, the Fourier coefficients \(A_n\) are determined through integration by parts. These coefficients modulate each term in the Fourier series that make up the solution for the initial temperature distribution. As time progresses, the exponential term in each Fourier series component dictates how each mode decays, reflecting how the temperature changes with time due to heat conduction.