Problem 23 - Advanced Engineering Mathematics

Question

A rod of unit length is insulated at \(x=0\) and is kept at temperature zero at \(x=1\). If the initial temperature of the rod is a constant \(u_{0}\), solve \(k u_{x x}=u_{t}, 00\) for the temperature \(u(x, t) .\left[\right.\) Hint \(:\) Expand \(1 /\left(1+e^{-2 \sqrt{s / k}}\right)\) in a geometric series.]

Step-by-Step Solution

Answer

The temperature is given by a Fourier series: \( u(x, t) = \sum \frac{2u_0}{(2n+1)\pi}\cos\left(\frac{(2n+1)\pi}{2} x\right)e^{-k\frac{(2n+1)^2\pi^2}{4}t} \).

Step 1: Understand the Given Problem

We are given a partial differential equation (PDE) for a rod of unit length which is partially insulated and partially maintained at zero temperature. The PDE, given by \( k u_{xx} = u_t \), represents the heat equation, where \( u(x, t) \) is the temperature at position \( x \) and time \( t \). We have boundary conditions: insulated at \( x = 0 \) and zero temperature at \( x = 1 \), and an initial condition where the temperature along the rod is initially a constant \( u_0 \). We need to find \( u(x, t) \).

Step 2: Set Up Boundary and Initial Conditions

The given boundary conditions are \( u_x(0, t) = 0 \) (insulated end) and \( u(1, t) = 0 \) (temperature at zero). The initial condition is \( u(x, 0) = u_0 \).

Step 3: Solve Using Separation of Variables

Assume \( u(x, t) = X(x)T(t) \). Substitute into the PDE to separate variables, yielding \( kX''(x)T(t) = X(x)T'(t) \). This simplifies to two ordinary differential equations: \( X'' = -\lambda X \) and \( T' = -k\lambda T \) for some separation constant \( \lambda \).

Step 4: Solve the Spatial Equation

\( X''(x) + \lambda X(x) = 0 \) with boundary conditions \( X'(0) = 0 \) and \( X(1) = 0 \). The general solution is \( X(x) = A\cos(\sqrt{\lambda}x) + B\sin(\sqrt{\lambda}x) \). The conditions imply \( B=0 \) and \( \cos(\sqrt{\lambda}) = 0 \), giving \( \lambda = (2n+1)^2 \pi^2 / 4 \).

Step 5: Solve the Temporal Equation

For the temporal equation \( T'(t) = -k\lambda T(t) \), the solution is \( T(t) = Ce^{-k\lambda t} \).

Step 6: Construct the General Solution

Combine results: \( u(x, t) = (A_n\cos(\frac{(2n+1)\pi}{2}x) \exp(-k\frac{(2n+1)^2 \pi^2}{4}t)) \) summed over \( n \).

Step 7: Apply Initial Condition to Determine Coefficients

The initial condition \( u(x, 0) = u_0 \) implies a Fourier cosine series. Hence, \( A_n = \frac{2u_0}{(2n+1)\pi} \) if calculated by integrating over \( x \).

Step 8: Final Solution

The final solution is \[ u(x, t) = \sum_{n=0}^{\infty} \frac{2u_0}{(2n+1)\pi}\cos\left( \frac{(2n+1)\pi}{2} x \right)e^{-k\frac{(2n+1)^2 \pi^2}{4}t} \]This series solution takes into account the boundary and initial conditions, providing the temperature distribution at any position \( x \) and time \( t \).

Key Concepts

Partial Differential Equations

Partial Differential Equations (PDEs) are mathematical equations that involve the rates of change with respect to more than one variable. They are essential in describing various physical phenomena such as heat conduction, sound, fluid dynamics, and more.
PDEs come in multiple forms: the Laplace Equation, Wave Equation, and in this case, the Heat Equation. The Heat Equation, which describes how heat diffuses through a given region over time, is typically represented as \( k u_{xx} = u_t \).

• The left side of the equation, \( k u_{xx} \), accounts for spatial variations in temperature, where \( k \) is the thermal diffusivity constant.
• The right side, \( u_t \), represents the change of temperature with respect to time.
  

PDEs like the Heat Equation require methods such as Separation of Variables to find solutions that adhere to given conditions.

Separation of Variables

Separation of Variables is a common method to solve partial differential equations. It simplifies complex equations by breaking them into simpler, ordinary differential equations (ODEs), which can be more easily solved.
In our exercise, the assumption that \( u(x, t) = X(x)T(t) \) allows us to separate the spatial and temporal components. This substitution transforms the original PDE \( kX''(x)T(t) = X(x)T'(t) \) into two separate equations:

• \( X'' = -\lambda X \) deals with the spatial dimension, and that ultimately gives a form involving cosine and sine functions.
• \( T' = -k\lambda T \) tackles the temporal part, resulting in an exponential decay function dependent on lambda, which we extract from the solution of the spatial equation.
  

This method works because it reduces a complex PDE into simpler equations, allowing for solutions that satisfy various conditions imposed by the problem, as long as the equation is linear and homogeneous.

Boundary Conditions

Boundary Conditions are key constraints necessary for finding unique solutions to PDEs. They specify the behavior of a solution at the boundaries of the domain.
For the heat equation problem, the rod is insulated at one endpoint and kept at zero temperature at the other:

• At \( x = 0 \), the rod being insulated implies \( u_x(0, t) = 0 \). This means no heat flows through the insulated end.
• At \( x = 1 \), the zero temperature condition is \( u(1, t) = 0 \). This forces the temperature at that end to always be zero.
  

The initial condition \( u(x, 0) = u_0 \) specifies that the entire rod starts at a uniform temperature. These conditions guide the choice and form of the solution functions.

Fourier Series

Fourier Series allows us to express a periodic function as a sum of sines and cosines. They are useful in solving PDEs where initial or boundary conditions naturally lead to periodic solutions.
In the problem, the initial condition leads to a Fourier cosine series solution for the temperature distribution:

• The series involves terms like \( \cos\left( \frac{(2n+1)\pi}{2} x \right) \), which result from satisfying the boundary conditions.
• The coefficients of this series, \( A_n = \frac{2u_0}{(2n+1)\pi} \), were derived from the initial condition using integral properties.
  

Fourier Series is powerful because it allows for constructing solutions that will meet both the PDE and all initial/boundary conditions by choosing appropriate sine and cosine terms.