Problem 16 - Advanced Engineering Mathematics

Question

Referring to the discussion in this section of the text we identify \(a=b=2, f(x)=0,\) $$g(x)=\left\\{\begin{array}{ll} x, & 0

Step-by-Step Solution

Answer

Combine series solutions for \(u_1\) and \(u_2\) to get \(u(x, y)\).

Step 1: Identify Problem Components

Begin by identifying the components given in the problem statement: We have \(a = b = 2\), \(f(x) = 0\), and \(F(y) = 0\). We also have the piecewise function for \(g(x)\) and \(G(y) = y(2-y)\). The Fourier coefficients \(A_n\) are mentioned to be \(0\).

Step 2: Analyze Given Equations for \(u_1(x, y)\)

The function \(u_1(x, y)\) is given by an infinite series. The coefficients \(B_n\) for this series depend on the integrals of \(g(x)\) over the intervals \((0,1)\) and \((1,2)\). This results in an expression involving the sum of hyperbolic sine and sine functions.

Step 3: Expression of \(B_n\) for \(u_1(x, y)\)

The expression for \(B_n\) was derived as:\[B_n = \frac{1}{\sinh n \pi} \left( \int_{0}^{1} x \sin \frac{n \pi}{2} x \, dx + \int_{1}^{2} (2-x) \sin \frac{n \pi}{2} x \, dx \right) = \frac{8 \sin \frac{n \pi}{2}}{n^{2} \pi^{2} \sinh n \pi}\]

Step 4: Analyze Given Equations for \(u_2(x, y)\)

The second part \(u_2(x, y)\) is similar to \(u_1(x, y)\) but involves integrating \(G(y)\). As \(A_n\) is zero, \(u_2(x, y)\) also relies on its specific \(B_n\).

Step 5: Expression of \(B_n\) for \(u_2(x, y)\)

For \(u_2(x, y)\), the coefficients \(B_n\) are given by integrating \(G(y)\), yielding:\[B_n = \frac{1}{\sinh n \pi} \int_{0}^{2} y(2-y) \sin \frac{n \pi}{2} y \, dy = \frac{16 [1-(-1)^n]}{n^3 \pi^3 \sinh n \pi}\]

Step 6: Combine Series to Find \(u(x, y)\)

Combine \(u_1(x, y)\) and \(u_2(x, y)\):\[u(x, y) = \frac{8}{\pi^2} \sum_{n=1}^{\infty} \frac{\sin \frac{n \pi}{2}}{n^2 \sinh n \pi} \sinh \frac{n \pi}{2} y \sin \frac{n \pi}{2} x \]\[+ \frac{16}{\pi^3} \sum_{n=1}^{\infty} \frac{(1-(-1)^n)}{n^3 \sinh n \pi} \sinh \frac{n \pi}{2} x \sin \frac{n \pi}{2} y. \]

Key Concepts

Fourier Series

The Fourier series is a mathematical tool used to express a function in terms of sine and cosine functions. It's particularly useful in solving partial differential equations (PDEs) and is instrumental in solving problems with periodic boundary conditions. Imagine trying to express a complex periodic waveform using simpler, familiar waves like sine and cosine.

A Fourier series works by breaking down any periodic function into a sum of these simpler functions. Each term of the series represents a different frequency, with coefficients that determine the amplitude of each sine and cosine wave involved. When solving PDEs, these coefficients become crucial, as they help to ensure solutions match the given boundary conditions.

In the given exercise, the function is expressed as a Fourier series with coefficients. These are cleverly determined by integrating the initial conditions or specific boundary values, like the piecewise function for a given function over its domain. Understanding and calculating these coefficients correctly is key to employing Fourier series effectively in boundary value problems.

Boundary Value Problems

Boundary value problems (BVPs) refer to differential equation problems where the solution is determined by the function's values at specific points, usually the boundaries of the domain. These types of problems are common in physics and engineering, where the behaviour of a system is defined by conditions at the edges or boundaries.

For the exercise we're discussing, the boundary conditions are given and used to form the equations that are solvable by Fourier series. This involves defining functions at specific boundary points such as 0 or 2 in the variable’s interval. By solving the PDEs under these conditions, we ensure solutions satisfy the required physical or theoretical state.

The piecewise function for the boundary, like our specific function, often guides the solution in cases where the function's behaviour changes across the domain. Identifying and applying these boundary conditions correctly is crucial to solving BVPs, as this aligns theoretical mathematical solutions with real-world systems.

Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions but for a hyperbola, similar to how sine and cosine are related to a circle. These functions include hyperbolic sine \(\sinh\) and hyperbolic cosine \(\cosh\), and they appear often in solutions to equations describing a variety of phenomena, especially in PDEs.

In this exercise, hyperbolic functions are prominent in the series solutions for the given PDE. The function \(\sinh\), for example, appears as part of each term in the series solution. This function helps describe the nature of the solution as it changes across the spatial domain.

• \(\sinh y\) can be visualized as halfway between \(e^y\) and \(-e^{-y}\).
• Similarly, \(\cosh y\) has a similar role to the cosine function but in the hyperbolic realm.

Understanding hyperbolic functions provides insight into how solutions to differential equations behave, particularly for BVPs requiring curves in their solution space. These functions' presence in a Fourier series expansion enriches the solution set, echoing familiar patterns in multiple physical domains.