Problem 14

Question

Using the method of images, solve $$ \nabla^{2} G=\delta\left(x-x_{0}\right) $$ in the first quadrant $(x \geqslant 0$ and $y \geqslant 0)$ with $G=0$ on the boundaries.

Step-by-Step Solution

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Answer

The Green's function in the positive quadrant $ G = -\frac{1}{2\pi} \log |r-r_0||r-r'_0| $, satisfying both the equation $ \nabla^{2} G = \delta\left(x-x_{0}\right) $ and the boundary condition $G=0$.

1Step 1: Conceptual Understanding

The method of images is commonly used in problems with specific symmetries and boundary conditions. The main idea behind the method of images is to replace the problem of interest with a simpler one which has the same boundary conditions. To begin, express the Laplace's equation in two dimensions and correctly identify the symmetries and boundaries which can be exploited.

2Step 2: Apply the Method of Images

In general, for a problem in the first quadrant, an image (symmetric) problem is constructed in the negative quadrants. Suppose the image problem's solution is $G'$, and by symmetry, $G' = 0$ on the boundary, and also satisfies $\nabla^{2} G' = 0$. Then, a single image point affects the solution due to the Dirac delta source at $x_0$. Hence, $G = G' + G_0$, where $G_0$ is the Green's function for an isolated point problem in infinite space without the quadrant boundary.

3Step 3: Solving for Green's Function

The Green's function for an isolated point problem ($G_0$) in infinite space can be written as $ G_0 = -\frac{1}{2\pi} \log |r-r_0| $, where $ r_0 $ and $ r $ are the source and field points, respectively. Using the symmetry characteristics of the problem, calculate $ G' = -\frac{1}{2\pi} \log |r-r'_0| $, where $ r'_0 $ is an image point source in the lower quadrants. Adding both, we will get the required Green's function $ G = -\frac{1}{2\pi} (\log |r-r_0| + \log |r-r'_0|) = -\frac{1}{2\pi} \log |r-r_0||r-r'_0| $, which satisfies $\nabla^{2} G = \delta\left(x-x_{0}\right)$ and $G=0$ on the boundaries.

Key Concepts

Green's functionLaplace's equationDirac delta functionSymmetry in partial differential equations

Green's function

Green's function is a remarkable tool used in the field of differential equations, especially when dealing with boundary value problems. It helps in finding the solution to partial differential equations (PDEs) by taking into account the effects of a point source.

Green's function acts like a 'response' from the system when excited at a specific point.
Mathematically, it is defined by the equation $abla^{2} G = \,$ Dirac delta function, $ \, \delta(x-x_0)$.
This implies that Green's function embodies the influence of the point source located at $(x_0, y_0)$ in the defined domain.

Green's function simplifies solving complex boundary value problems by representing the solution in terms of an integral involving boundary data and inhomogeneities. When this concept is applied, it becomes easier to handle different boundary conditions by focusing solely on the response of the system to specific inputs.
Its beauty lies in its applicability across various domains like electrostatics, acoustics, and thermal conduction, making it indispensable when dealing with linear PDEs.

Laplace's equation

Laplace's equation is a key player in the realm of partial differential equations. It is defined as $abla^{2} \, \psi = 0$, often encountered in various physical applications like gravitation, electrostatics, and fluid dynamics.

The equation indicates potential fields that are steady, i.e., they do not change with time.
Structured to detect "harmonic" functions, which are solutions that exhibit smoothness and the absence of sources or sinks within the domain.

In mathematics, understanding Laplace's equation revolves around grasping how solutions behave under specific boundary conditions:

These conditions tend to mimic real-world scenarios, such as setting the potential on a surface to zero (Dirichlet boundary condition).
Using Laplace's equation helps solve for potential functions that occur naturally in problems with symmetrical domains.

Within the exercise provided, Laplace's equation became essential for setting up a boundary value problem solvable by the method of images, as it relates the function's behavior to the boundary conditions and source terms involved.

Dirac delta function

The Dirac delta function is a mathematical construct that functions similarly to an "idealized point mass" or "point charge". It plays a crucial role in the study of partial differential equations.

It is symbolized by $\delta(x-x_0)$, signifying an infinitely high, infinitely narrow spike at $x_0$.
The essence of the function lies in its ability to isolate and represent point sources within a domain.
Even though it is not a function in the traditional sense, it makes integral equations particularly effective to solve.

In PDE problems like the exercise we examined, the Dirac delta function effectively pinpoints where the effect of the source needs to be considered.

Incorporating it allows transformations that simplify handling sources when calculating Green's function.
Despite its "singular" nature, it provides a way to conveniently incorporate concentrated influences in mathematical models.

Understanding the Dirac delta function helps in appreciating the need for constructing Green's functions that beautifully satisfy boundary conditions while addressing singularities in a discrete manner.

Symmetry in partial differential equations

Symmetry in partial differential equations is a fascinating concept that simplifies complex problems, particularly when considering boundary conditions and domains.

Symmetries provide known patterns or structures in equations that allow simpler representations and solutions.
This is often utilized in "the method of images" where symmetries create a mirror problem that remedies complexity without changing existing conditions.

For instance, in our specific exercise, the first quadrant's symmetry suggests that reflection or duplication of sources across boundaries helps maintain conditions while solving Green's function.

These symmetric characteristics are core to creating the image problem—another instance of the original problem across a boundary.
Utilizing symmetry helps in better engineering solutions that uphold boundary constraints appropriately.

Symmetry in PDEs and its strategic application streamline solving equations subject to boundary restrictions, making it invaluable in practical applications found in physics and engineering.

Problem 13

Problem 14

Other exercises in this chapter

Problem 12

The alternate modified Green's function (Neumann function) satisfies $$ \begin{aligned} &\frac{d^{2} G_{a}}{d x^{2}}=\delta\left(x-x_{0}\right) \\ &\frac{d G_{a

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Problem 13

Consider the one-dimensional infinite space wave equation with a periodic source of frequency $\omega$ : $$ \frac{\partial^{2} \psi}{\partial t^{2}}=c^{2} \fr

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Problem 14

Consider $L(u)=f(x)$ with $L=\frac{d}{d x}\left(p \frac{d}{d x}\right)+q$. Assume that the appropriate Green's function exists. Determine the representation

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Problem 15

Consider $L(G)=\delta\left(x-x_{0}\right)$ with $L=\frac{d}{d x}\left(p \frac{d}{d x}\right)+q$ subject to the boundary conditions \(G\left(0, x_{0}\right)=

View solution