Question

Bestimmen Sie mit Hilfe des Separationsansatzes Lösungen \(u(x, y)\) bzw. \(u(x, t)\) für die partiellen Differentialgleichungen (a) \(u_{x x}=4 u_{y}, u(0, y)=u(\pi, y)=0\), (b) \(a^{2} u_{x x}=u_{t t}, a>0\).

Short answer

For (a), the solution is a series: \( u(x, y) = \sum_{n=1}^{\infty} C_n \sin(2n x) \exp(-n^2 y) \). For (b), it is a superposition of solutions \( u(x, t) = \sum (A_n \cos(\sqrt{\lambda_n} x) + B_n \sin(\sqrt{\lambda_n} x))(C_n \cos(a\sqrt{\lambda_n} t) + D_n \sin(a\sqrt{\lambda_n} t)) \).

Step-by-step solution

Understanding the Problem

We are given two partial differential equations with boundary or initial conditions to solve using the method of separation of variables. We will solve them separately, each in their own context and with their specified conditions.

Separation of Variables for (a)

For the equation \( u_{xx} = 4u_{y} \) with conditions \( u(0, y) = u(\pi, y) = 0 \), assume a solution of the form \( u(x, y) = X(x)Y(y) \). Substituting this into the equation gives \( X''(x)Y(y) = 4X(x)Y'(y) \), simplifying to \( \frac{X''(x)}{4X(x)} = \frac{Y'(y)}{Y(y)} = -\lambda \) for separation constant \(-\lambda\).

Solve the Spatial Part X(x)

The equation for \( X(x) \) becomes \( X''(x) + 4\lambda X(x) = 0 \). With boundary conditions \( X(0) = X(\pi) = 0 \), solve to get \( X(x) = \sin(2n x) \) where \( \lambda = n^2 \) and \( n \) is a positive integer.

Solve the Vertical Part Y(y)

For \( Y(y) \), the equation is \( Y'(y) = -n^2 Y(y) \), solving this gives \( Y(y) = C \exp(-n^2 y) \).

General Solution for (a)

The general solution satisfying the given PDE and boundary conditions is \( u(x, y) = \sum_{n=1}^{\infty} C_n \sin(2n x) \exp(-n^2 y) \), where \( C_n \) are constants determined by initial or other boundary conditions.

Separation of Variables for (b)

For the equation \( a^2 u_{xx} = u_{tt} \), assume \( u(x, t) = X(x)T(t) \). Substituting gives \( a^2 X''(x)T(t) = X(x)T''(t) \), which separates as \( \frac{X''(x)}{X(x)} = \frac{T''(t)}{a^2 T(t)} = -\lambda \).

Solve the Spatial Equation for X(x)

The equation \( X''(x) + \lambda X(x) = 0 \) has solutions \( X(x) = A \cos(\sqrt{\lambda} x) + B \sin(\sqrt{\lambda} x) \).

Solve the Time Equation for T(t)

The equation \( T''(t) + a^2 \lambda T(t) = 0 \) has solutions \( T(t) = C \cos(a\sqrt{\lambda} t) + D \sin(a\sqrt{\lambda} t) \).

General Solution for (b)

The general solution for this equation is a superposition of solutions: \( u(x, t) = \sum (A_n \cos(\sqrt{\lambda_n} x) + B_n \sin(\sqrt{\lambda_n} x))(C_n \cos(a\sqrt{\lambda_n} t) + D_n \sin(a\sqrt{\lambda_n} t)) \).

Key concepts

Separation of Variables

The separation of variables is a powerful mathematical technique used to solve partial differential equations (PDEs). This method assumes that the solution can be expressed as the product of functions, each of which depends only on a single variable. For example, in the given exercise, we consider solutions of the form \( u(x, y) = X(x)Y(y) \) or \( u(x, t) = X(x)T(t) \).
By substituting these forms into the PDE, we can often transform a complex multi-variable problem into simpler, separate ordinary differential equations (ODEs). Each of these ODEs can then be solved independently.
Using separation of variables not only simplifies the problem but also allows us to exploit the benefits of linear algebra and eigenvalue problems, which are often essential when dealing with systems having boundary or initial conditions.

Boundary Conditions

Boundary conditions are crucial in determining a unique solution for differential equations. These conditions specify the values that a solution must satisfy at the boundary of the domain.
In part (a) of our problem, we have \( u(0, y) = u(\pi, y) = 0 \). These conditions tell us that the solution \( u(x, y) \) must be zero along the lines \( x = 0 \) and \( x = \pi \).
Boundary conditions can be of various types:

• Dirichlet (specifying function values),
• Neumann (specifying derivative values),
• or mixed conditions (a combination of both).

They play a pivotal role in forming the basis functions used in separation of variables, thus influencing the eigenvalues and eigenfunctions explored in the next concept.

Eigenvalue Problems

Eigenvalue problems emerge naturally in the process of using separation of variables for solving PDEs. When we separate variables, the resulting ordinary differential equations often have solutions only for particular values of the separation constant, known as eigenvalues.
For instance, in the spatial part \( X(x) \) for equation (a), we ended up solving \( X''(x) + 4\lambda X(x) = 0 \). With the boundary conditions \( X(0) = X(\pi) = 0 \), the allowed eigenvalues are \( \lambda = n^2 \), leading to solutions \( X(x) = \sin(2n x) \). Here \( n \) are integers representing the different modes of vibration.
These eigenvalues determine the behavior of the solution, helping transform infinite-dimensional problems into tractable series representations. Understanding eigenvalues helps in grasping the physical interpretation of vibrations, heat flow, and more, in mathematical models.

Wave Equation

The wave equation is a fundamental type of PDE used to describe wave phenomena such as sound, light, and water waves. It characterizes how waves propagate over time and space.
The classic form of the wave equation is \( a^2 u_{xx} = u_{tt} \), as seen in part (b) of our problem. This equation balances the spatial change with the temporal change of a wave based on the speed \( a > 0 \).
Solving the wave equation often involves applying the separation of variables, where the solution splits into a spatial part \( X(x) \) and a temporal part \( T(t) \). Each of these parts satisfies a simpler form of differential equation, resulting in solutions that can be sinusoidal in nature, such as \( \cos \) or \( \sin \) functions.
This method allows us to express wave behavior as a superposition of simpler oscillatory functions, capturing the essence of natural wave properties in mathematical terms.