Problem 5 - Advanced Engineering Mathematics

Question

In Problems 5 and 6 we try \(u(x, y, z)=X(x) Y(y) Z(z)\) to separate Laplace's equation in three dimensions: \\[ \begin{array}{c} X^{\prime \prime} Y Z+X Y^{\prime \prime} Z+X Y Z^{\prime \prime}=0 \\ \frac{X^{\prime \prime}}{X}=-\frac{Y^{\prime \prime}}{Y}-\frac{Z^{\prime \prime}}{Z}=-\alpha^{2} \end{array} \\] Then \\[ X^{\prime \prime}+\alpha^{2} X=0 \\] \\[ \begin{array}{c} \frac{Y^{\prime \prime}}{Y}=-\frac{Z^{\prime \prime}}{Z}+\alpha^{2}=-\beta^{2} \\\ Y^{\prime \prime}+\beta^{2} Y=0 \\ Z^{\prime \prime}-\left(\alpha^{2}+\beta^{2}\right) Z=0 \end{array} \\] The general solutions of equations \((\mathbf{4}),(\mathbf{5}),\) and \((\mathbf{6})\) are, respectively \\[ X(x)=c_{1} \cos \alpha x+c_{2} \sin \alpha x \\] \\[ \begin{array}{l} Y(y)=c_{3} \cos \beta y+c_{4} \sin \beta y \\ Z(z)=c_{5} \cosh \sqrt{\alpha^{2}+\beta^{2}} z+c_{6} \sinh \sqrt{\alpha^{2}+\beta^{2}} z .\end{array} \\] The boundary and initial conditions are $$\begin{array}{ll} u(0, y, z)=0, & u(a, y, z)=0 \\ u(x, 0, z)=0, & u(x, b, z)=0 \\ u(x, y, 0)=0, & u(x, y, c)=f(x, y) \end{array}$$ The conditions \(X(0)=Y(0)=Z(0)=0\) give \(c_{1}=c_{3}=c_{5}=0 .\) The conditions \(X(a)=0\) and \(Y(b)=0\) yield two sets of eigenvalues: \\[ \alpha=\frac{m \pi}{a}, m=1,2,3, \ldots \quad \text { and } \quad \beta=\frac{n \pi}{b}, n=1,2,3, \ldots \\] By the superposition principle \\[ u(x, y, t)=\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} A_{m n} \sinh \omega_{m n} z \sin \frac{m \pi}{a} x \sin \frac{n \pi}{b} y \\] where \\[ \omega_{m n}^{2}=\frac{m^{2} \pi^{2}}{a^{2}}+\frac{n^{2} \pi^{2}}{b^{2}} \\] and \\[ A_{m n}=\frac{4}{a b \sinh \omega_{m n} c} \int_{0}^{b} \int_{0}^{a} f(x, y) \sin \frac{m \pi}{a} x \sin \frac{n \pi}{b} y d x d y .\\]

Step-by-Step Solution

Answer

The solution is a series: \( u(x, y, z) = \sum_{m=1}^{\infty} \sum_{n=1}^{\infty} A_{mn} \sinh(\omega_{mn} z) \sin(\frac{m\pi}{a} x) \sin(\frac{n\pi}{b} y) \). Eigenvalues \( \alpha = \frac{m\pi}{a} \), \( \beta = \frac{n\pi}{b} \).

Step 1: Set Up the Problem

We begin with the separated form of Laplace's equation in 3D: \( u(x, y, z) = X(x)Y(y)Z(z) \). Substituting into Laplace's equation gives \( X'' Y Z + X Y'' Z + X Y Z'' = 0 \), leading to the separation of variables technique.

Step 2: Separate the Variables

Divide through by \( XYZ \) to obtain \( \frac{X''}{X} + \frac{Y''}{Y} + \frac{Z''}{Z} = 0 \). We choose \( \frac{X''}{X} = -\alpha^2 \), and hence \( X'' + \alpha^2 X = 0 \), similarly for \( Y \) and \( Z \) after eliminating one variable in terms of others using constants \( \alpha \) and \( \beta \).

Step 3: Solve the Individual PDEs

Solve \( X'' + \alpha^2 X = 0 \) to get \( X(x) = c_1 \cos(\alpha x) + c_2 \sin(\alpha x) \) and similarly \( Y(y) = c_3 \cos(\beta y) + c_4 \sin(\beta y) \). For \( Z(z) \, \), solve the modified equation \( Z'' - (\alpha^2 + \beta^2)Z = 0 \) giving \( Z(z) = c_5 \cosh(\sqrt{\alpha^2 + \beta^2} \, z) + c_6 \sinh(\sqrt{\alpha^2 + \beta^2} \, z) \).

Step 4: Apply Boundary Conditions

Use the conditions \( X(0) = 0 \), \( Y(0) = 0 \), \( Z(0) = 0 \) leading to \( c_1 = c_3 = c_5 = 0 \). Boundary conditions \( X(a) = 0 \) and \( Y(b) = 0 \) give the eigenvalues \( \alpha = \frac{m\pi}{a} \), \( \beta = \frac{n\pi}{b} \) where \( m, n \) are integers.

Step 5: Construct Complete Solution Using Superposition

By the principle of superposition, we write solution \( u(x, y, z) = \sum_{m=1}^{\infty} \sum_{n=1}^{\infty} A_{mn} \sinh(\omega_{mn} z) \sin\left(\frac{m\pi}{a}x\right) \sin\left(\frac{n\pi}{b}y\right) \), where \( \omega_{mn}^2 = \frac{m^2\pi^2}{a^2} + \frac{n^2\pi^2}{b^2} \).

Step 6: Determine Fourier Coefficients Using the Last Boundary Condition

Using \( u(x,y,c) = f(x,y) \) as a condition, we equate it to find \( A_{mn} = \frac{4}{ab\sinh(\omega_{mn}c)} \int_0^b \int_0^a f(x,y) \sin\left(\frac{m\pi}{a}x\right) \sin\left(\frac{n\pi}{b}y\right) dx \, dy \).

Key Concepts

Separation of Variables

Separation of variables is a crucial technique used to solve partial differential equations like Laplace's equation. The basic idea is to express the solution as a product of single-variable functions. For Laplace's equation in three dimensions, we assume a solution of the form \( u(x, y, z) = X(x) Y(y) Z(z) \). Each of these functions depends on only one of the variables: \( X \), \( Y \), and \( Z \) correspond to \( x \), \( y \), and \( z \) respectively.
Substituting this assumed form into the original equation allows the equation to be split into separate ordinary differential equations (ODEs), one for each variable. This decomposition simplifies the analysis as each ODE can be solved independently.
The technique is effective because it transforms a complex PDE problem into simpler, solvable ODEs. The approach requires carefully choosing constants to correctly separate the variables, often starting with assumptions such as \( \frac{X''}{X} = -\alpha^2 \), which then guides further separation for \( Y \) and \( Z \). By solving these, we find expressions for \( X(x) \), \( Y(y) \), and \( Z(z) \), which combine to form part of the complete solution.

Boundary Conditions

Boundary conditions ensure that the solution to a differential equation satisfies certain conditions at the boundaries of the domain. They are essential in finding specific solutions to Laplace's equation as they impose constraints that determine the values of constants and coefficients.
In the given exercise, we have conditions such as \( u(0, y, z) = 0 \) and \( u(a, y, z) = 0 \). These conditions are applied to functions \( X(x) \), \( Y(y) \), and \( Z(z) \) at specific points, simplifying or refining our solution. For example, \( X(0) = 0 \) leads to \( c_1 = 0 \), eliminating terms in the general solution that would violate the boundary requirement.
Once the homogeneous boundary conditions are applied, they lead to discrete allowable values for parameters or constants, known as eigenvalues, of the solutions. It is the boundary conditions that transform the general forms into a specific solution which fits the physical or geometrical constraints of the problem.

Eigenvalues

Eigenvalues play a crucial role in solving boundary value problems like Laplace's equation. These are specific values for constants that satisfy the boundary conditions, leading to non-trivial solutions for the system of equations.
In the context of this exercise, eigenvalues appear as constants \( \alpha \) and \( \beta \) in the solutions \( X(x) = c_1 \cos(\alpha x) + c_2 \sin(\alpha x) \) and \( Y(y) = c_3 \cos(\beta y) + c_4 \sin(\beta y) \). The boundary conditions \( X(a) = 0 \) and \( Y(b) = 0 \) require that these functions satisfy specific boundary conditions at \( x = a \) and \( y = b \).
This leads to the quantization of the constants as \( \alpha = \frac{m \pi}{a} \) and \( \beta = \frac{n \pi}{b} \), where \( m \) and \( n \) are positive integers. These values are crucial because they dictate the allowed "modes" of the system, defining distinct solutions aligned with the constraints imposed by the boundary conditions.

Superposition Principle

The superposition principle is a key concept in linear systems like those described by Laplace's equation. It states that if individual solutions can be found for each component of a problem, the overall solution is simply the sum of these individual solutions.
In the context of Laplace's equation treated here, the separation of variables results in multiple functions describing states of the system: \( X(x) \), \( Y(y) \), and \( Z(z) \). Each corresponds to different simple solutions governed by their respective eigenvalues and fulfills the boundary conditions. By the superposition principle, the general solution is a sum of products of these functions for various eigenvalues.
The complete solution is then presented as an infinite series, \( u(x, y, z) = \sum_{m=1}^{\infty} \sum_{n=1}^{\infty} A_{mn} \sinh(\omega_{mn} z) \sin\left(\frac{m\pi}{a}x\right) \sin\left(\frac{n\pi}{b}y\right) \). This series adds up all the possible modes that satisfy both the differential equation and the boundary conditions, providing a versatile solution capable of describing a broad array of physical scenarios.