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. $$ u(0, t)=T, \quad u_{x}(L, t)=0 $$

Short answer

Answer: The steady-state temperature distribution is constant throughout the region and is given by \(u(x) = T\).

Step-by-step solution

Rewrite the PDE as a BVP for the steady-state case

In the steady-state case, the temperature does not change with time, meaning \(u_t=0\). The heat conduction equation becomes: $$ \alpha^2 u_{xx}=0 $$

Solve the BVP using separation of variables

Suppose \(u(x)\) is the steady-state temperature distribution in the region. The steady-state BVP can be written as: $$ \frac{d^2 u(x)}{dx^2}=0 $$ Integrating twice with respect to x, we get: $$ u(x)=Ax+B $$ where A and B are constants.

Apply the boundary conditions

We are given the boundary conditions: \(u(0, t)=T\) and \(u_x(L, t)=0\). To find A and B, let's apply these conditions to the expression for the steady-state temperature distribution \(u(x)\). For the first boundary condition, \(u(0,t)=T\): $$ u(0) = A\cdot 0 + B = T \\ B = T $$ For the second boundary condition, \(u_x(L, t)=0\). First, find \(u_x(x)\): $$ u_x(x) = A $$ Now apply the second boundary condition: $$ u_x(L)=A=0 $$

Write down the steady-state solution

From Step 3, we found that \(A=0\) and \(B=T\). Therefore, the steady-state solution of the heat conduction equation \(\alpha^2 u_{xx}=u_t\) with given boundary conditions is: $$ u(x) = T $$

Key concepts

Heat Conduction Equation

The heat conduction equation is a fundamental partial differential equation (PDE) that helps us understand how heat disperses throughout a given region over time. It mathematically expresses how temperature changes as heat moves through a material. The equation often looks like this: \( \alpha^2 u_{xx} = u_t \), where \( u(x,t) \) represents the temperature at a position \( x \) and time \( t \). The constant \( \alpha^2 \) is related to the material's thermal diffusivity, which is essentially how rapidly heat can spread through the material.

In the steady-state scenario, as in the provided exercise, the term \( u_t \) becomes zero. This implies that the temperature isn’t changing with time anymore; thus, heating rates are equal to cooling rates, reaching an equilibrium. This simplification reduces the PDE to \( \alpha^2 u_{xx} = 0 \), meaning the second spatial derivative of temperature must equal zero. Solving this altered form of the equation helps us find the temperature distribution across the material once equilibrium is reached.

Boundary Conditions

Boundary conditions are crucial for solving differential equations like the heat conduction equation. They provide additional information needed to find a unique solution to the mathematical problem. These are like constraints or rules that our solution must adhere to; they describe what happens at the boundaries or edges of the region we're examining.

In the exercise, two boundary conditions are given:

• \( u(0, t) = T \): This implies that at position \( x = 0 \), the temperature is kept constant at \( T \).
• \( u_x(L, t) = 0 \): This tells us the rate of change of temperature is zero at \( x = L \), meaning the temperature gradient is flat, indicating an insulating boundary.

These conditions help determine the constants in the general solution to the PDE that we derived while assuming the steady-state scenario. By applying these specific rules to our more general mathematical solution, we can pinpoint the exact solution fitting those criteria and describe the equilibrium temperature distribution in the region.

Separation of Variables

Separation of variables is a common mathematical technique utilized to solve linear partial differential equations such as the heat conduction equation. The main idea is to assume that the solution can be broken down into simpler, separable parts, ideally making the problem easier to solve.

For steady-state problems, separation of variables involves expressing the temperature as a function of one variable at a time, often written as \( u(x,t) = X(x) \cdot T(t) \). However, since the exercise is a steady-state case where the solution doesn't change with time, we're simply concerned with the spatial component, \( u(x) \).

By isolating \( u(x) \) and integrating the simplified steady-state equation \( \frac{d^2u(x)}{dx^2} = 0 \), we derive the general solution \( u(x) = Ax + B \). The constants \( A \) and \( B \) are determined by applying the boundary conditions. This step of establishing a format using separation of variables is pivotal, allowing us to transform the original complex differential equation into more manageable parts, making it possible to obtain the unique steady-state solution.