Problem 4 - Advanced Engineering Mathematics

Question

A rod of length \(L\) coincides with the interval \([0, L]\) on the \(x\) -axis. Set up the boundary-value problem for the temperature \(u(x, t)\) There is heat transfer from the left end into a surrounding medium at temperature \(20^{\circ}\), and the right end is insulated. The initial temperature is \(f(x)\) throughout.

Step-by-Step Solution

Answer

The boundary-value problem involves the heat equation with specific boundary conditions: \(u(0, t) = 20\) and \(\frac{\partial u}{\partial x}(L, t) = 0\), and initial condition \(u(x, 0) = f(x)\).

Step 1: Understanding the Problem

We need to set up a boundary-value problem for the heat conduction in a rod of length \(L\) coinciding with the interval \([0, L]\) on the \(x\)-axis. The rod has specific boundary conditions and an initial condition.

Step 2: Constructing the Heat Equation

The temperature \(u(x, t)\) in the rod must satisfy the heat equation, which is given by \(\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}\), where \(\alpha\) is the thermal diffusivity of the rod.

Step 3: Applying the Boundary Conditions

For the left end at \(x=0\), there is heat transfer to a medium at a constant temperature of \(20^{\circ}\), which implies a boundary condition \(u(0, t) = 20\). The right end at \(x=L\) is insulated, meaning there is no heat flow, leading to the boundary condition \(\frac{\partial u}{\partial x}(L, t) = 0\).

Step 4: Stating the Initial Condition

The initial temperature throughout the rod is given by \(u(x, 0) = f(x)\), where \(f(x)\) represents the initial temperature distribution.

Step 5: Assembling the Boundary-Value Problem

The boundary-value problem is comprised of the heat equation: \(\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}\), with boundary conditions \(u(0, t) = 20\), \(\frac{\partial u}{\partial x}(L, t) = 0\), and initial condition \(u(x, 0) = f(x)\).

Key Concepts

Heat Equation

The heat equation is a fundamental partial differential equation in mathematics that deals with the distribution of heat in a given region over time. It is used to model the process of heat conduction in materials. In the context of the boundary value problem involving a rod, the heat equation is given by:

1. \(\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}\)

This equation describes how the temperature \(u(x, t)\), or the heat content at a point \(x\) at time \(t\), changes. The term \(\alpha\) represents the thermal diffusivity, which we will discuss shortly.

• \(\frac{\partial u}{\partial t}\) shows how the temperature changes with time.
• \(\frac{\partial^2 u}{\partial x^2}\) represents how the temperature changes in space along the rod.

Thus, the heat equation captures the essence of how heat spreads through the rod over time.

Boundary Conditions

To solve the heat equation, we need to establish boundary conditions that specify the behavior of the temperature at the boundaries of the rod. Boundary conditions are essential as they determine how the system interacts with its surroundings. For the rod in this problem:

• At \(x = 0\), where the left end of the rod is situated, it interacts with a medium at a constant temperature of \(20^{\circ}\). Therefore, we set the boundary condition as: \(u(0, t) = 20\).
• At \(x = L\), the right end is insulated. Insulation means no heat flows through this end, resulting in a condition called the Neumann boundary condition: \(\frac{\partial u}{\partial x}(L, t) = 0\).

These conditions are directly applied in mathematical calculations to ensure that the model accurately reflects the physical constraints of the system.

Thermal Diffusivity

Thermal diffusivity, represented by \(\alpha\), is a material-specific parameter that plays a crucial role in the heat equation. It defines how quickly heat spreads through a material.
The formula for thermal diffusivity is:

• \(\alpha = \frac{k}{\rho c}\)

where

• \(k\) is the thermal conductivity, showing how well a material conducts heat.
• \(\rho\) is the density of the material.
• \(c\) is the specific heat capacity, indicating the amount of heat per unit mass required to raise the temperature by one degree Celsius.

Thus, a material with high thermal diffusivity will disperse heat quickly, affecting how the temperature changes within the rod over time.

Initial Condition

To fully define the boundary value problem for the rod, we must also establish the initial condition, which describes the state of the system at time \(t = 0\). The initial condition provides a starting point for studying how the temperature evolves over time.
In this context, the initial temperature distribution across the rod is given by:

• \(u(x, 0) = f(x)\)

where \(f(x)\) is a function describing the temperature at every point \(x\) at the beginning of the observation. Initial conditions are crucial because they affect future states of the system and determine the trajectory of thermal change along the rod. Understanding the initial state is the foundation upon which the entire dynamic process of heat conduction is built.