Question

Find the solution \(u(r, \theta)\) of Laplace's equation $$ u_{r r}+(1 / r) u_{r}+\left(1 / r^{2}\right) u_{\theta \theta}=0 $$ outside the circle \(r=a\) also satisfying the boundary condition $$ u(a, \theta)=f(\theta), \quad 0 \leq \theta<2 \pi $$ on the circle. Assume that \(u(r, \theta)\) is single-valued and bounded for \(r>a\)

Short answer

The final form of the solution for the Laplace equation in polar coordinates, satisfying the given boundary condition, is: $$ u(r, \theta) = \sum_{n=0}^{\infty} \left[A_nr^{\alpha_1} + B_nr^{\alpha_2}\right] (C_n \cos{n \theta} + D_n \sin{n \theta}) $$

Step-by-step solution

Separation of Variables

Assume that the solution \(u(r, \theta)\) can be expressed as a product of two independent functions: \(u(r, \theta) = R(r) \Theta(\theta)\). Substitute this expression into Laplace's equation: $$ R''(r) \Theta(\theta) + \frac{1}{r} R'(r) \Theta(\theta) + \frac{1}{r^2} R(r) \Theta''(\theta) = 0 $$

Separate the Equations

Divide both sides of the equation by \(R(r) \Theta(\theta)\) to separate the variables: $$ \frac{R''(r)}{R(r)} + \frac{1}{r} \frac{R'(r)}{R(r)} = -\frac{\Theta''(\theta)}{r^2 \Theta(\theta)} $$ Since the left-hand side depends only on \(r\) and the right-hand side depends only on \(\theta\), both sides must be equal to a constant, which we denote by \(-k^2\): $$ \frac{R''(r)}{R(r)} + \frac{1}{r} \frac{R'(r)}{R(r)} = -k^2, \qquad \frac{\Theta''(\theta)}{\Theta(\theta)} = k^2 r^2 $$

Solve the Equation for \(\Theta(\theta)\)

To solve the equation for \(\Theta(\theta)\), notice that it is a second-order linear ordinary differential equation with constant coefficients. This equation has the general solution: $$ \Theta(\theta) = A \cos{k \theta} + B \sin{k \theta} $$ where \(A\) and \(B\) are constants.

Solve the Equation for \(R(r)\)

The equation for \(R(r)\) is a second-order linear ordinary differential equation with variable coefficients. To solve this equation, we use the substitution \(R(r) = r^\alpha\) and rewrite the equation as: $$ \alpha (\alpha - 1) r^{\alpha - 2} + \alpha r^{\alpha - 1} = -k^2 r^\alpha \quad \Rightarrow \quad \alpha^2 - \alpha + k^2 = 0 $$ The above equation is a quadratic equation, with one root \(\alpha_1 = \sqrt{k^2 + 1}\) and another root \(\alpha_2 = -\sqrt{k^2 + 1}\). We thus have two possible solutions for \(R(r)\), in the form of \(r^{\alpha_1}\) and \(r^{\alpha_2}\). The general solution \(R(r)\) can be written as a linear combination of the two solutions: $$ R(r) = C_1 r^{\alpha_1} + C_2 r^{\alpha_2} $$ where \(C_1\) and \(C_2\) are constants.

Combine the Solutions and Apply the Boundary Condition

Combine the solutions for \(R(r)\) and \(\Theta(\theta)\) to find the total solution for \(u(r, \theta)\): $$ u(r, \theta) = \left(C_1 r^{\alpha_1} + C_2 r^{\alpha_2}\right) \left(A \cos{k \theta} + B \sin{k \theta}\right) $$ Now, apply the given boundary condition: $$ u(a, \theta) = \left(C_1 a^{\alpha_1} + C_2 a^{\alpha_2}\right) \left(A \cos{k \theta} + B \sin{k \theta}\right) = f(\theta) $$ Since \(u(a, \theta)\) is expected to be a Fourier series, matching terms in the above series representation, we'll find the coefficients \(A_n\) and \(B_n\), and express \(u(a, \theta)\) as an infinite sum.

Rewriting the Series Representation

The series representation for the solution outside the circle can now be expressed as: $$ u(r, \theta) = \sum_{n=0}^{\infty} \left[A_nr^{\alpha_1} + B_nr^{\alpha_2}\right] (C_n \cos{n \theta} + D_n \sin{n \theta}) $$ This is the final form of the solution for the Laplace equation in polar coordinates, satisfying the given boundary condition.

Key concepts

Laplace's Equation

Laplace's Equation is a fundamental equation in the field of partial differential equations. It is widely used across physics and engineering to describe phenomena such as electrostatics, fluid dynamics, and more. The equation in its most basic form is given by \( abla^2 u = 0 \), where \( abla^2 \) is the Laplacian operator. This operator can be expanded in different coordinate systems, and in the context of polar coordinates, it appears as \( u_{rr} + (1/r) u_r + (1/r^2) u_{\theta\theta} = 0 \). The significance of Laplace's equation lies in its ability to model steady-state systems, where changes over time are not considered.
In the given problem, we are solving Laplace's equation outside a circle, which highlights the boundary value problem aspect. This means we're looking for a solution that satisfies given conditions on the boundary of the domain, here defined by the circle \( r = a \). Laplace's equation assumes that within these constraints, the solution will be smooth and well-behaved, offering a precise way to describe the potential field outside the circle.

Separation of Variables

Separation of Variables is a mathematical technique used to reduce complex partial differential equations into simpler, ordinary differential equations. It involves expressing the solution as a product of functions, each of which depends on only one of the variables. In the context of Laplace's equation in polar coordinates, we suppose the solution has the form \( u(r, \theta) = R(r) \Theta(\theta) \). By substituting this product into our equation, it allows us to separate the equation into two distinct parts: one depending solely on \( r \) and the other on \( \theta \).
The separation involves dividing each term by \( R(r) \Theta(\theta) \), leading to two independent equations: one for \( R(r) \) and one for \( \Theta(\theta) \). This split is possible because the equation sets ensure that both sides must equal a constant, denoted as \( -k^2 \). This method simplifies solving the original partial differential equation by converting it into two easier to handle ordinary differential equations.

Fourier Series

A Fourier Series is a way to represent a function as an infinite sum of sinusoidal functions, which are sines and cosines. This is particularly useful in boundary value problems where the function is periodic, as it allows us to express complicated periodic functions in terms of simple waves. In the given boundary condition \( u(a, \theta) = f(\theta) \), the function \( f(\theta) \) can be expressed as a Fourier series.
This series form is given by \( f(\theta) = \sum_{n=0}^\infty \left( A_n \cos{n\theta} + B_n \sin{n\theta} \right) \). Here, \( A_n \) and \( B_n \) are the Fourier coefficients, which can be determined by integrating over the interval \( 0 \leq \theta < 2\pi \). By expressing \( f(\theta) \) in this form, it becomes easier to determine the solution to the boundary value problem using superposition of solutions.

Polar Coordinates

Polar Coordinates offer a different way of representing points in a plane, particularly useful in problems with circular or radial symmetry. Unlike Cartesian coordinates \((x, y)\), polar coordinates use \( (r, \theta) \), where \( r \) is the radial distance from the origin and \( \theta \) is the angle measured from the positive x-axis.
In boundary value problems involving circular domains, such as the one described here, polar coordinates simplify the mathematics. The Laplace's equation in polar form \( u_{rr} + (1/r) u_r + (1/r^2) u_{\theta\theta} = 0 \) directly takes into account the symmetry. This makes it easier to apply boundary conditions and solve the equation using techniques like separation of variables. Polar coordinates are particularly effective when paired with boundary conditions on a circle, as it aligns naturally with the geometry of the problem.