Question

find the steady-state solution of the heat conduction equation \(\alpha^{2} u_{x x}=u_{t}\) that satisfies the given set of boundary conditions. $$ u(0, t)=30, \quad u(40, t)=-20 $$

Short answer

Answer: The steady-state solution of the heat conduction equation with the given boundary conditions is: $$ u(x) = \frac{-20\alpha^2 - 30\alpha^2}{40}x + 30\alpha^2 $$

Step-by-step solution

Write down the steady-state heat conduction equation

Since the steady-state temperature distribution no longer changes with time, the time derivative becomes zero. Thus, the given heat conduction equation becomes: $$ \alpha^2 u_{xx} = 0 $$

Solve the equation subject to the boundary conditions

To solve the above equation, we'll integrate it twice with respect to \(x\): 1st integration: $$ \int \alpha^2 u_{xx} dx = \int 0 dx \\ \Longrightarrow \alpha^2 u_x = C_1 $$ 2nd integration: $$ \int \alpha^2 u_x dx = \int C_1 dx \\ \Longrightarrow \alpha^2 u(x) = C_1x + C_2 $$ Now, we need to impose the given boundary conditions: 1. \(u(0, t) = 30\): This implies that \(u(0) = 30\). Plugging this into the equation, we get: $$ \alpha^2(30) = C_1(0) + C_2 \\ \Longrightarrow C_2 = 30\alpha^2 $$ 2. \(u(40, t) = -20\): This implies that \(u(40) = -20\). Plugging this into the equation, we get: $$ \alpha^2(-20) = C_1(40) + C_2 \\ \Longrightarrow C_1 = \frac{-20\alpha^2 - 30\alpha^2}{40} $$ Thus, the steady-state solution of the heat conduction equation with the given boundary conditions is: $$ u(x) = \frac{-20\alpha^2 - 30\alpha^2}{40}x + 30\alpha^2 $$

Key concepts

Heat Conduction Equation

The heat conduction equation, also known as the heat diffusion equation, is a fundamental concept in physics and engineering. It describes how heat flows within a given region over time. Generally, the equation is expressed as:

• \( \alpha^{2} u_{xx} = u_t \)

where:

• \( \alpha \) is the thermal diffusivity, a constant that indicates how fast heat diffuses through the material.
• \( u(x,t) \) represents the temperature distribution as a function of space \( x \) and time \( t \).
• \( u_{xx} \) denotes the second spatial derivative, indicating how the temperature changes with respect to location \( x \).
• \( u_t \) is the time derivative, representing how temperature changes over time.

In a steady-state solution, the temperature no longer changes with time, simplifying the equation to \( \alpha^{2} u_{xx} = 0 \). This equation forms the basis for finding the temperature distribution, given certain conditions.

Boundary Conditions

Boundary conditions are crucial in solving differential equations like the heat conduction equation. They specify the behavior of the solution at the boundaries of the region where the equation is defined. In our problem, the given boundary conditions are:

• \( u(0, t) = 30 \)
• \( u(40, t) = -20 \)

Essentially, boundary conditions can determine the temperature at the edges of the material. For instance:

• \( u(0, t) = 30 \) means that the temperature at position \( x = 0 \) is always 30, regardless of time.
• \( u(40, t) = -20 \) means that the temperature at \( x = 40 \) is consistently -20.

These conditions allow us to solve for unknown constants in our integration process later, helping us construct an accurate temperature distribution model.

Integration

Integration is a mathematical process used to solve differential equations, such as those arising in the heat conduction problem. Once the steady-state heat conduction equation is simplified to zero, it transforms to:

• \( \alpha^{2} u_{xx} = 0 \)

This means the second derivative of \( u \) with respect to \( x \) equals zero, implying the first derivative \( u_x \) must be a constant. This is calculated by integrating the equation:

• First integration: \( \alpha^2 u_x = C_1 \)

Carrying out a second integration with respect to \( x \), we find:

• Second integration: \( \alpha^2 u(x) = C_1 x + C_2 \)

Here, \( C_1 \) and \( C_2 \) are constants determined by the boundary conditions. Integration enables us to find these constants and subsequently provides the solution for the temperature distribution.

Temperature Distribution

The temperature distribution is the final solution of the heat conduction equation. It tells us how the temperature varies across a material at a steady state. Using the equation obtained from integration:

• \( \alpha^2 u(x) = C_1 x + C_2 \)

The boundary conditions are applied to solve for \( C_1 \) and \( C_2 \):

• At \( x = 0 \): \( C_2 = 30 \alpha^2 \)
• At \( x = 40 \): \( C_1 = \frac{-20 \alpha^2 - 30 \alpha^2}{40} \)

Substituting these values back into our equation gives the temperature distribution:

• \( u(x) = \frac{-20 \alpha^2 - 30 \alpha^2}{40}x + 30\alpha^2 \)

This equation describes how temperature varies within the material from one boundary to the next, guided by the established boundary conditions.