Question

(a) Sketch the graph of the given initial condition \(f(x)\) and determine its Fourier sine series. (b) For the given initial condition \(f(x)\), solve the initial-boundary value problem $$ \begin{aligned} &u_{t}=\kappa u_{x x}, \quad 0

Short answer

Question: Determine the solution of the given initial-boundary value problem with the function f(x) as the initial condition. Answer: The solution of the given initial-boundary value problem with the initial condition f(x) is: \[u(x,t) = \sum_{n=1}^\infty \frac{8l}{n\pi} \sin\left(\frac{n\pi}{4}\right) \sin\left(\frac{n\pi x}{l}\right) e^{-\kappa n^2\pi^2 t/l^2}\]

Step-by-step solution

Sketch the graph of \(f(x)\)

The graph of \(f(x)\) is a piecewise function and can be sketched as follows: - If \(0\leq x<\frac{l}{4}\), \(f(x)=0\) - If \(\frac{l}{4}\leq x\leq\frac{3l}{4}\), \(f(x)=1\) - If \(\frac{3l}{4}<x\leq l\), \(f(x)=0\) So, the function is simply a horizontal line segment of height 1 that starts from \(\frac{l}{4}\) and ends at \(\frac{3l}{4}\).

Determine the Fourier sine series of \(f(x)\)

We'll construct the Fourier sine series of \(f(x)\): \[f(x) = \sum_{n=1}^\infty b_n \sin\left(\frac{n \pi x}{l}\right)\] Using the general formula for calculating the Fourier coefficients \(b_n\), we have: \[b_n =\frac{2}{l} \int_0^l f(x)\sin\left(\frac{n\pi x}{l}\right) dx\] After the integration and simplification, we get: \[b_n = \frac{8l}{n\pi} \sin\left(\frac{n\pi}{4}\right)\] So, the Fourier sine series for \(f(x)\) is: \[f(x) = \sum_{n=1}^\infty\frac{8l}{n\pi} \sin\left(\frac{n\pi}{4}\right) \sin\left(\frac{n\pi x}{l}\right)\]

Solve the initial-boundary value problem using Fourier sine series

The given initial-boundary value problem is: \[\begin{cases} u_t = \kappa u_{xx}, &0<x<l, &0<t<\infty \\ u(0, t) = u(l, t) = 0, &0\leq t<\infty \\ u(x, 0) = f(x), & 0\leq x\leq l \end{cases}\] As we have found the Fourier sine series of the initial condition \(f(x)\), we can write the solution of this problem in the form of a Fourier sine series as follows: \[u(x,t) = \sum_{n=1}^\infty \frac{8l}{n\pi} \sin\left(\frac{n\pi}{4}\right) \sin\left(\frac{n\pi x}{l}\right) e^{-\kappa n^2\pi^2 t/l^2}\] This is the solution of the given initial-boundary problem with the initial condition \(f(x)\).

Key concepts

Fourier Sine Series

The Fourier Sine Series is a mathematical tool used to represent functions, particularly piecewise and periodic functions, using a sum of sine functions. This approach is incredibly useful for solving partial differential equations, especially with boundary conditions where the function is zero at the boundaries. The sine series is built upon the idea that complex functions can be broken down into simpler, harmonic components.

The series is represented as:

• \( f(x) = \sum_{n=1}^\infty b_n \sin\left(\frac{n \pi x}{l}\right) \)

To determine the coefficients \( b_n \), integration techniques are used, focusing on the symmetry and periodicity properties of sine functions. These coefficients capture the essential characteristics of the original function within a defined interval. In many applications, such as solving the heat equation, the Fourier sine series helps simplify the problem by transforming the partial differential equations into a more manageable form.

Initial-Boundary Value Problems

Initial-boundary value problems arise frequently in physics and engineering, where a system's behavior is described over time and within a certain spatial domain and conditions. Such a problem defines the input or starting conditions ('initial') and the restrictions or limitations at the edges of the domain ('boundary').

In mathematical terms, we deal with equations like:

• \( u_t = \kappa u_{xx}, \quad 0

With boundary conditions:

• \( u(0, t) = u(l, t) = 0, \quad 0 \leq t<\infty \)

And an initial function \( u(x, 0) = f(x) \), within the range \( 0 \leq x \leq l \). Here, the goal is to find the solution \( u(x, t) \) that respects both the initial and boundary conditions over time. Successfully solving these problems often involves using techniques like separation of variables, Fourier series, and transformations.

Heat Equation

The heat equation is a type of partial differential equation that models the distribution of heat (or variation in temperature) in a given region over time. It is expressed as:

• \( u_t = \kappa u_{xx} \)

Where \( u(x, t) \) represents the temperature at position \( x \) at time \( t \), and \( \kappa \) represents thermal diffusivity, indicating how quickly heat spreads through the material.

This equation is central in understanding how heat propagates and is crucial for thermal analysis in engineering and sciences. Using the Fourier sine series, solutions to the heat equation become manageable, enabling the transformation of complex spatial and temporal dependencies into simple, interpretable series. Applying this to initial-boundary value problems, engineers and scientists can predict temperature distribution accurately over time.

Piecewise Functions

Piecewise functions are mathematical expressions defined by multiple sub-functions, each applying to a specific interval of the main function's domain. This allows for complex, real-world phenomena to be represented in mathematical form by breaking them into simpler, understandable pieces for different regions of the domain.

For example:

• If \( 0 \leq x < \frac{l}{4} \), \( f(x) = 0 \)
• If \( \frac{l}{4} \leq x \leq \frac{3l}{4} \), \( f(x) = 1 \)
• If \( \frac{3l}{4} < x \leq l \), \( f(x) = 0 \)

This piecewise approach is beneficial for solving problems that involve abrupt changes or discontinuities, like the given initial condition function \( f(x) \). Such functions are often encountered in boundary value problems, where different conditions or states are observed within different sections of the domain. Using piecewise functions in Fourier analysis or differential equations enables a clearer depiction of scenarios with varying dynamics.