Question

Write \(u(x, t)=v(x)+w(x, t),\) where \(v\) and \(w\) are the steady-state and transient parts of the solution, respectively. State the boundary value problems that \(v(x)\) and \(w(x, t),\) respectively, satisfy. Observe that the problem for \(w\) is the fundamental heat conduction problem discussed in Section \(10.5,\) with a modified initial temperature distribution.

Short answer

Based on the given step-by-step solution, identify the boundary value problems (BVPs) that the steady-state part, \(v(x)\), and transient part, \(w(x, t)\), of the solution satisfy. The BVP for the steady-state part of the solution, \(v(x)\), is given by \(\frac{\partial^2 v}{\partial x^2} = 0\). The BVP for the transient part of the solution, \(w(x, t)\), is given by \(\frac{\partial w}{\partial t} = \alpha \frac{\partial^2 w}{\partial x^2}\) along with appropriate initial and boundary conditions as per the modified initial temperature distribution.

Step-by-step solution

Identify the given equation and determine the BVPs

We're given the equation \(u(x, t) = v(x) + w(x, t)\). The BVPs for \(v(x)\) will be derived from the steady-state PDE, and the BVPs for \(w(x, t)\) will be derived from the transient PDE. Step 2: Find the steady-state part of the solution

Derive the BVP for steady-state part, \(v(x)\)

The steady-state part, \(v(x)\), is obtained when the temperature has reached a steady equilibrium, which means that the rate of change of temperature with time is zero, i.e., \(\frac{\partial u}{\partial t} = 0\). Since \(u(x, t) = v(x) + w(x, t)\), the equation for \(v(x)\) is given by \(\frac{\partial v}{\partial x} + \frac{\partial w}{\partial x} = 0\). Since \(v\) does not depend on time, its BVP is determined by the spatial part of the equation. The BVP for \(v(x)\) is now given by: \(\frac{\partial^2 v}{\partial x^2} = 0\). Step 3: Find the transient part of the solution

Derive the BVP for transient part, \(w(x, t)\)

Since we know that the problem for \(w\) is the fundamental heat conduction problem, the BVPs for \(w\) are derived from the transient PDE. We can subtract the equation for \(v(x)\) from the main equation \(u(x, t)\) to get the equation for \(w(x, t)\): \(u(x, t) - v(x) = w(x, t)\). We also know that \(\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}\), so taking the time derivative of the equation above, we get \(\frac{\partial w}{\partial t} = \alpha \frac{\partial^2 w}{\partial x^2}\) since \(v(x)\) has no time component. Step 4: State the boundary value problems

Summarize the BVPs for \(v(x)\) and \(w(x, t)\)

The BVP for the steady-state part of the solution, \(v(x)\), is given by \(\frac{\partial^2 v}{\partial x^2} = 0\). The BVP for the transient part of the solution, \(w(x, t)\), is given by \(\frac{\partial w}{\partial t} = \alpha \frac{\partial^2 w}{\partial x^2}\) along with appropriate initial and boundary conditions as per the modified initial temperature distribution.

Key concepts

Steady-state solution

In the context of boundary value problems, a steady-state solution refers to a condition where the system variables no longer change over time. For a heat conduction problem, this implies that the temperature in a material reaches a uniform state across the object and remains constant. In mathematical terms, this is when the rate of change with respect to time is zero.

To identify the steady-state solution in our exercise, we consider only the spatial aspects of the equation, since time dependency is removed. This is mathematically represented by the equation \( \frac{\partial^2 v}{\partial x^2} = 0 \). Here, \(v(x)\) represents the temperature distribution in a stable state. Essentially, it means that any spatial changes in temperature are linear, such as from one end of a rod to the other.

This steady-state part, \(v(x)\), is a crucial element in understanding how systems react at equilibrium, and it provides clarity on the behavior of a temperature distribution once external influences have stabilized.

Transient solution

While the steady-state solution tells us about equilibrium, the transient solution explains how the system evolves over time to reach that equilibrium. In a heat conduction problem, transient solutions describe the time-dependent changes in temperature as the system progresses from an initial state to the steady state.

The transient solution \(w(x, t)\) is governed by the equation \( \frac{\partial w}{\partial t} = \alpha \frac{\partial^2 w}{\partial x^2} \). Here, \(\alpha\) is the thermal diffusivity of the material, a property that describes how quickly heat spreads through it. This equation models how heat flows and distributes itself over time, influenced by the initial temperature distribution and possibly boundary conditions.

• Time derivative \( \frac{\partial w}{\partial t} \) signifies the rate of change of temperature over time.
• Spatial derivative \( \frac{\partial^2 w}{\partial x^2} \) describes how temperature gradients drive the flow of heat.

This dynamic aspect of heat conduction is essential for understanding how systems respond over time, influenced by material properties and initial conditions.

Heat conduction problem

A heat conduction problem involves determining how heat flows through a material and how temperatures distribute themselves over time. These problems are often resolved by setting up partial differential equations (PDEs), like the ones seen in solving the boundary value problems for steady-state and transient solutions.

In our exercise, the analysis begins with the equation \(u(x, t) = v(x) + w(x, t)\), depicting overall temperature as a combination of steady-state and transient components. This separation is powerful because it isolates the fixed distribution from the time-dependent behavior of the system.

Such problems usually incorporate:

• Initial conditions detailing the temperature distribution at the beginning.
• Boundary conditions specifying how the environment interacts with the boundaries of the material.

In practice, solving a heat conduction problem allows for predicting temperature changes and planning for heat management, which is crucial in engineering, materials science, and climate studies.

With a clear understanding of both steady-state and transient solutions, one can effectively model thermal behavior within an object and leverage this understanding to optimize thermal conditions in real-world applications.