Question

Laplace's equation in three dimensions is \(u_{x x}(x, y, z)+u_{y y}(x, y, z)+u_{z z}(x, y, z)=0\). Assume a solution of the form \(u(x, y, z)=X(x) Y(y) Z(z)\). Repeat the separation of variables approach outlined in Exercise 21 to derive separation equations for \(X(x), Y(y)\), and \(Z(z)\). These equations will again involve two separation constants.

Short answer

Answer: The three separation equations are: 1. For \(X(x)\): $$X''(x) + k^2 X(x) = 0$$ 2. For \(Y(y)\): $$Y''(y) + l^2 Y(y) = 0$$ 3. For \(Z(z)\): $$Z''(z) + (k^2 + l^2)Z(z) = 0$$

Step-by-step solution

Insert the given form of the solution into Laplace's equation

We are given the Laplace's equation in three dimensions and the solution in the form of \(u(x, y, z) = X(x)Y(y)Z(z)\). Let's substitute this into the Laplace's equation: $$X''(x)Y(y)Z(z) + X(x)Y''(y)Z(z) + X(x)Y(y)Z''(z) = 0$$

Separate the variables

To separate the variables, we'll divide both sides of the equation by \(X(x)Y(y)Z(z)\): $$\frac{X''(x)}{X(x)} + \frac{Y''(y)}{Y(y)} + \frac{Z''(z)}{Z(z)} = 0$$ Now let's define the separation constants. We will use the same approach as in Exercise 21:

Define the separation constants and simplify the equations

Set the sum of the first two terms to a constant, say \(-k^2\): $$\frac{X''(x)}{X(x)} + \frac{Y''(y)}{Y(y)} = -k^2$$ Notice that now, we have two terms combined into one. We can define another separation constant, say \(l^2\), for the remaining term: $$\frac{Z''(z)}{Z(z)} = -(k^2 + l^2)$$

Write the separation equations for X(x), Y(y), and Z(z)

Now we have three separation equations: 1. For \(X(x)\): $$X''(x) + k^2 X(x) = 0$$ 2. For \(Y(y)\): $$Y''(y) + l^2 Y(y) = 0$$ 3. For \(Z(z)\): $$Z''(z) + (k^2 + l^2)Z(z) = 0$$ These are the three separation equations for the given solution of the three-dimensional Laplace's equation.

Key concepts

Separation of Variables

The method of separation of variables is a powerful technique employed to solve partial differential equations (PDEs), such as Laplace's equation. The central idea is to assume the solution to the PDE can be written as the product of functions, each depending on a single coordinate.

For a three-dimensional space, a function u(x, y, z) can be expressed as the product of three functions: X(x), Y(y), and Z(z). To apply this method, one substitutes this assumed form into the original PDE and reorganizes the equation to segregate variables. The goal is to divide the equation such that each side of the equation depends on a separate variable, allowing them to be set equal to constants, because a change in one variable should not affect the other.

In practice, after substituting the product into Laplace's equation, all terms involving X(x), Y(y), and Z(z) are divided across to yield an equation where each function's second derivative is divided by the function itself. This manipulation curtain raises the unique ability of separation of variables: turning a PDE into simpler, ordinary differential equations (ODEs) to solve.

Partial Differential Equations

Partial differential equations (PDEs) are equations that involve rates of change with respect to multiple variables. PDEs are fundamental in the arenas of physics, engineering, and mathematics because they describe patterns of flow, vibrations, heat, electric fields, and more – often in the form of a function dependent on several variables.

Laplace's equation is a prime example of a second-order linear PDE, where the solution presents potentials with no internal sources, such as gravitational or electrostatic potentials in free space. The equation asserts that the sum of the second partial derivatives of a potential function with respect to all spatial variables equals zero. This homogeneity is a clue that allows for techniques like separation of variables to be applied, as they capitalize on the equality to constants to disentangle the complex relationship between variables.

Understanding PDEs involves recognizing boundary conditions, type of equation (elliptic, parabolic, hyperbolic), and then applying appropriate strategies, like separation of variables, to find solutions that fit the physical or the geometrical constraints of the problem.

Boundary Value Problems

Boundary value problems (BVPs) arise when solving PDEs such as Laplace's equation under certain fixed conditions, known as boundary conditions. These types of problems require the determination of a solution that not only satisfies the differential equation but also meets the constraints dictated by the boundaries of the domain.

Using separation of variables, one can reduce the PDE to a set of ODEs. However, the journey doesn't end there; the next step involves integrating these ODEs considering the boundary conditions. The boundary conditions are critical – they might represent physical situations like the temperature at the surface of an object or the electrical potential at the edge of a region.

The boundary conditions render the problem unique, guiding the integration process by limiting the scope of possible solutions. They often result in quantizing the constants arrived at through the separation of variables, because not any value will satisfy the imposed edges of the problem. Solving BVPs thus becomes a blend of analytical rigor and careful consideration of the specific physical context outlined by the boundaries.