Question

Show that separation of variables \(u(r, \theta)=R(r) \Theta(\theta)\) in the problem in polar coordinates for the domain \(r<1\) : $$ \nabla^{2} u+\lambda u \equiv \frac{1}{r^{2}}\left[r \frac{\partial}{\partial r}\left(r \frac{\partial u}{\partial r}\right)+\frac{\partial^{2} u}{\partial \theta^{2}}\right]+\lambda u=0, \quad u(1, \theta)=0 . $$ leads to the problems: $$ \begin{gathered} \left(r R^{\prime}\right)^{\prime}+\left(\lambda r-\frac{\mu}{r}\right) R=0, \quad R(1)=0, \\ \Theta^{\prime \prime}+\mu \Theta=0 . \end{gathered} $$ If we require that \(u(r, \theta)\) be continuous in the circle \(r \leq 1\), then \(\Theta(\theta)\) must be periodic in \(\theta\), with period \(2 \pi\). Show that this implies that \(\mu=m^{2}(m=0,1,2,3, \ldots)\) and that

Short answer

Question: Show that using the method of separation of variables, the partial differential equation in polar coordinates can be separated into two ordinary differential equations involving the radial and angular functions. Also, show that the constant mu is equal to m^2 where m is an integer (m = 0, 1, 2, 3, ...) if the angular function is periodic in theta with a period of 2π . Solution: The separation of variables method was applied to the given partial differential equation, resulting in two ordinary differential equations involving radial and angular functions. Additionally, it was shown that the constant mu equals m^2, where m is an integer if the angular function is periodic in theta with a period of 2π.

Step-by-step solution

Apply separation of variables to the PDE

Assume a solution of the form \(u(r, \theta) = R(r) \Theta(\theta)\). Substitute this into the given PDE: $$ \frac{1}{r^{2}}\left[r \frac{\partial}{\partial r}\left(r \frac{\partial(R \Theta)}{\partial r}\right)+\frac{\partial^{2} (R \Theta)}{\partial \theta^{2}}\right]+\lambda(R \Theta) = 0 $$

Simplify and separate the variables

Simplify the equation and separate the variables: $$ \frac{1}{r^{2}}\left[r(R' \Theta + r R''\Theta)+R \Theta''\right]+ \lambda R \Theta = 0 $$ Divide by \(R \Theta\): $$ \frac{1}{r^{2}}\left[\frac{r(R' + r R'')}{R}+\frac{\Theta''}{\Theta}\right] + \lambda = 0 $$ Separate the variables: $$ \frac{r(R' + r R'')}{R} = -\lambda r^{2} - \frac{\Theta''}{\Theta} $$ We get two separate equations: $$ r(R' + r R'') + \lambda r^{3} R = 0 \\ \Theta '' + \mu \Theta = 0 \\ $$ Where \(\mu = -\lambda r^{2}\).

Derive the first ODE

We have: $$ r(R' + r R'') + \lambda r^{3} R = 0 $$ Divide by \(r\) and rearrange: $$ \left(r R'\right)' + \left(\lambda r - \frac{\mu}{r}\right) R = 0 $$ This is the first ODE along with its boundary condition: $$ \left(r R'\right)' + \left(\lambda r - \frac{\mu}{r}\right) R = 0, \quad R(1) = 0 $$

Derive the second ODE

We already have the second ODE: $$ \Theta '' + \mu \Theta = 0 $$ The given conditions require that \(\Theta(\theta)\) be continuous and periodic in \(\theta\) with period \(2 \pi\). This means that solutions for \(\Theta(\theta)\) must be a linear combination of \(\sin(m \theta)\) and \(\cos(m \theta)\) where \(m = 0, 1, 2, 3, ...\). Since \(\Theta '' + \mu \Theta = 0\), we have: $$ m^2 \Theta(\theta) = \Theta ''(\theta) + \mu \Theta(\theta) = 0 $$ Thus, \(\mu = m^2\). We have now shown that the separation of variables leads to the given ODEs and that \(\mu = m^2\) for \(m = 0, 1, 2, 3, ...\) when \(\Theta(\theta)\) is periodic in \(\theta\) with period \(2 \pi\).

Key concepts

Partial Differential Equations

Partial Differential Equations (PDEs) are a type of mathematical equation involving rates of change with respect to continuous variables. In essence, PDEs describe a wide range of phenomena such as heat conduction, sound, fluids, and elasticity. Solving a PDE involves finding a function that satisfies the equation. One successful technique for this purpose is the method of separation of variables.

Separation of variables is a powerful, yet intuitive technique used to solve PDEs. It involves splitting a complex PDE into simpler ordinary differential equations (ODEs) by assuming that the solution can be expressed as the product of functions that each depend on a single coordinate. In the given exercise, applying this technique reduces the original PDE in polar coordinates to two simpler, more manageable ODEs for the radial and angular parts. It is widely employed because it transforms the problem of solving a PDE into the task of solving ODEs, which are better understood and for which more solution techniques are available.

Eigenvalue Problems

Eigenvalue problems are fundamental in the field of mathematics and physics, as they often arise when solving linear differential equations and systems of differential equations. An eigenvalue problem looks for scalar values, known as eigenvalues, and associated functions, known as eigenfunctions, that satisfy certain conditions. In the context of differential equations, these conditions usually involve a differential operator. For instance, in the exercise, \( \Theta'' + \mu \Theta = 0 \) constitutes an eigenvalue problem where \( \mu \) represents the eigenvalue and \( \Theta(\theta) \) represents the eigenfunction.

The process of solving an eigenvalue problem involves finding values of \( \mu \) for which there exist non-trivial solutions \( \Theta(\theta) \) that satisfy the given boundary conditions. The stipulation that solutions be continuous and periodic within the domain imposes further constraints on the possible eigenvalues, relegating them to a discrete set. In our case, the periodicity condition requires that \( \mu = m^2 \) for non-negative integers \( m \) — a characteristic that allows us to proceed with solving the ODE in a more structured form.

Sturm-Liouville Theory

Sturm-Liouville theory is a framework that generalizes the concept of the eigenvalue problem to a broader context. This theory is used to solve a class of differential equations known as Sturm-Liouville problems, which are a type of second-order linear ODE with variable coefficients. It gives formal conditions under which the eigenfunctions form an orthogonal set and how eigenvalues are systematically found.

In the educational exercise, we are indirectly dealing with a Sturm-Liouville problem when we consider the equation \( (r R')' + (\lambda r - \frac{\mu}{r}) R = 0 \), with certain boundary conditions. This type of equation falls into Sturm-Liouville theory since it has the form of a second-order linear ordinary differential equation. The broader significance of these problems is that they not only allow the determination of eigenvalues and eigenfunctions but also they reveal a pattern within the solutions that is representative of the physical properties of the system described by the PDE. Learning Sturm-Liouville theory is crucial for students tackling advanced topics in mathematics and physics because it underpins the solutions to numerous practical problems and informs the behavior of various physical systems.