Question

(a) As in Example 2, use (14) and (15) to solve the initial value problem $$ \begin{aligned} &u_{t}(x, t)=u_{x x}(x, t), \quad 0

Short answer

Question: Find the solution to the given heat equation initial value problem with Neumann boundary conditions. Then evaluate the solution at \((x, t) = \left(\frac{l}{2}, 1\right)\), and show that \(\lim_{t\rightarrow \infty} u(x,t) = l^{-1}\int_{0}^{l}f(x)dx\). Solution: The solution to the heat equation initial value problem is \(u(x,t) = 3e^{-t}\cos x\). At \((x, t) = \left(\frac{l}{2}, 1\right)\), the solution is \(u\left(\frac{l}{2}, 1\right) = 0\). Also, the limit of the solution, \(\lim_{t\rightarrow \infty} u(x,t) = l^{-1}\int_{0}^{l}f(x)dx = 0\).

Step-by-step solution

Initial Value Problem and Boundary Conditions

Given, the heat equation initial value problem is $$ \begin{aligned} &u_{t}(x, t)=u_{x x}(x, t), \quad 0<x<l, \quad 0<t<\infty \\\ &u_{x}(0, t)=u_{x}(l, t)=0, \quad 0 \leq t<\infty \\\ &u(x, 0)=f(x), \quad 0 \leq x \leq l, \end{aligned} $$ where \(f(x)=3 \cos x\). The boundary conditions are Neumann, given by \(u_{x}(0, t)=u_{x}(l, t)=0\).

Applying Fourier Series Solution

Without the formulas (14) and (15) provided, we can't apply them directly. However, we can still apply a Fourier cosine series solution due to the Neumann boundary conditions. The solution can be represented as $$ u(x,t) = \sum_{n=0}^\infty a_n(t)\cos\left(\frac{n\pi x}{l}\right), $$ where \(a_n(t)\) are time-dependent coefficients.

Solving for Time-dependent Coefficients

Since we know \(u(x, 0)=f(x)=3\cos x\), the initial coefficients \(a_n(0)\) for the Fourier cosine series can be determined by $$ a_n(0) = \frac{2}{l} \int_0^l 3\cos x\cos\left(\frac{n\pi x}{l}\right) dx, $$ where the series coefficients can be evaluated by inspection. Using the given value of \(I=\pi\), we have \(l=\pi\) and \(n=1\). So, we just need to find the coefficient for \(a_1(t)\), as others will become zero due to orthogonality. Thus, the integral becomes $$ a_1(0) = \frac{2}{\pi} \int_0^\pi 3\cos x\cos(\pi x) dx = 3. $$ Now plug the series representation of \(u(x, t)\) back into the heat equation and solve for \(a_n(t)\). After some trigonometric manipulations, we have $$ a_n(t) = a_n(0)e^{-\left(\frac{n\pi}{l}\right)^2t}. $$

Finding the Solution

Now we can substitute \(a_n(0)\) and \(a_n(t)\) back into the series representation of \(u(x,t)\): $$ u(x,t) = 3e^{-t}\cos x. $$

Evaluating at (l/2,1)

Now, we need to evaluate the solution at \((x,t)=(l/2, 1)\): $$ u\left(\frac{l}{2}, 1\right) = 3e^{-1}\cos\left(\frac{l}{2}\right) = 3e^{-1}\cos\left(\frac{\pi}{2}\right) = 0. $$

Proving the Limit of u(x,t)

For part (c), we need to show \(\lim_{t\rightarrow \infty} u(x,t) = l^{-1}\int_{0}^{l}f(x)dx\). Using the solution we found $$ \lim_{t\rightarrow \infty} u(x,t) = \lim_{t\rightarrow \infty} 3e^{-t}\cos x = 0, $$ since the exponential term goes to zero as \(t\) goes to infinity. Now we need to evaluate the right-hand side of the equation, $$ l^{-1}\int_{0}^{l}f(x)dx = \frac{1}{\pi}\int_{0}^{\pi}3\cos xdx = \frac{3}{\pi}\left[\sin x\right]_0^{\pi} = \frac{3}{\pi}(0-0) = 0. $$ Since both limits are equal to 0, we conclude that $$ \lim_{t\rightarrow \infty} u(x,t) = l^{-1}\int_{0}^{l}f(x)dx. $$

Key concepts

Fourier Series

The Fourier series is a critical mathematical tool used to express a function as the sum of sinusoidal components. This method is particularly useful when solving partial differential equations like the heat equation. By decomposing complex waveforms into simpler sine and cosine waves, the Fourier series provides a way to apply boundary and initial conditions effectively.

In this exercise, we employ a Fourier cosine series due to the Neumann boundary conditions, which are derived from symmetry properties and the requirement that the derivatives at the boundary are zero. The Fourier series representation facilitates finding the time-dependent coefficients needed for the solution. These coefficients are determined by projecting the initial condition, in this case, a cosine function, onto the basis of the Fourier series.

This approach allows us to solve the heat equation by connecting physical phenomena (e.g., heat distribution) with mathematical expressions that can predict the system's behavior over time.

Initial Value Problem

An initial value problem consists of a differential equation coupled with a set of initial conditions. In our case, the heat equation, \(u_t(x, t) = u_{xx}(x, t)\), governs the change in temperature over time. The initial condition specified as \(u(x, 0) = f(x)\) indicates the temperature distribution at time \(t = 0\).

This type of problem requires finding a solution that satisfies both the differential equation and the initial value condition. Solving it involves integrating the equation in time while respecting the constraints provided by the initial values. Such problems are crucial for predicting how a system evolves from a known state, providing insights into future behavior based on past data.

Boundary Conditions

Boundary conditions are essential for defining the behavior of a system at its edges. They complement initial conditions in partial differential equations by specifying how the solution behaves on the boundaries of the defined domain. In this exercise, the boundaries of the spatial interval \(0 < x < l\) are where the heat distribution’s behavior is constrained.

There are various types of boundary conditions such as Dirichlet, Neumann, and Robin. In our problem, Neumann boundary conditions are used, where the derivative of the solution at the boundaries is set to zero. This represent physical situations where there is no heat flux across the boundaries, implying an insulated system. Correctly applying boundary conditions ensures the solution is unique and corresponds to the physical problem being modeled.

Neumann Boundary Conditions

Neumann boundary conditions specify that the derivative of the function at the boundary is zero. These are often used when modeling insulated boundaries where no flux occurs across the boundary. In mathematical terms, this condition is expressed as \(u_x(0, t) = u_x(l, t) = 0\).

In the context of the heat equation, this implies that there is no heat flowing through the ends of the rod, which might represent an isolated system. Implementing these conditions alters which types of Fourier series components are applicable, typically involving only cosine terms due to their even symmetry properties. Neumann conditions are naturally suited for problems involving conservation laws or where rates of change at the boundaries need specific control based on physical insights.