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 \(k=1.\) \(u(x, t)=A e^{-a^{2} t} \cos a x,\) for any real numbers \(a\) and \(A\)

Short answer

Answer: Yes, the function \(u(x, t)=A e^{-a^{2} t} \cos a x\) satisfies the heat equation with \(k=1\) for any real numbers \(a\) and \(A\).

Step-by-step solution

Find the partial derivative of u with respect to t

We are given the function: \(u(x, t)=A e^{-a^{2} t} \cos a x.\) Now we will find the partial derivative of \(u\) with respect to \(t:\) $$\frac{\partial u}{\partial t}=A \frac{\partial}{\partial t} \left( e^{-a^{2} t} \cos a x \right).$$ Now we use the product rule, treating \(x\) as a constant: $$\frac{\partial u}{\partial t}=A \left( \frac{\partial e^{-a^{2} t}}{\partial t} \cos a x + e^{-a^{2} t} \frac{\partial (\cos a x)}{\partial t} \right).$$ Since \(\cos a x\) doesn't depend on \(t,\) its derivative with respect to \(t\) is 0. Also, the derivative of \(e^{-a^{2} t}\) with respect to \(t\) is \(-a^{2} e^{-a^{2} t}.\) Therefore: $$\frac{\partial u}{\partial t}=-a^{2}A e^{-a^{2} t} \cos a x.$$

Find the second partial derivative of u with respect to x

Now, we will find the second partial derivative of \(u\) with respect to \(x:\) $$\frac{\partial^{2} u}{\partial x^{2}}=A \frac{\partial^{2}}{\partial x^{2}}\left( e^{-a^{2} t} \cos a x \right).$$ Since \(e^{-a^{2} t}\) doesn't depend on \(x,\) it can be treated as a constant. Therefore: $$\frac{\partial^{2} u}{\partial x^{2}}=A e^{-a^{2} t} \frac{\partial^{2}}{\partial x^{2}} (\cos a x).$$ The first derivative of \(\cos a x\) with respect to \(x\) is \(-a \sin a x.\) Now we find the second derivative: $$\frac{\partial^{2} u}{\partial x^{2}}=A e^{-a^{2} t} \frac{\partial}{\partial x} (-a \sin a x)=A e^{-a^{2} t}(-a^{2} \cos a x).$$

Substitute the partial derivatives into the heat equation

We are given that \(k=1.\) Substitute the values of the partial derivatives found in steps 1 and 2 into the heat equation: $$\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}.$$ Substituting the values, we get: $$-a^{2}A e^{-a^{2}t} \cos a x = A e^{-a^{2} t}(-a^{2}\cos a x).$$

Verify if the equation holds true

We can see that both sides of the equation are equal. Therefore, the function \(u(x, t)=A e^{-a^{2} t} \cos a x\) does indeed satisfy the heat equation with \(k=1\) for any real numbers \(a\) and \(A.\)

Key concepts

Partial Derivatives

In calculus, a partial derivative represents the derivative of a function with respect to one of its variables, while keeping other variables constant. This concept is essential in multivariable calculus and plays a vital role in solving physical problems involving several changing factors, such as temperature over time or space.

• Partial derivatives are denoted by the symbol \( \frac{\partial}{\partial x} \) for a derivative with respect to \( x \).
• In our context, \( u(x, t) = A e^{-a^{2} t} \cos ax \) represents temperature \( u \) at a location \( x \) on a conducting bar over time \( t \).

To analyze temperature changes on the bar, we need to compute the partial derivatives with respect to \( x \) and \( t \). The first partial derivative with respect to \( t \) indicates how the temperature changes over time, while keeping \( x \) constant. Meanwhile, the second partial derivative with respect to \( x \) tells us how the temperature distribution changes along the bar's length, assuming time is constant.
Calculating these derivatives accurately is crucial because they allow us to verify whether the provided function satisfies the heat equation, revealing valuable insights into heat distribution and behavior.

Conducting Bar

A conducting bar is a physical object which can transmit or conduct heat along its length. Understanding how heat behaves on such a bar is critical in various applications—ranging from engineering to environmental science.

• The heat equation \( \frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2} \) models this heat flow along the bar.
• Here, \( u(x, t) \) signifies the temperature at point \( x \) over time \( t \).
• The constant \( k \) is the material's thermal conductivity, indicating how rapidly the material can conduct heat.

In practical terms, when considering a thin conducting bar, we assume it is thin enough for heat transfer to occur solely along its length (1D). This differs significantly from heat transfer in thicker objects or higher dimensions, involving more complex equations. The conducting bar scenario is often used as a simpler model to understand and study fundamental principles of thermal conductivity.

Boundary Conditions

Boundary conditions determine how the heat equation solutions behave at the edges or boundaries of the region. They are crucial in physical contexts as they affect the overall outcome and stability of the heat distribution solutions.

• Common types include Dirichlet conditions (fixed temperatures) and Neumann conditions (fixed heat flux).
• In this problem, boundaries are assumed ideal unless otherwise noted with specific conditions.
• Boundary conditions typically depend on the specific scenario or environment being modeled (e.g., insulated ends, contact with an external heat source).

Proper application of boundary conditions ensures that the mathematical model accurately mirrors real-world heat transfer phenomena. Without considering boundary conditions, solutions can become unrealistic or unusable in practice. They are therefore fundamental to both theoretical derivations and practical implementations.

Trigonometric Functions

Trigonometric functions play a key role in the solutions to differential equations, like the heat equation. These functions often elegantly describe spatial periodicities, such as waveforms or heat distributions in this context.

• The cosine function \( \cos ax \) appears in the solution as part of the temperature function \( u(x, t) \).
• It introduces a periodic component, reflecting potential oscillations or evenly distributed temperatures along the bar.
• Trigonometric identities and properties help simplify the calculation of derivatives and the overall understanding of how these functions behave over space.

By incorporating trigonometric functions, the model can accurately represent repeating patterns or fluctuations in temperature distribution along the bar, essential for precise scientific and engineering predictions.