Question

Define the bounded formal solution of $$ \begin{aligned} u_{r r}+\frac{1}{r} u_{r}+\frac{1}{r^{2}} u_{\theta \theta}=0, & 0

Short answer

Answer: The general solution for a polar Laplace equation with boundary conditions is given by: $$ u(r, \theta) = \sum_{n=1}^{\infty}\left( A_n r^{n\pi/\gamma} \cos(n\pi \frac{\theta}{\gamma})\right) $$ where \(A_n\) are determined by the Fourier cosine series of the boundary function \(f(\theta)\).

Step-by-step solution

Separate Variables

Let us assume a product solution of the form \(u(r, \theta) = R(r)\Theta(\theta)\). Substitute this into the polar Laplace equation: $$ \frac{R''(r)\Theta(\theta)}{R(r)\Theta(\theta)} + \frac{1}{r} \frac{R'(r)\Theta(\theta)}{R(r)\Theta(\theta)} + \frac{1}{r^2}\frac{R(r)\Theta''(\theta)}{R(r)\Theta(\theta)} = 0 $$ Now, divide through by the product \(R(r)\Theta(\theta)\) to get: $$ \frac{R''(r)}{R(r)} + \frac{1}{r} \frac{R'(r)}{R(r)} + \frac{1}{r^2}\frac{\Theta''(\theta)}{\Theta(\theta)} = 0 $$ Notice that the first two terms have only radial dependence, while the last term has only angular dependence. Thus, we can separate the equation into two independent equations, one for \(R(r)\) and one for \(\Theta(\theta)\).

Solve for the Radial Part

Set the radial part of the separated equation equal to a constant \(-\lambda ^ 2\): $$ \frac{R''(r)}{R(r)} + \frac{1}{r} \frac{R'(r)}{R(r)} = -\lambda ^ 2 $$ To solve this equation for \(R(r)\), we can multiply by \(r^2 R(r)\) and then rewrite the equation as: $$ r^2 R''(r) + r R'(r) = -\lambda^2 r^2 R(r) $$ This is an Euler-Cauchy equation, which has solutions in the form of \(R(r) = r^\alpha\). Upon solving this equation, we find: $$ R(r) = A r^\lambda + B r^{-\lambda} $$ where \(A\) and \(B\) are constants of integration.

Solve for the Angular Part

Set the angular part of the separated equation equal to \(\lambda^2\): $$ \frac{\Theta''(\theta)}{\Theta(\theta)} = \lambda^2 $$ This is an ordinary differential equation, and the general solution can be written as: $$ \Theta(\theta) = C\cos(\lambda \theta) + D\sin(\lambda \theta) $$ where \(C\) and \(D\) are constants of integration.

Apply Boundary Conditions and Construct the General Solution

To satisfy the boundary conditions, we will first set \(B = 0\) to ensure the solution is bounded at \(r = 0\). Next, consider the boundary conditions for \(\Theta(\theta)\): $$ u_{\theta}(r, 0) = 0,\quad u(r,\gamma) = 0 $$ The first boundary condition is met by setting \(D = 0\). The second boundary condition is satisfied by having: $$ C\cos(\lambda \gamma) = 0 $$ This implies that \(\lambda \gamma = n\pi\) for integer values of \(n\). Thus, the solution for \(\Theta(\theta)\) becomes: $$ \Theta(\theta) = \sum_{n=1}^{\infty} C_n \cos(n\pi \frac{\theta}{\gamma}) $$ Finally, the boundary condition at \(r = \rho\) must be enforced: $$ u(\rho, \theta) = f(\theta) = \sum_{n=1}^{\infty} A_n \rho^\lambda \cos(n\pi \frac{\theta}{\gamma}) $$ Now, we obtain the coefficients \(A_n\) by taking the Fourier cosine series for the given function \(f(\theta)\): $$ A_n = \frac{2}{\gamma}\int_0^{\gamma}{f(\theta)\cos(\frac{n\pi \theta}{\gamma}) d \theta} $$ Finally, the general solution of the polar Laplace equation can be written as: $$ u(r, \theta) = \sum_{n=1}^{\infty}\left( A_n r^{n\pi/\gamma} \cos(n\pi \frac{\theta}{\gamma})\right) $$ where \(A_n\) are determined by the Fourier cosine series of the boundary function \(f(\theta)\).

Key concepts

Separation of Variables

The technique known as separation of variables is an extremely powerful method in mathematics, particularly useful for solving partial differential equations (PDEs). Imagine we have a complex problem that includes two or more variables, such as time and space.

Separation of variables involves assuming that the solution to a PDE can be written as the product of functions, each depending on a single variable. This approach transforms the original PDE into simpler, ordinary differential equations (ODEs) that can be solved independently. For the exercise given, we assume the solution to the polar Laplace equation can be expressed as the product of a radial function, R(r), and an angular function, Θ(θ).

By substituting this product into the PDE and organizing terms based on their variable dependence, we effectively separate variables. The power of this method lies in its ability to break down a seemingly insurmountable equation into manageable pieces, especially when dealing with symmetrical domains, as often encountered in physics and engineering problems involving circular or spherical coordinates.

Euler-Cauchy Equation

The Euler-Cauchy equation, also known as the Euler's differential equation, is a type of ordinary differential equation (ODE) with variable coefficients that appear in power-law form. It is of the form:

\[ x^n y^{(n)} + a_{n-1}x^{n-1}y^{(n-1)} + \dots + a_0 y = 0 \] where n denotes the order of the ODE and y^{(n)} represents the n-th derivative of y.

To tackle this kind of equation, as seen in the radial part of our polar Laplace equation, a common approach is to propose a solution of the form y = x^k, where k is a constant to be determined. It is essential to recognize this form of the ODE because it hints at specific solution strategies, notably looking for solutions that are powers of the independent variable.

The characteristic equation derived from the Euler-Cauchy equation helps us find the powers that will be part of the solution, and from there, we determine the general solution. In our exercise, the structural similarity to the Euler-Cauchy equation gives us a direct route to the correct form of the radial solution of the problem.

Fourier Cosine Series

The Fourier cosine series is a way to represent a function as an infinite sum of cosines. It's particularly useful in solving boundary value problems where the function is even and periodic.

Mathematically, a Fourier cosine series for a function f(x) defined on an interval [0, L] is written as:

\[ f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos\left(\frac{n\pi x}{L}\right) \] where the coefficients a_n are determined by the following integral: \[ a_n = \frac{2}{L} \int_0^{L} f(x) \cos\left(\frac{n\pi x}{L}\right) dx \]

In our exercise, the function f(θ) is given as the boundary condition of the solution along the arc of the circle at radius ρ. The Fourier cosine series allows us to express this function as a sum of cosine terms, which, when multiplied by the radial solution, gives us the full solution to the original PDE. The Fourier coefficients are precisely the A_n constants that tie together the radial and angular solutions to model the boundary behavior described by f(θ). This is crucial in achieving an accurate and complete representation of the physical phenomena we're modeling.