Question

Show that the substitution \(u=R(\rho) \Phi(\phi) \Theta(\theta)\) in the problem in spherical coordinates for the domain \(\rho<1\) : $$ \begin{gathered} \nabla^{2} u+\lambda u \equiv \frac{1}{\rho^{2} \sin ^{2} \phi}\left[\sin ^{2} \phi \frac{\partial}{\partial \rho}\left(\rho^{2} \frac{\partial u}{\partial \rho}\right)+\sin \phi \frac{\partial}{\partial \phi}\left(\sin \phi \frac{\partial u}{\partial \phi}\right)+\frac{\partial^{2} u}{\partial \theta^{2}}\right]+\lambda u=0, \\ u(\rho, \phi, \theta)=0 \quad \text { for } \rho=1 . \end{gathered} $$ leads to the separate Sturm-Liouville problems: $$ \begin{gathered} \left(\rho^{2} R^{\prime}\right)^{\prime}+\left(\lambda \rho^{2}-\alpha\right) R=0, \quad R=0 \quad \text { for } \rho=1 \\ \left(\sin \phi \Phi^{\prime}\right)^{\prime}+(\alpha \sin \phi-\beta \csc \phi) \Phi=0, \quad \Theta^{\prime \prime}+\beta \Theta=0 \end{gathered} $$ Here \(\alpha, \beta\), and \(\lambda\) are characteristic values to be determined. The condition that \(u\) be continuous throughout the sphere requires \(\Theta\) to have period \(2 \pi\), so that \(\beta=k^{2}(k=0\), \(1,2, \ldots)\) and \(\Theta_{k}(\theta)\) is a linear combination of \(\cos k \theta\) and \(\sin k \theta\). When \(\beta=k^{2}\), it can be shown that continuous solutions of the second problem for \(0 \leq \phi \leq \pi\) are obtainable only when \(\alpha=n(n+1), k=0,1, \ldots, n\), and \(\Phi\) is a constant times \(P_{n, k}(\cos \phi)\), where $$ P_{n, k}(x)=\left(1-x^{2}\right)^{\frac{1}{2}} \frac{d^{k}}{d x^{k}} P_{n}(x) $$ and \(P_{n}(x)\) is the \(n\)th Legendre polynomial. When \(\alpha=n(n+1)(n=0,1,2, \ldots)\), the first problem has a solution continuous for \(\rho=0\) only when \(\lambda\) is one of the roots \(\lambda_{n+\lfloor 1,1}, \lambda_{n+\frac{1}{2}, 2}, \ldots\) of the function \(J_{n+\frac{1}{!}}(\sqrt{x})\), where \(J_{n+\frac{1}{2}}(x)\) is the Bessel function of order \(n+\frac{1}{2}\); for each such \(\lambda\) the solution is a constant times \(\rho^{-\frac{1}{2}} J_{n+\frac{1}{2}}(\sqrt{\lambda} \rho)\). Thus one obtains the characteristic functions \(\rho^{-\frac{1}{2}} J_{n+\frac{1}{1}}(\sqrt{\lambda} \rho) P_{n, k}(\cos \phi) \cos k \theta, \rho^{-\frac{1}{2}} J_{n+\frac{1}{2}}(\sqrt{\lambda} \rho) P_{n, k}\) \((\cos \phi) \sin k \theta\), where \(n=0,1,2, \ldots, k=0,1, \ldots, n\), and \(\lambda\) is chosen as above for each \(n\). It can be shown that these functions form a complete orthogonal system for the spherical region \(\rho \leq 1\).

Short answer

Short Answer: By substituting the given function into the Laplacian, we were able to separate the variables and rewrite the equation into three separate equations. We analyzed each equation and found conditions for continuity and orthogonal properties. Specifically, we showed that when \(\beta=k^2\) and \(\alpha=n(n+1)\), the given functions form a complete orthogonal system for the spherical region \(\rho \leq 1\).

Step-by-step solution

Substitution

Begin by substituting \(u = R(\rho)\Phi(\phi)\Theta(\theta)\) into the given equation: $$ \nabla^2 u + \lambda u \equiv \frac{1}{\rho^2 \sin^2\phi} \bigg( \sin^2 \phi \frac{\partial}{\partial \rho} \left( \rho^2 \frac{\partial (R\Phi\Theta)}{\partial \rho} \right) + \sin\phi \frac{\partial}{\partial \phi} \left( \sin\phi \frac{\partial (R\Phi\Theta)}{\partial \phi} \right) + \frac{\partial^2 (R\Phi\Theta)}{\partial \theta^2} \bigg) + \lambda (R\Phi\Theta) = 0 $$

Step 2:Separate Variables

Separate the variables by dividing through by \(R(\rho)\Phi(\phi)\Theta(\theta)\): $$\frac{1}{\rho^2 \sin^2\phi} \bigg(\sin^2 \phi\left(\frac{\partial}{\partial \rho}\left( \rho^2 \frac{\partial R / \partial \rho}\right)\right)\frac{\Phi\Theta}{R}+\sin\phi \left(\frac{\partial}{\partial \phi}\left(\sin\phi\frac{\partial \Phi / \partial \phi}\right)\right)\frac{R\Theta}{\Phi}+\frac{\Theta''}{\Theta}\bigg)+\lambda=0$$

Rewrite Equation

We will rewrite the equation in the following way: $$\frac{1}{R} \left(\rho^2 R'\right)' + \lambda \rho^2 = \alpha$$, $$\frac{1}{\sin\phi\Phi}\left(\sin\phi\Phi'\right)'+\frac{\alpha\sin\phi}{\sin\phi}=\beta$$, $$\Theta'' = -\beta \Theta$$

Analyze the Third Equation

Start by solving for \(\beta\) in the third equation: $$\Theta'' + \beta \Theta = 0$$ Since the solution must be continuous and \(\Theta\) have a period of \(2\pi\), this gives us the condition: $$\beta = k^2 (k=0, 1, 2, \dots)$$

Analyze the First Equation

Proceed with analyzing the first equation, finding the characteristic values \(\alpha\) to get: $$(\rho^2 R')' + (\lambda \rho^2 - \alpha)R = 0$$

Analyze the Second Equation

For the second equation, finding the characteristic values \(\alpha\), we get: $$(\sin\phi\Phi')' + (\alpha\sin\phi - \beta\csc\phi)\Phi = 0$$

Discuss the Conditions

From our analysis, we conclude that when \(\beta = k^2\) and \(\alpha = n(n+1)\), we get that \(\Phi\) is a constant times \(P_{n, k}(\cos \phi)\). Similarly, when \(\alpha = n(n+1)\) and the first equation is continuous for \(\rho = 0\), we find that \(\lambda\) is one of the roots as described in the exercise. Thus, we obtain a complete orthogonal system for the spherical region \(\rho \leq 1\) as stated in the exercise.

Key concepts

Spherical Coordinates

In mathematics and physics, spherical coordinates are a system of coordinates that are used to define the position of a point in three-dimensional space. This system is analogous to cylindrical coordinates, but with an extra layer of complexity due to the spherical symmetry.

The spherical coordinates \( (\rho, \phi, \theta) \) consist of:

• \(\rho\): The radial distance from the origin to the point.
• \(\phi\): The polar angle measured from the positive z-axis.
• \(\theta\): The azimuthal angle measured from the positive x-axis in the x-y plane.

They are particularly useful in solving problems that have inherent spherical symmetry, like those involving the gravitational or electric fields around spherical objects. In the context of Sturm-Liouville problems, spherical coordinates allow us to express the Laplace's equation in a form that lets us apply the method of separation of variables effectively.

Laplace's Equation

Laplace's equation is a second-order partial differential equation (PDE) that appears frequently in physics and engineering, especially in the areas of electromagnetism and fluid dynamics. The equation is given by \[ abla^{2} u = 0 \.\]

It is a special case of the more general Poisson's equation, which includes a source term. The absence of the source term in Laplace's equation implies that it describes a potential field in free space.

In spherical coordinates, Laplace's equation can be written specifically for problems with spherical symmetry. This formulation is crucial because it brings out the separable nature of the equation when expressed in terms of \( \rho, \phi, \theta \) coordinates, which is the first step in finding the solution to many physical problems.

Separation of Variables

The technique of separation of variables is a method to solve certain partial differential equations, by breaking them into simpler, solvable ordinary differential equations. The basic idea is to assume that the solution can be written as a product of functions, each depending only on one of the independent variables.

This assumption transforms the original PDE into a set of ODEs. After the separation, each ODE can be solved separately, and the solutions to the original problems are constructed from the solutions of the ODEs. This approach is particularly effective for linear PDEs with separable variables, such as Laplace's and heat equations. It is a key step in solving Sturm-Liouville problems, which include a broad class of important eigenvalue problems in mathematical physics.

Legendre Polynomials

Legendre polynomials are a series of orthogonal polynomials that arise in solving the Laplace's equation in spherical coordinates using the separation of variables method. They are denoted as \( P_n(x) \) and the nth polynomial is of degree n.

The polynomials have many important properties. One is their orthogonality: \[ \int_{-1}^{1} P_m(x) P_n(x)\,dx = 0 \.\] if \( m eq n \.\) Another is that they satisfy the Legendre differential equation, which appears when solving Laplace's equation on a sphere. These polynomials are essential tools for expressing the angular part of solutions to problems in quantum mechanics, gravitational fields, and electric potentials around spherically symmetric bodies.

Bessel Functions

Bessel functions, denoted by \( J_n(x) \) for integer orders n, emerge from the radial part of Laplace's equation in cylindrical and spherical coordinates under separation of variables. They are a central part of the solution in many physical problems, particularly those involving vibrational modes, heat conduction in cylindrical objects, and electromagnetic wave propagation.

The Bessel functions are solutions to Bessel's differential equation, which is encountered when solving radial parts of PDEs. They are also oscillatory and exhibit behavior similar to sine and cosine functions for large arguments. Their orthogonality and completeness make them crucial in expanding functions in a manner analogous to Fourier series expansions.

Orthogonal Functions

In mathematics, functions are considered orthogonal if their inner product in a specified interval equals zero. Just as orthogonal vectors in a plane or three-dimensional space are perpendicular to each other, orthogonal functions represent a generalization of this concept to function spaces.

Orthogonal functions play a vital role in mathematical physics and engineering because they can be used to decompose complex problems. In Sturm-Liouville problems, the eigenfunctions obtained through separation methods form an orthogonal basis for the space of solutions. This property is essential to expanding functions in terms of basis functions like Legendre polynomials and Bessel functions, similar to Fourier expansions. This concept paves the way for solving PDEs by transforming them into series of simpler, solvable terms.