Problem 19

Question

Consider the boundary-value problem $$ \begin{aligned} &k \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial u}{\partial t}, 00 \\ &u(0, t)=u_{0}, u(L, t)=u_{1} \\ &u(x, 0)=f(x) \end{aligned} $$ that is a model for the temperature $u$ in a rod of length $L$. If $u_{0}$ and $u_{1}$ are different nonzero constants, what would you intuitively expect the temperature to be at the center of the rod after a very long period of time? Prove your assertion.

Step-by-Step Solution

Verified

Answer

At the center of the rod, the temperature approaches $(u_0 + u_1)/2$ after a long time.

1Step 1: Understand the boundary-value problem

The given problem is a heat equation representing the temperature distribution $u(x,t)$ along a rod with length $L$. The thermal conductivity is represented by $k$. The boundary conditions $u(0,t)=u_0$ and $u(L,t)=u_1$ indicate constant temperatures at both ends of the rod. The initial condition $u(x,0)=f(x)$ specifies the initial temperature distribution along the rod.

2Step 2: Consider the long-term behavior

After a very long period of time, the system typically reaches a steady state where the temperature no longer changes with time. This means $\frac{\partial u}{\partial t}=0$, reducing the PDE to $k \frac{\partial^{2} u}{\partial x^{2}}=0$. This implies $\frac{\partial^{2} u}{\partial x^{2}}=0$, leading to a linear temperature distribution along the rod.

3Step 3: Solve for the steady-state solution

The equation $\frac{\partial^{2} u}{\partial x^{2}}=0$ simplifies to $u(x)=Ax+B$, where $A$ and $B$ are constants determined by the boundary conditions. Applying $u(0)=u_0$, we have $B=u_0$. Applying $u(L)=u_1$, we solve $AL+u_0=u_1$ to get $A=(u_1-u_0)/L$.

4Step 4: Find the temperature at the center of the rod

The center of the rod is at $x=L/2$. Substituting into the linear equation from Step 3: $u(L/2) = A(L/2) + B = ((u_1-u_0)/L)(L/2) + u_0$. Simplifying, this gives $u(L/2) = (u_1+u_0)/2$.

5Step 5: Conclude with the intuitive expectation

Given that $u_0$ and $u_1$ are constants denoting the temperatures at the ends, over time, the rod's temperature distribution becomes linear. Therefore, intuitively, the temperature at the center $x=L/2$ approaches the average of the two boundary temperatures $(u_0+u_1)/2$ in the steady state.

Key Concepts

Boundary Value ProblemSteady-State SolutionTemperature Distribution

Boundary Value Problem

The Boundary Value Problem (BVP) in the context of the heat equation focuses on understanding how the temperature varies along a rod. The conditions at the boundaries, or ends of the rod, influence this distribution.
In our problem, the rod has length $L$ and the heat equation is given by $k \frac{\partial^{2} u}{\partial x^{2}} = \frac{\partial u}{\partial t}$. Here, $u(x,t)$ is the temperature at point $x$ on the rod at time $t$.

The boundary conditions are critical:

$u(0, t) = u_0$: This indicates a constant temperature $u_0$ at $x=0$.
$u(L, t) = u_1$: This specifies a constant temperature $u_1$ at $x=L$.

These conditions determine how heat flows and distributes along the rod over time. Additionally, the initial condition $u(x, 0) = f(x)$ tells us the initial temperature profile of the rod.
Solving the BVP involves finding a temperature function that satisfies the heat equation and these boundary conditions.

Steady-State Solution

A Steady-State Solution occurs when a system, such as our rod, reaches a condition where temperature no longer changes with time.
In mathematical terms, this means $\frac{\partial u}{\partial t} = 0$.

In the heat equation, this results in a simplified second-order differential equation $k \frac{\partial^{2} u}{\partial x^{2}} = 0$. Solving this differential equation gives $u(x) = Ax + B$, a linear relationship, where $A$ and $B$ are constants.

These constants are determined using the boundary conditions:

At $x = 0$, $u(0) = u_0$ gives $B = u_0$.
At $x = L$, $u(L) = u_1$ leads to $AL + u_0 = u_1$, which can be rearranged to $A = \frac{u_1 - u_0}{L}$.

This linear solution reflects the state of the rod after an extended period, showing how its temperature evens out from the initial state.

Temperature Distribution

Temperature Distribution describes how temperature values are spread across the rod in the situation given.
Initially, the temperature distribution is defined by $u(x, 0) = f(x)$. As time goes on, the rod's temperature changes due to thermal conductivity and the imposed boundary conditions.

Once the rod reaches a steady state, the temperature distribution becomes linear. This is mathematically defined by $u(x) = Ax + B$, where $A$ and $B$ have been calculated using the boundary conditions.
At any point $x$ along the rod, the temperature $u(x)$ is found by substituting $x$ into our linear equation.

For instance, the temperature at the center, $x = L/2$, is $u(L/2) = \frac{u_1 + u_0}{2}$.

This value is the average of the boundary temperatures, as expected. This distribution reflects the balance achieved in the system over time, where heat spreads uniformly due to the boundaries' influence.

Problem 18

Problem 19

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