Problem 80 - Calculus Early Transcendentals

Question

The flow of heat along a thin conducting bar is governed by the one- dimensional heat equation (with analogs for thin plates in two dimensions and for solids in three dimensions) $$\frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}},$$ where \(u\) is a measure of the temperature at a location \(x\) on the bar at time t and the positive constant \(k\) is related to the conductivity of the material. Show that the following functions satisfy the heat equation with \(k=1.\) $$u(x, t)=10 e^{-t} \sin x$$

Step-by-Step Solution

Answer

Answer: Yes, the function \(u(x,t) = 10 e^{-t} \sin x\) does satisfy the heat equation with \(k=1\).

Step 1: Find the partial derivative of u with respect to t

We need to find \(\frac{\partial u}{\partial t}\). Since \(u(x, t) = 10 e^{-t} \sin x\), we partially differentiate with respect to \(t\) as follows: $$\frac{\partial u}{\partial t} = -10 e^{-t} \sin x$$

Step 2: Find the second partial derivative of u with respect to x

We need to find \(\frac{\partial^2 u}{\partial x^2}\). First, let's find the first partial derivative with respect to x: $$\frac{\partial u}{\partial x} = 10 e^{-t} \left(\cos x\right)$$ Now, let's differentiate the expression above with respect to x once more: $$\frac{\partial^2 u}{\partial x^2} = -10 e^{-t} \left(\sin x\right)$$

Step 3: Substitute derivatives into the heat equation

We have the heat equation given as \(\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}\). In this case \(k=1\). Now, let's substitute \(\frac{\partial u}{\partial t}\) and \(\frac{\partial^2 u}{\partial x^2}\) into the equation: $$-10 e^{-t} \sin x = \left(1\right) \left(-10 e^{-t} \sin x\right)$$

Step 4: Check if the equation is satisfied

Looking at the equation, we can see that both sides are equal, which means the given function \(u(x, t) = 10 e^{-t} \sin x\) does satisfy the heat equation with k = 1.

Key Concepts

Partial Derivatives

In mathematics, partial derivatives are a fundamental concept in the study of multivariable functions. They allow us to understand how a function changes as one of its variables changes, while keeping the others constant. This is particularly important in equations like the heat equation where multiple variables, such as time \(t\) and position \(x\), are involved.
A partial derivative with respect to one variable is represented by differentiating the function while treating other variables as constants. In the heat equation \(\frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}}\), we focus both on \(\frac{\partial u}{\partial t}\) which represents the rate of change of temperature with respect to time, and \(\frac{\partial^2 u}{\partial x^2}\) which is the rate of change of the rate of change of temperature along the spatial dimension. This second derivative signifies how the temperature curve bends along the bar.

• The notation \(\frac{\partial}{\partial t}\) indicates the rate of change with time.
• The notation \(\frac{\partial^2}{\partial x^2}\) indicates the curvature in the temperature profile.

Partial derivatives provide a powerful tool to analyze changes occurring in different directions and are critical in many areas of physics and engineering.

Conductivity

Conductivity is a measure of a material's ability to conduct heat. It plays a crucial role in the heat equation, denoted by the constant \(k\). This parameter defines how quickly heat is transferred through a material, dictating how efficiently a material can bring a system to thermal equilibrium.
Higher conductivity means a material can conduct heat more swiftly, while lower conductivity implies slower heat conduction. This affects how temperature distributions evolve over time, especially in dynamic systems like the heat equation. In the given problem, \(k=1\), which simplifies calculations but still maintains the theoretical understanding of how heat flows.

• Materials with high conductivity examples include metals like copper and aluminum.
• Insulating materials, like rubber or wood, have low conductivity.

Understanding conductivity is essential to designing thermal management systems in electronics, buildings, and even understanding geological or atmospheric processes.

Temperature Distribution

Temperature distribution describes how temperature varies within a material or space over time. In the context of the heat equation, temperature distribution along a bar is determined by the function \(u(x, t)\), where \(x\) and \(t\) indicate position and time, respectively.
In the example provided, the solution \(u(x, t) = 10 e^{-t} \sin x\) epitomizes the temperature distribution that changes over time. As time progresses, the influence of the \(e^{-t}\) term ensures the temperature diminishes exponentially, simulating the physical phenomenon of heat dissipating over time.
The \(\sin x\) term offers a cyclical spatial component reflecting how temperature might alternate at different points along the bar. This can help us analyze thermal waves or oscillations in the distribution, providing a robust model for thermal analysis.
Temperature distribution analysis helps:

• Understand how heat spreads in objects.
• Predict temperature in engineering and natural systems.
• Design efficient thermal management systems.

Thus, evaluating temperature distribution is key in many scientific and engineering applications.