Question

Show that the separation of variables solution for the Dirichlet problem (30) is $$ u(r, \theta)=\frac{a_{0}}{2}+\frac{A_{0}}{2} \ln r+\sum_{n=1}^{\infty} r^{n}\left(a_{n} \cos n \theta+b_{n} \sin n \theta\right)+\sum_{n=1}^{\infty} r^{-n}\left(A_{n} \cos n \theta+B_{n} \sin n \theta\right) . $$

Short answer

#Solution Summary# We started with the Laplace equation in polar coordinates, performed separation of variables, solved the resulting ODEs, and combined the radial and angular solutions to obtain the separated solution for the Dirichlet problem. The solution is given by the sum of all possible values of n: $$ u(r, \theta)=\frac{a_{0}}{2}+\frac{A_{0}}{2} \ln r+\sum_{n=1}^{\infty} r^{n}\left(a_{n} \cos n \theta+b_{n} \sin n \theta\right)+\sum_{n=1}^{\infty} r^{-n}\left(A_{n} \cos n \theta+B_{n} \sin n \theta\right) . $$

Step-by-step solution

Identify the given PDE and boundary conditions

The Dirichlet problem is the Laplace equation with Dirichlet boundary conditions in a polar coordinate system. The Laplace equation in polar coordinates is given by: $$ \nabla^2 u = \frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial u}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} u}{\partial \theta^{2}}=0. $$ The Dirichlet boundary condition states that the solution u(r, θ) is specified along the boundary of the domain.

Perform separation of variables on the PDE

We assume the solution can be written as a product of radial and angular parts, i.e., \(u(r, \theta) = R(r)\Theta(\theta)\). Substituting this into the Laplace equation and dividing by \(R(r)\Theta(\theta)\) yields: $$ \frac{1}{R}\left[\frac{\partial}{\partial r}\left(r \frac{\partial R}{\partial r}\right)\right]+\frac{1}{r^{2}}\frac{1}{\Theta}\frac{\partial^{2} \Theta}{\partial \theta^{2}}=0. $$ This equation is now separable, with the radial part depending only on r and the angular part depending only on θ.

Solve the resulting ODEs

We can separate the radial and angular parts by setting the angular term equal to \(-n^2\) (where n is an integer), while the radial part equals to \(r^{2}n^2\): $$ \frac{1}{\Theta}\frac{\partial^{2} \Theta}{\partial \theta^{2}}=-n^{2}, $$ which has the general solution: $$ \Theta_n(\theta) = A_n \cos(n\theta) + B_n \sin(n\theta). $$ For the radial part, $$ \frac{1}{R}\frac{\partial}{\partial r}\left(r \frac{\partial R}{\partial r}\right) = r^{2}n^{2}, $$ which has the general solution: $$ R_n(r) = a_n r^{n} + b_n r^{-n}. $$ We can combine the solutions for the radial and angular parts into a single expression: $$ u_n(r, \theta) = R_n(r)\Theta_n(\theta) = r^{n}\left(a_{n} \cos n \theta+b_{n} \sin n \theta\right)+r^{-n}\left(A_{n} \cos n \theta+B_{n} \sin n \theta\right). $$

Construct the solution using the method of superposition

Using the method of superposition, the solution for the Dirichlet problem is given by the sum of all possible values of n: $$ u(r, \theta)=\frac{a_{0}}{2}+\frac{A_{0}}{2} \ln r+\sum_{n=1}^{\infty} r^{n}\left(a_{n} \cos n \theta+b_{n} \sin n \theta\right)+\sum_{n=1}^{\infty} r^{-n}\left(A_{n} \cos n \theta+B_{n} \sin n \theta\right) . $$ This is the desired separated solution for the Dirichlet problem (30).

Key concepts

Laplace Equation in Polar Coordinates

The Laplace equation is a fundamental equation in mathematical physics that describes the behavior of scalar potential fields, such as electrostatic and gravitational potentials. In polar coordinates, the Laplace equation is reformulated to take advantage of the symmetry in problems that are naturally circular or radial.

When dealing with the Laplace equation in polar coordinates, we use the form
\[abla^2 u = \frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial u}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} u}{\partial \theta^{2}}=0.\]This expression incorporates the partial derivatives with respect to the radial distance, \( r \), and the angle, \( \theta \), and equates the sum to zero. Problems in circular domains or with radial symmetry often lead to simpler, more tractable solutions when expressed in polar coordinates.

When you encounter a problem requiring the use of the Laplace equation in polar coordinates, remember that it is a restatement of the more familiar Cartesian coordinates version, adapted to better suit problems with rotational symmetry.

Separation of Variables

Separation of variables is a mathematical method used to solve partial differential equations (PDEs). The key idea is to assume that the solution can be broken down into a product of functions, each dependent on a single coordinate.

In the context of the Laplace equation in polar coordinates, we express the solution as \(u(r, \theta) = R(r)\Theta(\theta)\). This assumption allows one to transform the original PDE into two ordinary differential equations (ODEs), each in a single variable. By separating the variables, the complex PDE problem is turned into simpler, more manageable pieces.

Mathematically, this is justifiable when the resulting expressions only depend on their respective variables, which means we can equate each part to a constant. This approach leads to solutions that are sinusoidal or exponential in nature, characterized by the boundary conditions of the problem.

Boundary Value Problems

Boundary value problems are a class of mathematical problems in which we seek a solution to a differential equation under a set of constraints specified on the boundary of the domain. The Dirichlet problem, for example, requires that the solution to the Laplace equation satisfies prescribed values on the boundary.

It's essential to understand that the boundary conditions significantly influence the uniqueness and existence of a solution. For the Dirichlet problem in polar coordinates, boundary conditions are given for radial and angular coordinates at certain limits, such as the edge of a disk.

To successfully approach a boundary value problem, one must carefully analyze the stated conditions and accordingly modify the form of the solution. It involves a subtle mix of intuition and mathematical rigor to apply suitable methods, such as separation of variables, to find satisfactory solutions that respect the imposed boundary behavior.

Method of Superposition

The method of superposition is a crucial concept in linear systems, which holds that the total response in a linear system to multiple stimuli is the sum of the responses that would have been caused by each stimulus individually.

In solving the Dirichlet problem for the Laplace equation, we use the method of superposition to construct the general solution by adding the individual solutions corresponding to each value of \(n\) obtained from the separation of variables. Each term in the solution represents a mode of the physical system, and by summing them, we cater to all possible variations in the boundary conditions.

The resulting infinite series, which includes terms for all the integer values of \(n\), encompasses all the necessary components to match the boundary values given by the problem. Adjusting the coefficients \(a_n, b_n, A_n,\) and \(B_n\), allows the series to converge to the exact shape required by the boundary conditions. This manipulation is what eventually leads to the specific function that satisfies both the Laplace equation and the boundary constraints.