/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 Define the formal solution of ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 1

Define the formal solution of $$ \begin{array}{c} u_{r r}+\frac{1}{r} u_{r}+\frac{1}{r^{2}} u_{\theta \theta}=0, \quad \rho_{0}

Short Answer

Expert verified
Question: Find the formal solution of the partial differential equation (PDE) in polar coordinates with boundary conditions given by: $$ u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta}=0, \qquad u(\rho_0, \theta) = f(\theta), \qquad u(\rho, \theta) = 0, \qquad -\pi \leq \theta < \pi$$ Answer: The formal solution of the given PDE with the provided boundary conditions is given by: $$ u(r, \theta) = \sum_{n=1}^\infty \left[C_n J_n(\sqrt{\lambda_n}r)\right]\left[A_n\cos(n\theta) + B_n\sin(n\theta)\right] $$ where \(J_n\) are Bessel functions of the first kind, \(n\) is an integer, and the coefficients \(A_n\), \(B_n\), and \(C_n\) are determined by expanding \(f(\theta)\) as a Fourier series and matching the coefficients.

Step by step solution

01

Separate the variables

Begin by assuming that the function \(u(r, \theta)\) can be separated into a product of two functions, \(R(r)\) and \(T(\theta)\): $$ u(r, \theta) = R(r)T(\theta) $$ Substitute this into the given PDE: $$ R''(r) T(\theta) + \frac{1}{r} R'(r) T(\theta) + \frac{1}{r^2} R(r) T''(\theta) = 0 $$
02

Rearrange the equation

Divide both sides of the equation by \(R(r)T(\theta)\), and rearrange: $$ \frac{R''(r)}{R(r)} + \frac{1}{r} \frac{R'(r)}{R(r)} = -\frac{T''(\theta)}{T(\theta)r^2} $$ Since the left side of the equation depends only on \(r\), and the right side depends only on \(\theta\), each side must equal a constant, which we'll call \(\lambda\): $$ \frac{R''(r)}{R(r)} + \frac{1}{r} \frac{R'(r)}{R(r)} = \lambda = -\frac{T''(\theta)}{T(\theta)r^2} $$
03

Solve the \(T(\theta)\) ODE

Now we have two separated ODEs. Let's start with the \(\theta\)-dependent equation: $$ T''(\theta) + \lambda r^2 T(\theta) = 0 $$ The general solution of this ODE will be: $$ T(\theta) = A\cos(\sqrt{\lambda}r\theta) + B\sin(\sqrt{\lambda}r\theta) $$
04

Solve the \(R(r)\) ODE

Now let's move onto the \(r\)-dependent equation: $$ R''(r) + \frac{1}{r} R'(r) - \lambda R(r) = 0 $$ This equation is a Bessel's equation, and its general solution is: $$ R(r) = C J_n(\sqrt{\lambda}r) + D Y_n(\sqrt{\lambda}r) $$ Here, \(J_n\) and \(Y_n\) are Bessel functions of the first and second kind, respectively, and \(n\) is an integer.
05

Apply the boundary conditions

We can now apply the boundary conditions to the general solution of the separated ODEs: $$ u(r, \theta) = R(r)T(\theta) = \left[C J_n(\sqrt{\lambda}r) + D Y_n(\sqrt{\lambda}r)\right]\left[A\cos(\sqrt{\lambda}r\theta) + B\sin(\sqrt{\lambda}r\theta)\right] $$ Apply the boundary condition \(u(\rho, \theta) = 0\): $$ u(\rho, \theta) = \left[C J_n(\sqrt{\lambda}\rho) + D Y_n(\sqrt{\lambda}\rho)\right]\left[A\cos(\sqrt{\lambda}\rho\theta) + B\sin(\sqrt{\lambda}\rho\theta)\right] = 0 $$ To satisfy this condition, we require \(J_n(\sqrt{\lambda}\rho)=0\). As \(J_n\) has an infinite number of zeros, we will have an infinite series of separated solutions. Finally, apply the boundary condition \(u(\rho_0, \theta) = f(\theta)\): $$ u(\rho_0, \theta) = \sum_{n=1}^\infty \left[C_n J_n(\sqrt{\lambda_n}\rho_0)\right]\left[A_n\cos(n\theta) + B_n\sin(n\theta)\right] = f(\theta) $$
06

Determine coefficients

To determine the coefficients \(A_n\), \(B_n\), and \(C_n\), we need to expand \(f(\theta)\) as a Fourier series and match the coefficients. This will ultimately provide us with the formal solution of the given PDE with the given boundary conditions. The final solution will be in the form: $$ u(r, \theta) = \sum_{n=1}^\infty \left[C_n J_n(\sqrt{\lambda_n}r)\right]\left[A_n\cos(n\theta) + B_n\sin(n\theta)\right] $$

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Partial Differential Equations
Partial differential equations (PDEs) are fundamental to modeling various physical phenomena, from heat conduction and fluid flow to sound and elasticity. A PDE involves derivatives with respect to multiple variables, and solving a PDE means finding a function that satisfies the equation across a specified domain.

PDEs can be classified based on order, linearity, and homogeneity. The order is the highest derivative present, while linearity pertains to whether the equation is a linear combination of the function and its derivatives. Homogeneity involves whether the PDE equals zero (homogeneous) or a non-zero function (non-homogeneous).

In the exercise provided, you came across a homogeneous, linear second-order PDE with derivatives with respect to two variables, radius \(r\) and angle \(\theta\). The solution approach employed 'separation of variables', a technique where one assumes that the solution can be expressed as a product of functions, each depending on a single variable. This simplification facilitates solving PDEs by breaking them down into ordinary differential equations (ODEs).
Grasping the Essence of Bessel Functions
Bessel functions, denoted by \(J_n\) and \(Y_n\), are canonical solutions to Bessel's differential equation. This equation arises naturally in problems with cylindrical or spherical symmetry, such as in heat conduction, wave propagation, and in our exercise, the vibrational modes of a circular membrane.

Bessel functions of the first kind \(J_n\) are used when dealing with finite solutions at the origin, which is crucial for physical problems where the solution must be bounded. On the other hand, Bessel functions of the second kind \(Y_n\) are often ruled out in physical applications due to their singularity at the origin. These functions form a complete set, allowing us to express solutions to PDEs in terms of an infinite series involving \(J_n\).

In our problem, a Bessel function solution emerges when applying the separation of variables to the radial part of the PDE. The boundary conditions lead us to seek zeros of these functions since they must satisfy certain conditions at the boundaries of the defined domain, \(\rho_0 < r < \rho\).
Decomposing Functions with Fourier Series
Fourier series decompose periodic functions into an infinite sum of sine and cosine functions, each multiplied by a coefficient. This powerful tool enables us to express complex, periodic functions in terms of simple, harmonic components. These series play a pivotal role in solving PDEs by assisting in boundary condition applications and within the process of separation of variables.

In our solution, after separating the variables, the angular part of the solution is represented by a Fourier series. When applying the given boundary conditions, the Fourier series helps in determining the coefficients involved in the solution. By matching coefficients from the series with those of the boundary condition \(f(\theta)\), we ensure that the solution to the PDE adheres to the specified constraints at the domain edges.

The method of Fourier series helps to encapsulate the initial condition \(f(\theta)\) into an analytical form that aligns with the separated solution, resulting in a complete and comprehensive description of the wave profile over the domain, neatly incorporating the boundary data.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In Exercises \(1-16\) apply the definition developed in Example 1 to solve the boundary value problem. (Use Theorem 11.3 .5 where it applies.) Where indicated by \(\mathrm{C}\), graph the surface \(u=u(x, y), 0 \leq x \leq a\), \(0 \leq y \leq b\) $$ \begin{array}{l} \text { C } u_{x x}+u_{y y}=0, \quad 0

In Exercises \(1-16\) apply the definition developed in Example 1 to solve the boundary value problem. (Use Theorem 11.3 .5 where it applies.) Where indicated by \(\mathrm{C}\), graph the surface \(u=u(x, y), 0 \leq x \leq a\), \(0 \leq y \leq b\) $$ \begin{array}{l} u_{x x}+u_{y y}=0, \quad 0

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

In Exercises \(17-28\) define the formal solution of $$ u_{x x}+u_{u y}=0, \quad 0

In Exercises \(1-16\) apply the definition developed in Example 1 to solve the boundary value problem. (Use Theorem 11.3 .5 where it applies.) Where indicated by \(\mathrm{C}\), graph the surface \(u=u(x, y), 0 \leq x \leq a\), \(0 \leq y \leq b\) $$ \begin{array}{l} u_{x x}+u_{y y}=0, \quad 0
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.