Problem 5 - Calculus for the Life Sciences: A Modeling Approach Volume II

Question

a. Show that $$ u(x, t)=20 e^{-t} \sin \pi x, \quad 0 \leq x \leq 1, \quad 0 \leq t $$ solves $$ u_{t}(x, t)=\frac{1}{\pi^{2}} u_{x x}(x, t), \quad u(x, 0)=20 \sin \pi x, \quad \text { and } \quad u(0, t)=u(1, t)=0 $$ b. Describe a physical problem for which this is a solution. c. What is the 'eventual' value of \(u(x, t)\) (what is \(\left.\lim _{t \rightarrow \infty} u(x, t)\right) ?\) d. At what time, \(t,\) will the maximum value of \(u(x, t)\) for \(0 \leq x \leq 1\) be \(20 ?\)

Step-by-Step Solution

Answer

a: Yes. b: Heat conduction in a rod. c: 0. d: \(t = 0\).

Step 1: Verify the initial condition

To verify the initial condition, substitute \(t=0\) into the given function \(u(x, t)\):\[ u(x, 0) = 20 e^{0} \sin \pi x = 20 \sin \pi x. \]This matches the initial condition \(u(x, 0)=20 \sin \pi x\). Thus, the initial condition is satisfied.

Step 2: Verify the boundary conditions

For the boundary conditions, evaluate \(u(x, t)\) at \(x=0\) and \(x=1\):- At \(x=0\), \[ u(0, t) = 20 e^{-t} \sin (0) = 0. \] - At \(x=1\), \[ u(1, t) = 20 e^{-t} \sin (\pi \times 1) = 0. \] Thus, the boundary conditions \(u(0, t)=0\) and \(u(1, t)=0\) are satisfied.

Step 3: Compute \(u_t(x, t)\)

Differentiate \(u(x, t) = 20 e^{-t} \sin \pi x\) with respect to \(t\):\[ u_t(x, t) = \frac{d}{dt} [20 e^{-t} \sin \pi x] = -20 e^{-t} \sin \pi x. \]

Step 4: Compute \(u_{xx}(x, t)\)

Differentiate \(u(x, t) = 20 e^{-t} \sin \pi x\) twice with respect to \(x\):- First derivative: \[ u_x(x, t) = 20 e^{-t} \pi \cos \pi x. \]- Second derivative: \[ u_{xx}(x, t) = -20 e^{-t} \pi^2 \sin \pi x. \]

Step 5: Verify the differential equation

Substitute \(u_t(x, t)\) and \(u_{xx}(x, t)\) into the differential equation \(u_t = \frac{1}{\pi^2}u_{xx}\):\[ -20 e^{-t} \sin \pi x = \frac{1}{\pi^2}(-20 e^{-t} \pi^2 \sin \pi x). \]This simplifies to \(-20 e^{-t} \sin \pi x = -20 e^{-t} \sin \pi x\), confirming that the differential equation is satisfied.

Step 6: Describe a physical problem

This solution corresponds to the heat equation, which models the temperature distribution along a one-dimensional rod of length 1 over time. The initial temperature distribution is given by \(20 \sin \pi x\), and the ends of the rod are kept at 0 temperature.

Step 7: Determine eventual value of \(u(x, t)\)

Compute the limit as \(t\) approaches infinity:\[ \lim_{t \rightarrow \infty} u(x, t) = \lim_{t \rightarrow \infty} 20 e^{-t} \sin \pi x = 0. \]As \(t\) gets large, the exponential term \(e^{-t}\) approaches 0, so \(u(x, t)\) tends to 0.

Step 8: Find time when maximum value is 20

The maximum value of \(u(x, t)\) occurs when \(\sin \pi x = 1\), which is at \(x = 0.5\). The given value to reach is 20, thus:\[ u(0.5, t) = 20 e^{-t} (1) = 20. \]Solving the equation \(20 e^{-t} = 20\) gives:\[ e^{-t} = 1, \quad \Rightarrow \quad t = 0. \]

Key Concepts

Initial Condition

The initial condition of a problem refers to the state of a system at the very beginning of the time period being considered. This is essential as it sets the stage for how the system will evolve over time.

For this problem, the initial condition is given by:

• \( u(x, 0) = 20 \sin \pi x \)

This indicates that at time \( t = 0 \), the temperature distribution along the rod is provided by a sine wave function. The sine function ensures that the temperature starts at zero at point \( x = 0 \), rises to a maximum at \( x = 0.5 \), and drops back to zero at \( x = 1 \).

This initial temperature distribution is fundamental as it impacts the trajectory of the temperature changes over time. Understanding this helps in predicting how the system will behave as it reaches equilibrium.

Boundary Conditions

Boundary conditions are critical in defining how the solution to a differential equation should behave at the edges of the domain. They ensure that the solution is physically viable and aligns with real-world constraints.

In our problem, the boundary conditions are:

• \( u(0, t) = 0 \)
• \( u(1, t) = 0 \)

These conditions specify that the temperature at both ends of the rod remains zero for all time \( t \), simulating a scenario where both ends are maintained at a constant, freezing temperature. This makes sense physically, as these fixed temperatures at the boundaries drive the heat transfer throughout the rod, aligning the problem with practical situations, such as cooling a rod by attaching it to ice trays at both ends.

Boundary conditions help shape the entire behavior of the temperature distribution, ensuring a realistic simulation of thermal dynamics.

Temperature Distribution

The temperature distribution describes how temperature varies along the rod over time. This is key to understanding heat flow in the system.

The given function:

• \( u(x, t) = 20 e^{-t} \sin \pi x \)

illustrates how the initial temperature distribution changes with time. The component \( 20 \sin \pi x \) defines the spatial temperature pattern at \( t = 0 \), and the term \( e^{-t} \) governs the exponential decay of temperature over time.

The exponential decay factor \( e^{-t} \) indicates that as time progresses, the temperature across the entire rod decreases until it eventually stabilizes at zero. This suggests that without an external input of heat, the temperature equalizes due to dissipation. Understanding the temperature distribution over time helps in predicting system behavior and ensures that one can derive conclusions about how quickly the system reaches thermal equilibrium.

Differential Equation

A differential equation is a mathematical expression that relates a function to its derivatives, describing how that function changes. In this problem, the differential equation is:

• \( u_{t}(x, t) = \frac{1}{\pi^{2}} u_{xx}(x, t) \)

This equation is a form of the heat equation, which is a partial differential equation that models the diffusion of heat in materials.

The term \( u_{t}(x, t) \) represents the rate of change of temperature with respect to time, while \( u_{xx}(x, t) \) reflects how temperature changes with respect to space. The right side of the equation, \( \frac{1}{\pi^{2}} u_{xx}(x, t) \), indicates that the rate of temperature change is directly proportional to its spatial curvature, meaning that steeper temperature gradients (higher curvatures) result in faster heat flow.

Solving this differential equation provides insights into how the rod's temperature evolves over time, crucial for applications involving thermal management and material design.