Question

Find a solution \(u(r, \theta)\) of Laplace's equation inside the circle \(r=a,\) also satisfying the boundary condition on the circle $$ u_{r}(a, \theta)=g(\theta), \quad 0 \leq \theta<2 \pi $$ Note that this is a Neumann problem, and that its solution is determined only up to an arbitrary additive constant. State a necessary condition on \(g(\theta)\) for this problem to be solvable by the method of separation of variables (see Problem 10 ).

Short answer

In summary, we have investigated the problem of finding a solution to Laplace's equation inside a circle of radius 'a' with a Neumann boundary condition. We applied the method of separation of variables and found that the problem is solvable if the given boundary condition is compatible with the separated variables solutions. The necessary condition for the problem to be solvable is that the average value of the boundary function \(g(\theta)\) over the interval from 0 to \(2\pi\) should be zero: $$ c_0 = \frac{1}{2\pi}\int_{0}^{2\pi}g(\theta)d\theta = 0 $$

Step-by-step solution

Write Laplace's equation in polar coordinates

Laplace's equation in polar coordinates is given by: $$ \frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial u}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 u}{\partial \theta^2}=0 $$ We will be using the method of separation of variables to seek solutions of the form \(u(r, \theta) = R(r)\Theta(\theta)\).

Apply separation of variables

Let's substitute \(u(r, \theta) = R(r)\Theta(\theta)\) into Laplace's equation: $$ \frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial (R\Theta)}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 (R\Theta)}{\partial \theta^2}=0 $$ Now, we separate the variables by dividing the equation by \(R(r)\Theta(\theta)\): $$ \frac{1}{rR}\frac{\partial}{\partial r}\left(r\frac{\partial R}{\partial r}\right) + \frac{1}{r^2\Theta}\frac{\partial^2 \Theta}{\partial \theta^2}=0 $$ Subtract the second term of the equation from the first term: $$ \frac{1}{rR}\frac{\partial}{\partial r}\left(r\frac{\partial R}{\partial r}\right) - \frac{1}{r^2\Theta}\frac{\partial^2 \Theta}{\partial \theta^2}=0 $$ Since both terms are functions of different independent variables, both terms should be equal to a separation constant. Let's call it \(-\lambda^2\): $$ -\frac{1}{rR}\frac{\partial}{\partial r}\left(r\frac{\partial R}{\partial r}\right) = \frac{1}{r^2\Theta}\frac{\partial^2 \Theta}{\partial \theta^2} = \lambda^2 $$ Thus, we obtain two ordinary differential equations: $$ r\frac{d}{dr}\left(r\frac{dR}{dr}\right) + \lambda^2rR = 0 $$ $$ \frac{d^2\Theta}{d\theta^2} + \lambda^2\Theta = 0 $$

Solve the \(\Theta\)-equation

The \(\Theta\)-equation is a simple harmonic oscillator equation, and its general solution is given by: $$ \Theta(\theta) = A\cos(\lambda \theta) + B\sin(\lambda \theta) $$ Since the solution must be periodic with a period of \(2\pi\), the eigenvalues \(\lambda\) must be integers: \(\lambda = n\), where \(n\) is an integer.

Solve the \(R\)-equation

The \(R\)-equation is a Bessel's equation of order \(n\). The general solution is given by: $$ R(r) = C_nJ_n(\lambda r) + D_nY_n(\lambda r) $$ However, inside the circle, the solution must be regular, so \(D_n=0\). Thus, we have: $$ R(r) = C_nJ_n(\lambda r) $$

Apply the boundary condition

The boundary condition given is \(u_r(a,\theta)=g(\theta)\). Let's differentiate \(u(r,\theta)\) with respect to \(r\) and then set \(r=a\): $$ \frac{\partial u}{\partial r}\bigg|_{r=a} = \frac{\partial (R\Theta)}{\partial r}\bigg|_{r=a} = a\frac{dR}{dr}\bigg|_{r=a}\Theta(\theta) $$ We know that \(\frac{dR}{dr}\bigg|_{r=a}=\frac{d}{dr} (J_n(n r))\bigg|_{r=a}\), and \(\Theta(\theta)=A\cos(n \theta) + B\sin(n \theta)\). Thus, we have: $$ a\frac{dR}{dr}\bigg|_{r=a}\Theta(\theta) = g(\theta) $$ We need to find an expression for \(g(\theta)\) in terms of \(a\frac{dR}{dr}\bigg|_{r=a}\Theta(\theta)\), so we can find the necessary condition for the problem to be solvable.

Fourier series representation of \(g(\theta)\) and necessary condition

Let's represent \(g(\theta)\) as a Fourier series: $$ g(\theta) = \sum_{n=-\infty}^{\infty} c_ne^{in\theta} $$ The coefficients \(c_n\) are given by: $$ c_n = \frac{1}{2\pi}\int_{0}^{2\pi}g(\theta)e^{-in\theta}d\theta $$ For the problem to be solvable, the boundary condition must be compatible with the separated variables solutions that we have found. In other words, the function \(g(\theta)\) must be expressible as a Fourier series with coefficients corresponding to the eigenvalues and Bessel functions. If the Fourier coefficients do not match, the problem cannot be solved using separation of variables. Since the solution is determined only up to an arbitrary additive constant, we can set \(c_0=0\). Thus, the necessary condition for the problem to be solvable by the method of separation of variables is: $$ c_0 = \frac{1}{2\pi}\int_{0}^{2\pi}g(\theta)d\theta = 0 $$

Key concepts

Neumann Boundary Condition

Understanding the Neumann boundary condition is crucial in solving partial differential equations (PDEs) like Laplace's equation. Unlike Dirichlet boundary conditions which specify the values a solution must take on the boundary, Neumann boundary conditions are concerned with the gradient of the solution.

In the context of our exercise, we’re provided with the boundary condition \( u_{r}(a, \theta) = g(\theta) \), for a function \(u(r, \theta)\) inside a circle of radius \(a\). This condition specifies the derivative of \(u\) with respect to \(r\) (the radial direction) at every point along the boundary \(r=a\). Essentially, it's telling us how the function is changing as we move outward from the circle's edge.

The peculiar aspect of a Neumann problem is that the solution is defined only up to an arbitrary constant. This is because if you have a solution \(u\), then \(u+C\), where \(C\) is any constant, will also satisfy the Neumann condition since derivatives ignore constants. Therefore, the Neumann boundary condition does not uniquely determine the solution; other methods or additional information are necessary to find a unique solution.

The Neumann condition also sets a necessary criterion for \(g(\theta)\) when using separation of variables to solve Laplace’s equation. Specifically, the integral of \(g(\theta)\) over a full period should be zero. This corresponds to having no net flux across the boundary, making the problem well-posed for the separation of variables technique.

Separation of Variables

The method of separation of variables is a powerful technique for solving PDEs, particularly when we’re dealing with problems in a region with regular geometry, like the circle in our exercise.

The method works by assuming that a multi-variable function, such as \(u(r, \theta)\), can be expressed as the product of functions each in a single variable, \(u(r, \theta) = R(r)\Theta(\theta)\). This assumption transforms a PDE into a set of ordinary differential equations (ODEs), which are often easier to solve.

By substituting this assumed solution form back into the original PDE and separating the variables, we arrive at two ODEs—one for \(r\) and another for \(\theta\), each equated to a separation constant. This process ultimately simplifies the problem and helps in finding solutions that fit both the PDE and the boundary conditions.

In relation to the exercise, we use separation of variables to turn Laplace’s equation into two simpler equations. The function \(g(\theta)\), which defines the Neumann boundary condition, plays a vital role in determining the separation constant and the overall solution format. However, for this method to be applicable, \(g(\theta)\) has to satisfy certain integrality conditions, reinforcing that not every PDE with a Neumann boundary condition can be solved using separation of variables.

Fourier Series Representation

Fourier series are a tool to represent periodic functions as an infinite sum of sines and cosines, or complex exponentials. They are crucial in solving PDEs with periodic boundary conditions, like our Neumann problem.

In our exercise, we express \(g(\theta)\), which defines the Neumann boundary condition, as a Fourier series. This mathematical representation allows us to decompose the boundary condition into an infinite series of sinusoidal components, each associated with a specific frequency. In essence, \(g(\theta)\) gets written as \(\sum_{n=-\infty}^{\infty} c_ne^{in\theta}\), with \(c_n\) representing the amplitude of the component oscillating at the frequency \(n\).

Finding the correct Fourier series representation is essential for aligning the boundary condition with the solutions obtained from the separation of variables. The ability to represent \(g(\theta)\) in this way is particularly reliant on the discrete nature of the eigenvalues that arise from the separated angular equation. This series, when matched with the boundary condition, helps us determine the coefficients for the solution.

From our problem, for a solution to exist, the Fourier coefficient corresponding to the zeroth term (average value of the function over the period) should be zero (\(c_0=0\)). This reflects the physical necessity that there must be no net flow across the boundary, meshing nicely with the interpretation of the Neumann condition within the context of physical problems like heat flow or fluid dynamics.