Question

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

Short answer

Question: Write a short-answer response based on the given solution steps describing how to find the solution to Laplace's equation under specific boundary conditions. Answer: To find the solution to Laplace's equation under specific boundary conditions, we follow a series of steps: 1) Separate variables by assuming the solution u(x, y) can be represented as a product of two functions, one depending on x and the other on y; 2) Solve the resulting ordinary differential equations for X(x) and Y(y) with given boundary conditions; 3) Combine the solutions for X and Y into a general solution; 4) Determine the coefficients B_n in the general solution using the Fourier sine series representation of the given function g(x); 5) Find the solution u(x, y) for the given g(x) by substituting the coefficients B_n; 6) Plot the solution u(x, y) for different values of x and y using software like MATLAB, Python, or Mathematica, with given values of a and b.

Step-by-step solution

Separation of Variables

To solve Laplace's equation using separation of variables, assume that the solution u(x, y) can be represented as a product of two functions: one depending on x and the other depending on y: \(u(x, y) = X(x)Y(y)\). Substituting this into Laplace's equation and dividing by \(X(x)Y(y)\), we get $$\frac{X''(x)}{X(x)} + \frac{Y''(y)}{Y(y)} = 0$$

Solving the resulting ordinary differential equations

Since the left side of the equation depends only on x whereas the right side depends only on y, both sides must be equal to a constant, say, \(-k^2\). This gives us two ordinary differential equations, $$X''(x)=-k^2X(x), \quad Y''(y)=k^2Y(y)$$

Solving the X and Y ordinary differential equations

Solve the X ordinary differential equation with the given boundary conditions: \(u(0,y)=0\) and \(u(a,y)=0\). This implies that \(X(0)=0\) and \(X(a)=0\). Solving this gives, $$X_n(x) = \sin(\frac{n\pi x}{a})$$ with \(n=1,2,3,...\) Next, solve the Y ordinary differential equation with the given boundary conditions: \(u(x,0)=0\) and \(u(x,b)=g(x)\). This implies that \(Y(0)=0\) and we have to find \(Y_n(b)\) such that \(g(x)=u(x,b)=\sum\limits_{n=1}^\infty B_n \sin(\frac{n\pi x}{a})\). The \(Y''(y)=k^2Y(y)\) equation has solution \(Y_n(y) = \sinh(\frac{n\pi y}{b})\).

Combining X and Y solutions and determining B_n

Now that we have found the solutions for X and Y, we can write the general solution to Laplace's equation as $$u(x,y)=\sum\limits_{n=1}^{\infty} B_n \sin(\frac{n\pi x}{a})\sinh(\frac{n\pi y}{a})$$ To find the coefficients \(B_n\), we need to determine the Fourier sine series representation of g(x): $$g(x)=\sum\limits_{n=1}^{\infty} B_n \sin(\frac{n\pi x}{a})$$

Find the solution u(x, y) for the given g(x)

Now, we find the coefficients \(B_n\) for the given \(g(x)\). Since g(x) is a piecewise function, we need to integrate over two intervals. $$B_n=\frac{2}{a}\left[\int_0^{\frac{a}{2}} x\sin\left(\frac{n\pi x}{a}\right)dx + \int_{\frac{a}{2}}^a (a-x)\sin\left(\frac{n\pi x}{a}\right)dx\right]$$ Solving these integrals, we get $$B_n=\frac{8}{n^3\pi^3}\left[1-(-1)^n\right]$$ Finally, our solution for u(x, y) with the given g(x) becomes $$u(x,y)=\sum\limits_{n=1}^{\infty} \frac{8}{n^3\pi^3}\left[1-(-1)^n\right] \sin(\frac{n\pi x}{a})\sinh(\frac{n\pi y}{a})$$

Plotting the solution u(x,y)

Now that we have our general solution u(x, y), we can plot it for different values of x and y, as required in parts (c) and (d). To do this, you can use software like MATLAB, Python, or Mathematica to evaluate the series for different values of x and y and create the desired plots. Remember to use the given values of a=3 and b=1 for these plots.

Key concepts

Boundary Value Problem

The concept of a boundary value problem (BVP) is at the core of solving many physical phenomena described by differential equations, including the one in our exercise—Laplace's equation. In essence, a BVP is a differential equation coupled with a set of constraints called boundary conditions. For our problem, we need to find a function u(x, y) that satisfies Laplace's equation within a rectangle and meets specific conditions along the rectangle's edges. These conditions are provided by the manner in which the physical system of interest behaves at its boundaries—in this case, the values that the potential u takes on the rectangle's perimeter.A well-posed BVP yields a unique solution that represents the steady-state behavior of a system. In the context of our problem, it means finding a unique potential distribution within the rectangle that adheres to the prescribed boundary conditions, and this can be quite complex, as it demands considering the system as a whole.

Separation of Variables

One of the most powerful techniques for solving BVPs, especially when dealing with partial differential equations (PDEs), is separation of variables. The idea behind this method is to reduce a PDE into one or more ordinary differential equations (ODEs), which are typically much easier to solve.In our exercise, we assume that the solution u(x, y) can be expressed as the product of two functions, each solely dependent on a single variable: u(x, y) = X(x)Y(y). By inserting this assumption into Laplace's equation and rearranging, we obtain two separate ODEs for X and Y.Applying this technique allows us to convert a complex problem into simpler, more manageable parts. It is an analogy to tackling a challenge by breaking it down into smaller tasks, thereby making the solution approach more intuitive and structured.

Fourier Series

The extension of our problem involves the Fourier series, a tool immensely useful for representing periodic functions as infinite sums of sines and cosines. When we apply boundary conditions such as g(x) provided in our exercise, we need to express this function as a sum of sinusoidal components.Specifically, in our problem, we use the Fourier sine series because of the way our boundary conditions are defined (with u equalling zero at both x=0 and x=a). Hence, we only need the sine terms as they vanish at these points, satisfying our boundary conditions. We determine the coefficients of this series by integrating over the interval, as shown in the steps for g(x). These coefficients, B_n, are crucial as they provide the amplitude of each sine function in the series and thus are key to the final solution of our BVP.Understanding Fourier series helps us decompose complex, irregular functions into fundamental frequency components, a principle widely applied across physics, engineering, and even in digital signal processing today.

Ordinary Differential Equations

At the heart of many mathematical models used in science and engineering are ordinary differential equations (ODEs), which describe how a quantity changes with respect to another. When solving BVPs, we often end up with ODEs as part of the problem—as seen in our exercise.After applying separation of variables, we derived two second-order ODEs for X and Y, one for each dimension of the problem. These ODEs describe how the potential varies along the x-axis and y-axis independently. By solving these ODEs following the boundary conditions, we could find explicit forms for X(x) and Y(y), which play a significant role in constructing the solution to the original PDE.The real artistry in using ODEs lies in the ability to take real-life problems, often intricate and multidimensional, and translate them into equations that describe the essence of the physical behavior or changes observed, providing a pathway to solutions that explain the world around us.