91Ó°ÊÓ

The Laplace equation is a critical concept in mathematical physics and engineering, inherently used to model steady-state heat distribution, fluid flow, electrostatics, and more. This particular class of Partial Differential Equations (PDEs) is characterized by the absence of time as a variable and possess the form:

• \[abla^2 u = 0 = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\]

Laplace's equation is widely used due to its inherent property of describing phenomena in equilibrium states. The given problem is a two-dimensional Laplace equation, where we seek a function \(u(x,y)\) that satisfies this form within a specified domain of \(0 < x < \pi\) and \(0 < y < \pi\). By solving the Laplace equation under these constraints, we can determine the behavior of potential fields like electrostatic potentials or temperature distributions. This is the groundwork needed to apply more sophisticated solving techniques, like Separation of Variables, for finding specific solutions.

The method of Separation of Variables is invaluable for tackling PDEs like the Laplace equation. This technique assumes that the solution can be represented as a product of functions, each depending only on one of the variables. In our case, we express the solution as:

• \[u(x,y) = X(x)Y(y)\]

By substituting this assumption into the Laplace equation and performing some algebraic manipulation, the problem simplifies into two ordinary differential equations (ODEs) that are easier to solve. These independent ODEs take the form:

• \[\frac{X''(x)}{X(x)} = -\lambda^2 \,\text{and}\, \frac{Y''(y)}{Y(y)} = \lambda^2\]

Here, \( \lambda \) represents a constant that arises from the separation process. The technique's strength lies in converting a more complex PDE into simpler, univariate ODEs. This approach hinges on successfully applying Boundary Conditions to identify specific solutions that meet the context of the given physical scenario.

Boundary Conditions are crucial in determining a unique solution to a PDE such as the Laplace equation. They specify the behavior of the solution on the edges of the defined domain. In this problem, the boundary conditions provided are:

• \(u(0, y) = u(\pi, y) = u(x, \pi) = 0\)
• \(u(x, 0) = \sin^2 x\)

Theses conditions enforce that the function \(u(x, y)\) vanishes along the edges of the domain except along \(y = 0\), where it must match the function \(\sin^2x\). Applying these conditions to the separated ODEs allows us to solve for specific functions \(X(x)\) and \(Y(y)\), which combine to form the general solution:

• \[u(x, y) = \sum_{n=1}^{\infty} A_n \sin(nx) \sinh(ny)\]

The boundary condition \(u(x, 0) = \sin^2 x\) is particularly used to determine the Fourier coefficients \(A_n\), which are found by matching the solution to a given form using the Fourier Series.

Fourier Series is a powerful mathematical tool used to represent periodic functions as a sum of sines and cosines. In solving the Laplace equation problem, the Fourier Series is used to express \(\sin^2x\) in terms of sines. This allows us to find the Fourier coefficients necessary to adapt the general solution \(\sum_{n=1}^{\infty} A_n \sin(nx) \sinh(ny)\) to satisfy the boundary condition \(u(x, 0) = \sin^2x\).In practical terms, it works by expanding a periodic function into a series of harmonically related sinusoids. For the function \(\sin^2x\), the Fourier sine series is computed as:

• \[\sin^2(x) = \frac{1}{2} - \frac{1}{2} \cos(2x)\]

Through matching terms and solving, the Fourier coefficients \(A_n\) are identified as:

• \(A_1 = \frac{1}{2}, A_3 = -\frac{1}{2}\,\text{and}\,A_n = 0\,\text{for other}\, n\)

The resultant solution is then pieced together using these Fourier coefficients, thus arriving at the specific solution that satisfies all original boundary conditions.