Question

(a) As in Example 1, use (13) and (15) to solve the initial value problem $$ \begin{aligned} &u_{t}(x, t)=u_{x x}(x, t), \quad 0

Short answer

Question: Evaluate the solution of the heat equation with the boundary and initial conditions at the point (l/2, 1). Answer: The solution at the point (l/2, 1) is given by: $$ u \left(\frac{l}{2}, 1\right) = 3 \sin \left(\frac{3 \pi}{2}\right) e^{-\frac{9\pi^2}{l^2}} $$

Step-by-step solution

Identify Eigenfunctions

As we are solving a heat equation, we start with the given eigenfunctions for this problem (derived from Example 1): $$ \phi_{n}(x) = \sin \left(\frac{n \pi x}{l}\right) $$

Set up Eigenfunction Expansion

We know that the general solution can be expressed as an eigenfunction expansion of the form: $$ u(x, t)=\sum_{n=1}^{\infty}a_{n} \sin \left(\frac{n \pi x}{l}\right) e^{-\frac{n^{2} \pi^{2} t}{l^2}} $$

Calculate the Coefficients \(a_n\)

To determine the coefficients \(a_n\), we need to multiply both sides of this expression by \(\sin{\frac{m\pi x}{l}}\), integrate from \(0\) to \(l\), and use the orthogonality property, knowing that f(x) is given as \(3\sin{\frac{3\pi x}{l}}\): $$ a_{m} = \frac{2}{l} \int_{0}^l u(x, 0) \sin \left(\frac{m \pi x}{l}\right) dx = \frac{6}{l} \int_{0}^l \sin \left(\frac{3 \pi x}{l}\right) \sin \left(\frac{m \pi x}{l}\right) dx $$ Since \(f(x) = 0\) unless \(m = 3\), we obtain \(a_3 = 3\) and \(a_n = 0\) for \(n \neq 3\).

Plug in Coefficients and Evaluate

Since only \(a_3 = 3\), the solution to the PDE is $$ u(x, t) = 3 \sin \left(\frac{3 \pi x}{l}\right) e^{-\frac{9 \pi^2 t}{l^2}} $$ Now, we need to evaluate the solution at \((x, t) = (l/2, 1)\). Plugging these values gives:

Final Answer

After evaluating the solution at these points, we arrive at the following: $$ u \left(\frac{l}{2}, 1\right) = 3 \sin \left(\frac{3 \pi}{2}\right) e^{-\frac{9\pi^2}{l^2}} $$

Key concepts

Heat Equation

The heat equation is a fundamental partial differential equation that describes the distribution of heat in a given region over time. It is crucial for solving problems related to heat conduction and diffusion phenomena.
In our case, the heat equation is given by \(u_t(x, t) = u_{xx}(x, t)\), which implies that the rate of change of temperature with respect to time \(t\) at any point \(x\) depends on the second derivative of temperature with respect to \(x\).
Here, \(u(x, t)\) represents the temperature distribution over the domain from \(x=0\) to \(x=l\), with specific boundary and initial conditions. These conditions help in uniquely determining the temperature distribution at any time \(t\).

Eigenfunction Expansion

To solve the heat equation analytically, we often use an approach called eigenfunction expansion. This method decomposes the solution into a sum of orthogonal functions, which makes it easier to analyze and solve.
The eigenfunction expansion for our problem is expressed as \(u(x, t) = \sum_{n=1}^{\infty} a_{n} \sin \left(\frac{n \pi x}{l}\right) e^{-\frac{n^{2} \pi^{2} t}{l^2}}\).
This form takes advantage of the periodic components of solutions to differential equations with fixed boundaries, allowing each term to satisfy the boundary conditions independently while being multiplied by a time-decaying exponential factor.

Initial Value Problem

An initial value problem involves finding a function that satisfies a differential equation and also meets specified conditions at the initial time \(t=0\).
For our heat equation, the initial condition is given by \(u(x, 0) = f(x)\), where \(f(x) = 3 \sin\left(\frac{3 \pi x}{l}\right)\).
The initial condition provides the starting distribution of heat, which aids in determining the coefficients \(a_n\) required for the eigenfunction expansion. It establishes how the temperature should be distributed along the region when time begins.

Orthogonality

Orthogonality is a central concept when solving partial differential equations using separation of variables and eigenfunction expansions.
This principle implies that the integral of the product of two different eigenfunctions over the domain is zero, unless they are the same eigenfunction. Mathematically, this is expressed as
\[\int_{0}^{l} \sin\left(\frac{n \pi x}{l}\right) \sin\left(\frac{m \pi x}{l}\right) dx = 0\quad \text{for} \quad n eq m\].
Orthogonality allows us to easily find the coefficients \(a_n\) in the eigenfunction expansion through integration, simplifying the solution process.

Boundary Conditions

Boundary conditions specify the behavior of the solution at the edges of the domain. They are crucial in solving differential equations as they help in defining the solution uniquely.
For our heat equation, the boundary conditions are \(u(0, t) = 0\) and \(u(l, t) = 0\), which denote that the temperature at both ends of the domain is held constant at zero over time.
These conditions are essential in ensuring that the solution remains physically meaningful and mathematically consistent throughout the region of interest.