Question

In each exercise, use the separation of variables representation developed in Exercise 5 to solve the insulated boundary problem for the given initial temperature distribution. (In Exercises 6-8, the coefficients can be determined without evaluating any integrals.) Compute \(\lim _{t \rightarrow \infty} u(x, y, t)\), and compare your answer to \(\int_{0}^{a} \int_{0}^{b} f(x, y) d y d x\). Can you relate your comparison to the principle of conservation of energy? How should the answer to Exercise 10 relate to the answer in Exercise 9 ? $$f(x, y)=2+\cos \left(\frac{\pi x}{a}\right)+3 \cos \left(\frac{\pi x}{a}\right) \cos \left(\frac{\pi y}{b}\right)$$

Short answer

Question: Calculate the solution \(u(x, y, t)\) and compute the limit \(\lim _{t \rightarrow \infty} u(x, y, t)\) for the given initial temperature distribution \(f(x, y)=2+\cos \left(\frac{\pi x}{a}\right)+3 \cos \left(\frac{\pi x}{a}\right) \cos \left(\frac{\pi y}{b}\right)\), and relate the comparison to the principle of conservation of energy. Additionally, discuss the relation of this problem to Exercise 10. Solution: Following the 7 steps to solve the problem, we find that the solution for \(u(x,y,t)\) can be written as: \(u(x, y, t) = 2 + \sum_{m=1}^{\infty} \sum_{n=1}^{\infty} A_{m,n} \cos \left(\frac{m \pi x}{a}\right) \cos \left(\frac{n \pi y}{b}\right) e^{-k\pi^2\left(\frac{m^2}{a^2}+\frac{n^2}{b^2}\right)t}\) Where the coefficients can be computed as: \(A_{1,0} = 1\), \(A_{1,1} = 3\), and \(A_{m,n} = 0\) for all other pairs \((m,n)\). The limit of \(u(x, y, t)\) as \(t \rightarrow \infty\) is: \[\lim_{t \rightarrow \infty} u(x, y, t) = 2 + 1\cos\left(\frac{\pi x}{a}\right) + 3\cos\left(\frac{\pi x}{a}\right)\cos\left(\frac{\pi y}{b}\right)\] Comparing the limit to the double integral: \[\int_{0}^{a} \int_{0}^{b} f(x, y) dy dx = 2ab\] We see that as time goes to infinity, the temperature distribution approaches the initial temperature distribution. This relation is in accordance with the principle of conservation of energy since the energy (temperature) is conserved in the system for the insulated boundaries. In Exercise 10, if the conditions are similar to Exercise 9, then the overall solution should have a similar behavior as time goes to infinity. There might be some differences in the specific coefficients or terms depending on the given initial temperature distribution, but the overall idea of the limit approaching the initial condition and following the conservation of energy should still hold. It is ultimately dependent on the specific conditions given in Exercise 10.

Step-by-step solution

Applying the Separation of Variables Representation

In Exercise 5, we found a separation of variables representation for the solution to the insulated boundary problem. To solve for the coefficients, we can simply substitute the given initial temperature distribution \(f(x, y)\) into the equation and solve for them.

Computing the Coefficients

The given initial temperature distribution is \(f(x, y)=2+\cos \left(\frac{\pi x}{a}\right)+3 \cos \left(\frac{\pi x}{a}\right) \cos \left(\frac{\pi y}{b}\right)\). In the general separation of variables representation, the coefficients will involve double summation and can be determined without evaluating any integrals.

Calculate the Solution \(u(x, y, t)\)

Using the computed coefficients from Step 2, plug them into the separation of variables representation to determine the solution \(u(x, y, t)\), which would be a function of \(x\), \(y\), and \(t\).

Compute the Limit \(\lim _{t \rightarrow \infty} u(x, y, t)\)

To compute the limit, we first have to examine the terms involving \(t\) in the solution \(u(x, y, t)\). The terms containing exponential or trigonometric functions will be important in determining the limit as \(t\) approaches infinity. Examine the terms and simplify the solution to calculate the limit.

Compare to the Double Integral of \(f(x, y)\)

Compute the double integral \(\int_{0}^{a} \int_{0}^{b} f(x, y) d y d x\), and compare this value to the limit you computed in Step 4.

Relate Comparison to the Principle of Conservation of Energy

Discuss how the comparison between the limit in Step 4 and the double integral in Step 5 relates to the principle of conservation of energy. Provide an explanation for the results you observed.

Discuss Relation to Exercise 10

Without solving Exercise 10, discuss how the answer to Exercise 10 should relate to the answer in Exercise 9 based on your computations and explanations in the previous steps. Answer if there should be similarities or differences in the two problems, and provide a reason why this should be the case.

Key concepts

Insulated Boundary Problem

An insulated boundary problem in the context of heat transfer refers to a scenario where a region does not lose or gain heat through its boundaries. This is akin to wrapping an object in a perfect thermal insulator. Physically, it means that the heat flux across the boundary is zero, and mathematically, this translates to specific conditions being imposed on the boundaries in our differential equations.

In solving such problems, the separation of variables technique is a powerful tool. It assumes that the solution can be represented as the product of functions, each depending on a single coordinate. This method helps simplify complex partial differential equations (PDEs) by reducing them into a set of ordinary differential equations (ODEs) which are easier to solve.

Initial Temperature Distribution

The initial temperature distribution is a function that describes the temperature at every point within a region at the initial time, often denoted as t=0. It serves as a starting condition for understanding how heat will flow and distribute over time.

In the context of the PDEs used for heat conduction problems, and specifically for the insulated boundary problem, the initial temperature distribution is crucial as it kicks off the separation of variables process. By applying the initial distribution function, we can determine the coefficients needed for the series representation of the solution.

Conservation of Energy Principle

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In heat transfer problems, this principle indicates that whatever heat is present initially in a system will be redistributed within the system, but the total amount of heat will remain constant over time, assuming no heat is added or removed.

When relating this principle to the insulated boundary problem with a given initial temperature distribution, the total heat content can typically be represented by the integral of the temperature function over the entire domain. As time progresses, although the distribution of temperature changes, the total integrated heat should remain the same if the boundaries are indeed perfectly insulated.

Solving Differential Equations

Differential equations, especially PDEs, are equations that involve rates of change with respect to multiple variables. They are vital in describing many natural phenomena, including heat transfer. Solving these types of equations requires special methods, one of which is separation of variables.

This technique simplifies the problem significantly, but it requires a keen understanding of boundary and initial conditions to apply it correctly. Once the problem is solved and the solution is obtained, it reflects how the temperature changes with not only time but also spatial coordinates, giving us a complete picture of heat evolution in the system.

Limit of u(x, y, t) as t approaches infinity

The limit of the temperature distribution function, described as u(x, y, t), as time t approaches infinity, reveals the eventual steady-state distribution of temperature within the domain. Assuming no additional heat sources or sinks and perfectly insulated boundaries, we would expect that the heat will eventually distribute itself evenly.

The formal calculation of this limit involves looking at terms in the solution involving time-dependent factors, such as exponentials, which typically decay to zero, leaving behind a steady-state part of the solution that is independent of time – a clear sign of the system reaching thermal equilibrium.

Double Integral Comparison

A double integral comparison involves integrating the initial temperature distribution function, f(x, y), over the entire spatial domain. The result of this computation gives the total initial heat content of the system. By comparing this value to the steady-state solution reached as t approaches infinity, we expect consistency due to conservation of energy.

If these two values are equal, it supports the premise that the initially present heat has been redistributed without any loss or gain, adhering to the principle of conservation of energy. Discrepancies in this comparison would typically indicate an issue with the boundary conditions, such as an imperfect insulation or an error in the solution process.