Question

(a) Find the solution \(u(x, y)\) of Laplace's equation in the rectangle \(0

Short answer

#Question# Provide a brief explanation of the method used in solving Laplace's equation within a rectangular domain and satisfying the given boundary conditions.

Step-by-step solution

Apply Separation of Variables technique to Laplace's Equation

To solve the Laplace equation \(\nabla^2 u(x,y) = 0\), we assume that the solution can be written as \(u(x, y) = X(x)Y(y)\).

Rewrite Laplace's equation

Substituting the assumed solution u(x, y) = X(x)Y(y) into \(\nabla^2 u(x,y) = 0\), we obtain: $$ X''(x)Y(y) + X(x)Y''(y) = 0, $$

Separate the variables

We can now write the equation as: $$ \frac{X''(x)}{X(x)} = - \frac{Y''(y)}{Y(y)}. $$ Since the left-hand side (LHS) only depends on variable x, and the right-hand side (RHS) only depends on variable y, the equation must be equal to a constant (say, -\(\lambda_n^2\)). We now have two ordinary differential equations (ODEs): $$ X''(x) + \lambda_n^2 X(x) = 0, $$ $$ Y''(y) - \lambda_n^2 Y(y) = 0. $$

Solve the ODEs and apply boundary conditions

For \(X''(x) + \lambda_n^2 X(x) = 0\), the boundary conditions are \(X'(0) = 0\) and \(X'(a) = 0\). The general solution for X(x) is: $$ X_n(x) = A_n \cos(\lambda_n x) + B_n \sin(\lambda_n x). $$ Applying the boundary conditions, we find \(A_n = 0\), which leaves: $$ X_n(x) = B_n \sin(\lambda_n x). $$ For \(Y''(y) - \lambda_n^2 Y(y) = 0\), the boundary conditions are \(Y(0) = 0\) and \(Y(b) = g_n\). The general solution for Y(y) is: $$ Y_n(y) = C_n \cosh(\lambda_n y) + D_n \sinh(\lambda_n y), $$ and applying the condition \(Y(0) = 0\), we find \(C_n = 0\). So, $$ Y_n(y) = D_n \sinh(\lambda_n y). $$ ***Part (b) - Finding the solution with the given g(x)***

Use Fourier series expansion to represent g(x)

We need to find coefficients \(G_n\) such that: $$ g(x) = 1 + x^2(x - a)^2 = \sum_{n=1}^{\infty} G_n \sin \left( \frac{n \pi x}{a} \right). $$ We can now find each \(G_n\) using the Fourier series formula: $$ G_n = \frac{2}{a} \int_0^a \left[1 + x^2(x - a)^2\right] \sin\left(\frac{n \pi x}{a}\right) dx. $$ Now, we can write the solution for \(u(x, y)\) as: $$ u(x, y) = \sum_{n=1}^{\infty} G_n \sin\left(\frac{n \pi x}{a}\right) \sinh\left(\frac{n \pi y}{a}\right), $$ with G_n found from the integral above. ***Part (c) - Plotting the solution with \(a = 3\) and \(b = 2\)***

Compute the solution with given \(a\) and \(b\) values

Substitute \(a = 3\) and \(b = 2\) into the solution: $$ u(x, y) = \sum_{n=1}^{\infty} G_n \sin\left(\frac{n \pi x}{3}\right) \sinh\left(\frac{n \pi y}{3}\right), $$ where \(G_n\) is calculated as: $$ G_n = \frac{2}{3} \int_0^3 \left[1 + x^2(x - 3)^2\right] \sin\left(\frac{n \pi x}{3}\right) dx. $$

Plot the solution

Using appropriate software (e.g., Mathematica, Matlab, or Python), plot the solution u(x, y) up to a certain number of terms in the Fourier series (e.g., 20 terms). Visualizations can be done with contour plots, surface plots, or 3D mesh plots.

Key concepts

Separation of Variables

Separation of Variables is a powerful mathematical technique commonly used to solve partial differential equations (PDEs), such as Laplace's equation. Imagine trying to solve a puzzle; it's usually easier to tackle one piece at a time rather than the whole thing at once. Similarly, separation of variables allows us to break down complex problems into simpler, more manageable parts, transforming a PDE with multiple variables into a set of Ordinary Differential Equations (ODEs) that depend on a single variable each.

The process begins by assuming that the solution to a PDE can be represented as the product of functions, each dependent on a single variable. In the case of Laplace's equation, we assumed the solution as a product of two functions, one that only varies with 'x' and the other that varies with 'y'. It's a bit like deciding to deal with the horizontal and vertical aspects of a problem separately—you simplify the complex interaction into individual, straightforward parts that can be tackled in isolation. This clever strategy is the starting point for finding a solution that adheres to the given boundary conditions.

Fourier series

Fourier series is like a mathematical symphony, decomposing complex periodic functions into a sum of simple sines and cosines. This concept comes in handy when we deal with functions that have repeating patterns, like the boundary conditions in part (b) of our exercise, where we have a given function, g(x), describing the condition on the boundary.

To represent g(x), we express it as an infinite series of sine functions—each a different note in our symphony. By doing so, we account for the complexity of g(x) using a harmonic series. The Fourier coefficients, which are analogous to the amplitude of each note, are calculated through integrals. They tell us how much of each sine wave is present in our original function. When put all together, these sine waves mimic the original function with remarkable precision.

The use of a Fourier series isn't just a neat math trick; it's essential for transforming functions that can be applied to our original PDE, converting it into a form that matches the solution we're developing through separation of variables. It's math's way of taking a complicated, wave-like problem and using the concept of superposition—adding up simpler waves—to recreate the complex pattern.

Ordinary Differential Equations (ODEs)

Ordinary Differential Equations (ODEs) are the cornerstones of calculus, helping us to describe how physical quantities change over a single variable, usually time or one-dimensional space. In our exercise, after applying Separation of Variables, we derive two ODEs, each tied to a single spatial dimension. One equation tells us how our solution changes in the 'x' direction, and the other tells us about changes in the 'y' direction.

Solving these ODEs is akin to understanding the essence of change within each variable's domain. The solutions to these ODEs aren't just random guesses; they are dictated by the initial conditions or boundary conditions, that is, the known states of the system at certain points or edges. By solving the ODEs with these conditions, we craft a solution that not only satisfies the universal laws dictated by the differential equation, but also aligns with the specific, given circumstances of our unique problem.

Boundary Conditions

Boundary Conditions act as the rules of the game when solving differential equations—they outline the must-haves at the edges of the domain we're looking at. In a physical sense, boundaries can represent the walls of a room or the surface of an object. Mathematically, they're the constraints that shape our solution, ensuring that it reflects reality.

For our Laplace's equation exercise, we dealt with a rectangle and had to satisfy the conditions at each of its edges. These conditions give us the necessary information to determine arbitrary constants in our general solution. Without boundary conditions, we'd have an infinite number of potential solutions—like having an unlimited choice of outfits without knowing the dress code for an event. By applying the boundary conditions such as a derivative being zero (reflecting symmetry) or the function taking on a specific value (as with g(x) at y=b), we're narrowing down to the one solution that 'fits' perfectly into the scenario provided by the problem.

Mathematical Physics

Mathematical Physics is the stage where the abstract world of mathematics performs to help us understand the complexities of the physical universe. It provides us with the language and tools to turn physical concepts and phenomenon, like the flow of heat or the propagation of electromagnetic waves, into equations that can be analyzed and solved. In problems like Laplace's equation, we're seeing this in action: the equation itself is a mathematized version of a physical law describing how potential fields behave in space without any sources or sinks.

As we've seen with the boundary value problem given in the exercise, mathematical physics isn't just about finding a solution. It's about finding the right solution that complies with both the universal laws of nature (like how potential behaves) and the very particular conditions of our situation (the specific values along the edges of a rectangle). Through applying mathematical concepts like Separation of Variables, Fourier series, solving ODEs, and enforcing Boundary Conditions, we blend abstract mathematics with the real world to create a potent tool for solving complex physical problems.