Question

In this exercise take it as given that the infinite series \(\sum_{n=1}^{\infty} n^{p} e^{-q n}\) converges for all \(p\) if \(q>0\), and, where appropriate, use the comparison test for absolute convergence of an infinite series. Let $$ u(x, y)=\sum_{n=1}^{\infty} \alpha_{n} \frac{\sinh n \pi(b-y) / a}{\sinh n \pi b / a} \sin \frac{n \pi x}{a} $$ where $$ \alpha_{n}=\frac{2}{a} \int_{0}^{a} f(x) \sin \frac{n \pi x}{a} d x $$ and \(f\) is piecewise smooth on \([0, a]\). (a) Verify the approximations $$ \frac{\sinh n \pi(b-y) / a}{\sinh n \pi b / a} \approx e^{-n \pi y / a}, \quad y

Short answer

Question: Verify the given approximations, and find the partial derivatives of \(u(x, y)\) with respect to \(x\) and \(y\). Solution: (a) The approximations were verified in the given solution, and the expressions for large \(n\) match the required approximations. (b) It is shown that \(u(x, y)\) is defined for \((x, y)\) such that \(0<y<b\) since the infinite series converges. (c) The partial derivative of \(u(x, y)\) with respect to \(x\) is given by $$ u_{x}(x, y)=\frac{\pi}{a} \sum_{n=1}^{\infty} n \alpha_{n} \frac{\sinh n\pi(b-y) / a}{\sinh n \pi b / a} \cos \frac{n \pi x}{a} $$ (d) The second partial derivative of \(u(x, y)\) with respect to \(x\) is given by $$ u_{x x}(x, y)=-\frac{\pi^{2}}{a^{2}} \sum_{n=1}^{\infty} n^{2} \alpha_{n} \frac{\sinh n\pi(b-y) / a}{\sinh n \pi b / a} \sin \frac{n \pi x}{a} $$ (e) The partial derivative of \(u(x, y)\) with respect to \(y\) is given by $$ u_{y}(x, y)=-\frac{\pi}{a} \sum_{n=1}^{\infty} n \alpha_{n} \frac{\cosh n\pi(b-y) / a}{\sinh n \pi b / a} \sin \frac{n \pi x}{a} $$ (f) The second partial derivative of \(u(x, y)\) with respect to \(y\) is given by $$ u_{y y}(x, y)=\frac{\pi^{2}}{a^{2}} \sum_{n=1}^{\infty} n^{2} \alpha_{n} \frac{\sinh n\pi(b-y) / a}{\sinh n \pi b / a} \sin \frac{n \pi x}{a}, \quad 0<y<b $$

Step-by-step solution

(a) Verify the approximations

To verify the approximations, we need to show that for large \(n\), these approximations hold. $$ \frac{\sinh n \pi(b-y) / a}{\sinh n \pi b / a} \approx e^{-n \pi y / a}, \quad y<b $$ For large \(n\), \(\sinh n \pi(b-y) / a\) is approximately equal to \(e^{n \pi(b-y) / a}\). And \(\sinh n \pi b / a\) is approximately equal to \(e^{n \pi b / a}\). Thus, their ratio is approximately equal to \(e^{-n \pi y / a}\) as required. Similarly, for the second approximation, $$ \frac{\cosh n \pi(b-y) / a}{\sinh n \pi b / a} \approx e^{-n \pi y / a}, \quad y<b $$ For large \(n\), \(\cosh n \pi(b-y) / a\) is approximately equal to \(e^{n \pi(b-y) / a}\), and \(\sinh n \pi b / a\) is approximately equal to \(e^{n \pi b / a}\). Their ratio is approximately equal to \(e^{-n \pi y / a}\).

(b) Show that \(u\) is defined for \((x, y)\) such that \(0

Based on part (a), we can substitute the approximation for \(\frac{\sinh n \pi(b-y) / a}{\sinh n \pi b / a}\) and rewrite the expression for \(u(x,y)\) as $$ u(x, y)=\sum_{n=1}^{\infty} \alpha_{n} e^{-n\pi y/a} \sin \frac{n \pi x}{a} $$ Since it converges, it is defined for \((x, y)\) such that \(0<y<b\).

(c) Find \(u_x(x, y)\)

Based on part (a), we can use the approximation for large \(n\) and Theorem 12.1.2 with \(z = x\) to differentiate \(u(x, y)\) with respect to \(x\) and obtain $$ u_{x}(x, y)=\frac{\pi}{a} \sum_{n=1}^{\infty} n \alpha_{n} \frac{\sinh n\pi(b-y) / a}{\sinh n \pi b / a} \cos \frac{n \pi x}{a} $$

(d) Find \(u_{xx}(x, y)\)

Starting from the result of part (b), we use our approximation for large \(n\) and Theorem 12.1.2 with \(z = x\) to differentiate again with respect to \(x\), and obtain $$ u_{x x}(x, y)=-\frac{\pi^{2}}{a^{2}} \sum_{n=1}^{\infty} n^{2} \alpha_{n} \frac{\sinh n\pi(b-y) / a}{\sinh n \pi b / a} \sin \frac{n \pi x}{a} $$

(e) Find \(u_y(x, y)\)

For fixed but arbitrary \(x\), we use the approximation from part (a) and Theorem 12.1.2 with \(z = y\) to differentiate \(u(x, y)\) with respect to \(y\). We obtain $$ u_{y}(x, y)=-\frac{\pi}{a} \sum_{n=1}^{\infty} n \alpha_{n} \frac{\cosh n\pi(b-y) / a}{\sinh n \pi b / a} \sin \frac{n \pi x}{a} $$ Since \(y_0\) can be chosen arbitrarily small, the conclusion holds for all \(y\) in \((0, b)\).

(f) Find \(u_{yy}(x, y)\)

Starting from the result of part (e), we use our approximation for large \(n\) and Theorem 12.1.2 to differentiate again with respect to \(y\). We obtain $$ u_{y y}(x, y)=\frac{\pi^{2}}{a^{2}} \sum_{n=1}^{\infty} n^{2} \alpha_{n} \frac{\sinh n\pi(b-y) / a}{\sinh n \pi b / a} \sin \frac{n \pi x}{a}, \quad 0<y<b $$

Key concepts

Fourier Series

Fourier series are a way to represent a function as an infinite sum of sines and cosines. This method is extremely useful in solving differential equations, especially those involving periodic boundary conditions. The Fourier series allows functions to be broken down into fundamental frequency components. Here’s how it works:

• Basic Form: A Fourier series represents a function using the sum of sine and cosine terms. Each term in the series represents a harmonic of the original function.
  
• Coefficients: The coefficients of sine and cosine terms, often denoted as \(a_n\) and \(b_n\), are determined through integration over the period of the function.
  
• Convergence: Fourier series converge to the original function provided certain conditions, such as piecewise continuity, are met.
  

In our exercise, the function \(u(x, y)\) is expressed as a series involving sine functions, showing a relationship similar to a Fourier series. The sine integral component, \(\alpha_n\), helps determine the influence of each term.

Infinite Series

An infinite series is a sum of infinitely many terms. These series often converge to a finite value, making them powerful tools in various mathematical and physical applications. In the context of this exercise, we explore the series \(\sum_{n=1}^{\infty} n^{p} e^{-q n}\).

• Convergence: A critical concept in infinite series is convergence. This series converges for all \(p\) if \(q > 0\), meaning the terms get closer and closer to a specific value as we add more terms.
  
• Comparison Test: This test helps determine convergence by comparing with a known convergent series. If our series behaves similarly to a known one, it also converges absolutely.
  

In the exercise, convergence ensures the validity of functions like \(u(x, y)\) within certain constraints, allowing us to work with infinite sums safely.

Hyperbolic Functions

Hyperbolic functions, like \(\sinh\) and \(\cosh\), are analogs of the trigonometric functions but for a hyperbola. These functions often appear in solutions to differential equations relating to boundary value problems and are useful in various applications from physics to engineering.

• Sinh and Cosh: Defined as \(\sinh(x) = \frac{e^x - e^{-x}}{2}\) and \(\cosh(x) = \frac{e^x + e^{-x}}{2}\). They are smooth and even functions.
  
• Approximations for Large n: In this exercise, for large \(n\), \(\sinh\) and \(\cosh\) functions can be approximated using exponentials since they grow similarly as \(e^{x}\).
  

These approximations simplify solving boundary value problems, as seen when we express terms in \(u(x, y)\) and \(u_y(x, y)\) using hyperbolic functions.

Boundary Value Problems

Boundary value problems (BVPs) involve differential equations where solutions must satisfy specific conditions at the boundaries. These types of problems are common in physics, especially in areas like heat distribution, vibrations, and fluid dynamics.

• Setup: BVPs require finding a function \(u(x, y)\) satisfying a differential equation with given conditions at the edges of the domain.
  
• Application: In our exercise, the expression for \(u(x, y)\), along with conditions for \(y < b\), indicates solving a BVP. The challenge is ensuring these solutions adhere to the boundary constraints given.
  

The results for \(u_x\), \(u_y\), and their derivatives help to check that solutions meet boundary conditions and can be manipulated with smooth functions, an essential task in validating proper solutions to these problems.