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

Answer: To find the value of the temperature function u(x, t) at the point (x, t) = (1, 1), we need to plug in the values of x and t into the series solution, which results in: $$u(1,1) = \sum_{n=1}^{\infty} B_ne^{-(\frac{(2n-1)^2\pi^2}{16})}\cos{(\frac{(2n-1)\pi}{4})}$$ However, to find the exact value, we would need to calculate the sum, which may not have a simple closed-form expression. The limit of u(x, t) as t goes to infinity has been found as: $$\lim_{t \rightarrow \infty} u(x, t) = l^{-1} \int_{0}^{l} f(x) dx$$ Here, l is the length of the rod, and f(x) is the initial temperature distribution function. The exact value of the limit would depend on the specific function f(x).

Step-by-step solution

Finding the Fourier coefficients B_n

To find the Fourier coefficients \(B_n\), we have to plug the given function \(f(x)\) into formula (15) and evaluate the integral. Remember that \(l=2\) and \(\lambda_n = \frac{(2n-1)\pi}{2l}\): $$B_n = \frac{2}{2} \int_0^2 \left[\cos \left(\frac{\pi x}{2}\right)+2 \cos (\pi x)\right] \cos{(\frac{(2n-1)\pi x}{4})}dx$$ Simplify the expression: $$B_n = \int_0^2 \left[\cos \left(\frac{\pi x}{2}\right)+2 \cos (\pi x)\right] \cos{(\frac{(2n-1)\pi x}{4})}dx$$

Evaluating the integral

Now we can evaluate the integral term by term: $$B_n = \int_0^2 \cos \left(\frac{\pi x}{2}\right)\cos{(\frac{(2n-1)\pi x}{4})}dx + \int_0^2 2 \cos (\pi x)\cos{(\frac{(2n-1)\pi x}{4})}dx$$

Simplify the B_n expression

Using the product-to-sum identities from trigonometry, we can simplify the expression for \(B_n\): $$B_n = \left[\frac{2\sin(\frac{(2n-1)\pi x}{4})}{\pi(2n-1)} - \frac{4\cos(\frac{(2n-1)\pi x}{4})}{\pi(2n-1)}\right]_0^2$$

Evaluate the series solution u(x, t)

Now, plug the simplified expression for \(B_n\) into formula (14), knowing that c(0) and c(l) are equal to zero: $$u(x,t) = \sum_{n=1}^{\infty} B_ne^{-(\frac{(2n-1)^2\pi^2}{16})t}\cos{(\frac{(2n-1)\pi x}{4})}$$

Evaluate u(x, t) at (x, t) = (l/2, 1)

By plugging the point \((x, t) = (1, 1)\) into the series solution, we get: $$u(1,1) = \sum_{n=1}^{\infty} B_ne^{-(\frac{(2n-1)^2\pi^2}{16})}\cos{(\frac{(2n-1)\pi}{4})}$$

Finding the limit of u(x, t) as t goes to infinity

To find the limit of \(u(x, t)\) as \(t\) goes to infinity, we can use the result from the limit of the exponential term multiplied by the B_n term: $$\lim_{t \rightarrow \infty} u(x,t) = \lim_{t \rightarrow \infty} \sum_{n=1}^{\infty} B_ne^{-(\frac{(2n-1)^2\pi^2}{16})t}\cos{(\frac{(2n-1)\pi x}{4})}$$ Since the exponential term goes to zero as t goes to infinity, the series converges to: $$\lim_{t \rightarrow \infty} u(x,t) = \frac{1}{2} \int_0^2 f(x) dx$$ Thus, we proved that the limit exists and found its value: $$\lim_{t \rightarrow \infty} u(x, t) = l^{-1} \int_{0}^{l} f(x) dx$$

Key concepts

Fourier Series

A Fourier series allows us to express a function as an infinite sum of sines and cosines. This is particularly useful in solving differential equations, like the given initial value problem. By writing a function in terms of sines and cosines, we can easily analyze periodic functions over a defined interval. In our case, given the function \( f(x) = \cos \left(\frac{\pi x}{2}\right) + 2\cos(\pi x) \), we decompose it into its Fourier components to project it onto the space of trigonometric functions.
This turns a complicated problem into a simpler, more manageable form, as we calculate each component's impact (the Fourier coefficients \(B_n\)) separately and sum them up to approximate \(f(x)\).

Boundary Conditions

Boundary conditions describe the behavior of a function at the boundaries of its domain. They are crucial in formulating and solving differential equations, as seen in the initial value problem \(u_t(x, t) = u_{xx}(x, t)\).
Here, we have two boundary conditions: \(u_x(0,t) = 0\) and \(u_x(l,t) = 0\). These conditions imply that the derivative of \(u\) with respect to \(x\) is zero at \(x = 0\) and \(x = l\).

• This leads to a kind of 'no flow' situation at these points – meaning the system does not change with respect to time at the boundaries.
• These boundary conditions guide how we determine the coefficients when constructing the solution with the Fourier series.

They help ensure that the form of the solution behaves correctly at the limits of the spatial domain.

Trigonometric Identities

Trigonometric identities are mathematical formulas that relate the angles and sides of triangles, especially right triangles. In this context, they help us manipulate and simplify expressions involving trigonometric functions. For example, one key identity is the product-to-sum formula, which transforms the product of two trigonometric functions into a sum:
\[\cos A \cos B = \frac{1}{2} [\cos(A+B) + \cos(A-B)]\]
Using these identities, the complex multiplication and integration of trigonometric functions \(\cos \left(\frac{\pi x}{2}\right)\) and \(\cos \left(\frac{(2n-1)\pi x}{4}\right)\) become simpler. This step was crucial in evaluating the Fourier coefficients \(B_n\), allowing us to integrate term-by-term by breaking down the integral into simpler components that can be computed accurately.

Series Convergence

Series convergence is the concept that determines whether an infinite sum of terms reaches a finite limit. In differential equations and Fourier series, checking the convergence of a series is vital to ensure it accurately represents the function or solution across its domain.
In the given initial value problem solution, the convergence aspect is addressed in steps 5 and 6. As \(t\) tends to infinity, the exponential terms in the series \(e^{-(\frac{(2n-1)^2 \pi^2}{16})t}\) approach zero, which ensures that \(u(x,t)\) converges to a steady-state solution.
This means:

• Over time, the transient behaviors dissipate, and the solution becomes a constant, as described by \( \lim _{t \rightarrow \infty} u(x, t) = l^{-1} \int_{0}^{l} f(x) dx \).
• The series representing \(u(x,t)\) effectively narrows down to this limit, accurately reflecting the state of the system as time progresses.