Question

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

Short answer

Question: Determine the solution u(x, y) of Laplace's equation inside a rectangle with given boundary conditions. Answer: The solution u(x, y) can be written as: $$u(x, y) = \sum_{n=1}^{\infty} \frac{2}{a} \sin{\frac{n \pi x}{a}} \left(\int_{0}^{a} h(\xi) \sin{\frac{n \pi \xi}{a}} d\xi \right) (\cosh{\frac{n \pi y}{a}})$$ where a is the length of the rectangle, h(x) is the boundary condition function and the sum runs over all positive integers n.

Step-by-step solution

Apply the separation of variables

Assume that the solution u(x, y) can be written as the product u(x, y) = X(x)Y(y). Then substitute this into Laplace's equation: $$\nabla^2 u(x, y) = \frac{\partial^2}{\partial x^2}(X(x)Y(y)) + \frac{\partial^2}{\partial y^2}(X(x)Y(y)) = 0$$ Now, dividing by X(x)Y(y), we get: $$\frac{X''(x)}{X(x)} + \frac{Y''(y)}{Y(y)} = 0$$

Separate the equations

Since the left side of the equation depends on x and the right side depends on y, each term has to be equal to a constant, say -k^2: $$\frac{X''(x)}{X(x)} = -k^2 \quad \text{and} \quad \frac{Y''(y)}{Y(y)} = k^2$$

Solve the differential equations

Now we have two second-order ordinary differential equations: $$X''(x) + k^2 X(x) = 0$$ $$Y''(y) - k^2 Y(y) = 0$$ The general solutions for these equations can be written as: $$X(x) = A \cos{kx} + B \sin{kx}$$ $$Y(y) = C \cosh{ky} + D \sinh{ky}$$ Here, A, B, C, and D are constants.

Apply the boundary conditions

Now we need to apply the given boundary conditions to determine the constants: 1) u(0, y) = 0: $$X(0)Y(y) = (A \cos{0} + B \sin{0}) Y(y) = AY(y) = 0$$ Which means A = 0, so, $$X(x) = B \sin{kx}$$ 2) u(a, y) = 0: $$X(a)Y(y) = B \sin{ka} Y(y) = 0$$ Applying the sine condition to avoid trivial solution: $$k_n = \frac{n \pi}{a}, n \in \mathbb{N}$$ Thus, $$X(x) = B_n \sin{\frac{n \pi x}{a}}$$ 3) u(x, 0) = h(x): $$X(x)Y(0) = X(x)(C \cosh{0} + D \sinh{0}) = C X(x) = h(x)$$ So, $$C X_n(x) = h_n(x)$$ 4) u(x, b) = 0: $$X(x)Y(b) = X(x)(C_n \cosh{\frac{n \pi b}{a}} + D_n \sinh{\frac{n \pi b}{a}}) = 0$$ Since X(x) is not zero, the term in the parenthesis should be zero: $$D_n \sinh{\frac{n \pi b}{a}} = 0$$ But the trivial solution (D_n = 0) is useless, so we have no other choice but to combine harmonics (Fourier series):

Determine the Fourier series

Now, we will determine the Fourier series for h(x): $$h(x) = \sum_{n=1}^{\infty} C_n X_n(x) = \sum_{n=1}^{\infty} C_n B_n \sin{\frac{n \pi x}{a}}$$ Calculate the Fourier coefficients: $$C_n = \frac{2}{a} \int_{0}^{a} h(x) \sin{\frac{n \pi x}{a}} dx$$ Now we have all the constants and we are ready to write down the complete solution.

Write down the solution

The complete solution u(x, y) can be written as: $$u(x, y) = X(x)Y(y) = \sum_{n=1}^{\infty} C_n B_n \sin{\frac{n \pi x}{a}} (C_n \cosh{\frac{n \pi y}{a}} + D_n \sinh{\frac{n \pi y}{a}}) = \sum_{n=1}^{\infty} \frac{2}{a} \sin{\frac{n \pi x}{a}} \left(\int_{0}^{a} h(\xi) \sin{\frac{n \pi \xi}{a}} d\xi \right) (\cosh{\frac{n \pi y}{a}})$$ This is the solution u(x, y) for Laplace's equation inside the rectangle defined by the boundary conditions in the problem statement.

Key concepts

Separation of Variables

Separation of variables is a technique used to solve partial differential equations, like Laplace's equation, which can be quite complex due to their reliance on multiple variables. Simplifying these problems by splitting the equation into two or more separate ordinary differential equations allows us to tackle each one individually. This involves assuming that the solution to the equation can be expressed as a product of functions, each dependent on only one variable. For instance, considering the solution of Laplace's equation, we presume that the unknown function

can be written as X(x)Y(y), which proposes that the spatial variables can be isolated.

When we insert this product into Laplace's equation, we derive a system where each side is functionally dependent on a singular variable—either x or y. To satisfy the equation for any value of x and y, each side must equate to the same constant, which allows us to divide the problem into two ordinary differential equations that are more straightforward to solve compared to the original partial differential equation.

Boundary Conditions

Boundary conditions are essential to solving physical problems as they describe the behavior of a solution at the boundaries of its domain. They provide the necessary constraints to determine the unique solution to a differential equation. In the context of Laplace's equation, we are given conditions on the edge of a rectangle, which dictate the values the solution must assume on those edges.

Incorporating these conditions into our separated solutions allows us to find specific constants that make our general solution a valid one within the physical context. For example, we observed that u(0, y) = 0, implying the function X(x) must be zero when x = 0. This eliminates certain mathematical possibilities and steers us towards a solution that conforms with physical reality. Boundary conditions anchor the abstract form of our equations to the practical problem we seek to understand, forming an indispensable bridge between mathematical theory and applied science.

Fourier Series

The Fourier series expands upon the idea of expressing functions as a sum of sinusoidal components. This approach is particularly useful in solving differential equations with periodic boundary conditions. After applying separation of variables and finding solutions that respect the boundary conditions, we may end up with a series of sine and cosine functions—these represent the Fourier series of the solution.

The coefficients of these series are determined by integrating the function over its period, which in our case, is derived from the boundary condition involving h(x). Through this process, we convert a spatial function into a spectrum of frequencies, each with a specific amplitude. The power of Fourier series lies in its ability to synthesize complex, irregular phenomena into harmonics that can be analyzed and understood separately. Applying this technique in the context of Laplace's equation enables us to construct a complete and precise solution by piecing together a series of simpler, oscillating functions that respect the geometry and conditions of the problem.