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Chapter 6: Problem 1

Consider the heat equation in a two-dimensional rectangular region, \(00) .\)

Short Answer

Expert verified
Upon analyzing the heat equation under the given condition (a), it can be concluded that with time, the heat will disperse throughout the rectangular region, leading to eventually a uniform temperature distribution of 0.

Step by step solution

01

Setting up the equation under a given condition. (Considering condition (a) here)

For condition (a), the boundary conditions are \(u(0, y, t) = 0\) and \(u(x, 0, t) = 0\). Now, the general solution of the heat equation can be written as a Fourier series: \(u(x, y, t) = \sum_{m, n} B_{m, n} e^{-k(m^2/L^2 + n^2/H^2)t} sin(m \pi x / L) sin(n \pi y / H)\), where \(B_{m, n}\) are the coefficients to be determined from the initial condition. Apply this to the given boundary condition.
02

Applying the initial condition

We can use the initial condition to determine \(B_{m, n}\). This gives us: \(B_{m, n} = \frac{4}{LH} \int_{0}^{L} \int_{0}^{H} f(x, y) sin(m \pi x / L) sin(n \pi y / H) dx dy\).
03

Analyzing Temperature as \(t \rightarrow \infty\)

As \(t \rightarrow \infty\), the exponential term in the solution goes to 0, and hence \(u(x, y, t) \rightarrow 0\). This indicates that the heat disperses over time as per condition (a). Similar process to be followed for other conditions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Partial Differential Equations
Partial differential equations (PDEs) are a type of equation widely used in fields such as physics, engineering, and finance, to model processes involving functions of several variables and their partial derivatives. The heat equation given in the exercise is a classic example of a PDE, which describes the distribution of heat (or variation in temperature) in a given region over time.

At the heart of solving a PDE like the heat equation is finding a function that satisfies both the equation itself and any given conditions, which may include initial conditions and boundary conditions. In our case, the heat equation is second-order, as it involves the second derivatives of the temperature function with respect to both space and time variables.
The Role of Boundary Conditions
Boundary conditions are constraints necessary for uniquely solving a PDE and usually represent the physical state at the borders of the considered domain. For the heat equation, these conditions could specify temperature, heat flux, or a combination of both on the boundaries of the rectangular region.

In the problem, different sets of boundary conditions are provided, ranging from fixed temperature conditions (Dirichlet conditions) to insulated edges where the heat flux is zero (Neumann conditions). The correct application of these conditions is crucial, as they influence the temperature distribution and flow within the domain.
Fourier Series Solutions to Heat Equation
The Fourier series solution is a mathematical technique used to express functions as a sum of sinusoidal waveforms, which is highly useful in solving PDEs with periodic boundary conditions. In our exercise, the Fourier series spreads out the initial temperature function as an infinite series of sine waves, each with specific coefficients that represent the amplitude of that mode in the temperature distribution.

These coefficients, denoted by Bm,n, are determined by integrating the initial temperature distribution against the sine functions that form the basis of our solution. This process essentially projects the initial condition onto the different modes described by the series, capturing how each mode contributes to the overall solution.
Solving the Initial Value Problem
The initial value problem in the context of the heat equation involves finding the temperature distribution at future times, given an initial temperature distribution throughout the domain. It is called an initial value problem because the entire future behavior of the system is determined by the initial state.

After obtaining the Fourier coefficients from the initial condition, we can write out the full solution that satisfies both the heat equation and the initial condition. The exponential factor in the solution indicates that the influence of each mode on the temperature distribution decays over time, and as time goes to infinity (t 鈫� 鈭�), the temperature in the domain ultimately approaches a steady state or uniformly cools down, depending on the situation.

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Most popular questions from this chapter

For the following differential equations, what is the expected approximate behavior of all solutions near \(x=0\) ? (a) \(x^{2} \frac{d^{2} y}{d x^{2}}+(x-6) y=0\) (b) \(x^{2} \frac{d^{2} y}{d x^{2}}+\left(x^{2}+\frac{3}{16}\right) y=0\) (c) \(x^{2} \frac{d^{2} y}{d x^{2}}+\left(x+x^{2}\right) \frac{d y}{d x}+4 y=0\) (d) \(x^{2} \frac{d^{2} y}{d x^{2}}+\left(x+x^{2}\right) \frac{d y}{d x}-4 y=0\) (e) \(x^{2} \frac{d^{2} y}{d x^{2}}-4 x \frac{d y}{d x}+\left(6+x^{3}\right) y=0\) (f) \(x^{2} \frac{d^{2} y}{d x^{2}}+\left(x+\frac{1}{4}\right) y=0\)

Consider the heat equation in a three-dimensional box-shaped region, \(0

Consider the displacement \(u(r, \theta, t)\) of a membrane whose shape is a \(90^{\circ}\) sector of an annulus, \(a

Solve the heat equation $$ \frac{\partial u}{\partial t}=k \nabla^{2} u $$ inside a quarter-circular cylinder \((0<\theta<\pi / 2\) with radius \(a\) and height \(H)\) subject to the initial condition $$ u(r, \theta, z, 0)=f(r, \theta, z) . $$ Briefly explain what temperature distribution you expect to be approached as \(t \rightarrow \infty\). Consider the following boundary conditions (a) \(\begin{aligned} u(r, \theta, 0) &=0, & u(r, 0, z) &=0, \\ u(r, \theta, H) &=0, & u(r, \pi / 2, z) &=0, \\ \text { (b) } \frac{\partial u}{\partial z}(r, \theta, 0) &=0, & \frac{\partial u}{\partial \theta}(r, 0, z) &=0, & \frac{\partial u}{\partial r}(a, \theta, z) &=0 \end{aligned}\) \(\frac{\partial u}{\partial z}(r, \theta, H)=0\), (c) \(u(r, \theta, 0)=0\)

Derive an expression for $$ \iint[u L(v)-v L(u)] d x d y $$ over a two-dimensional region \(R\), where $$ L=\nabla^{2}+q(x, y) \quad\left[\text { i.e., } L(u)=\nabla^{2} u+q(x, y) u\right] . $$
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