91Ó°ÊÓ

In mathematics, boundary conditions are constraints that are necessary to uniquely determine the solutions to differential equations. For our specific problem of Laplace's equation, these conditions define how the temperature behaves along the edges of the semi-infinite plate. Here are the boundary conditions utilized:

• Left and Right Boundaries: Since the temperature at the left & right sides of the plate is held at a constant 0°C, this is mathematically expressed as: when \( x = 0 \) and \( x = 1 \), \( u(x, y) = 0 \).
• Bottom Edge: The temperature at the bottom of the plate (\( y = 0 \)) is provided by the function \( f(x) = 3 \sin 2\pi x - \sin \pi x \). By setting \( u(x, 0) = f(x) \), we establish how the temperature enters the plate.
• Boundedness Condition: The solution must remain bounded as \( y \) increases, which implies that any divergences are physically impossible, keeping our solution realistic.

Boundary conditions allow us to tailor our general solution to actual, physically meaningful scenarios.

The method of separation variables is a technique used to transform a complex partial differential equation into simpler, solvable ordinary differential equations. For Laplace's equation in two variables, we assume that the solution \( u(x, y) \) can be written as a product of two single-variable functions \( X(x) \) and \( Y(y) \). This leads to the assumption:

• \( u(x, y) = X(x)Y(y) \)

Substituting this into the Laplace equation gives us:\[ X''(x)Y(y) + X(x)Y''(y) = 0 \]Dividing through by \( XY \), each term becomes equal to a common constant, \( -\lambda \), creating two ordinary differential equations:

• \( X'' + \lambda X = 0 \)
• \( Y'' - \lambda Y = 0 \)

These equations are much simpler and more manageable than the original partial differential equation, demonstrating how separation of variables effectively simplifies our work.

Fourier series allow us to express a function as a sum of sines and cosines. In solving Laplace's equation, once the variable separation is done, we need to apply the boundary conditions at \( y = 0 \). Here, the Fourier series becomes essential:Our temperature distribution along the bottom boundary is given as \( f(x) = 3 \sin 2\pi x - \sin \pi x \). The aim is to express this condition in terms of a Fourier sine series:\[ u(x, 0) = \sum_{n=1}^{\infty} D_n \sin(n\pi x) = 3 \sin 2\pi x - \sin \pi x \] Through comparison, coefficients \( D_n \) are determined:

• For \( n = 1 \), \( D_1 = -1 \)
• For \( n = 2 \), \( D_2 = 3 \)
• All other coefficients \( D_n = 0 \) for \( n > 2 \)

This results in an understanding of which modes contribute to the steady-state temperature at \( y = 0 \), helping us build the overall solution.

The final solution represents the steady-state temperature distribution of the plate. This means it describes the temperature when it no longer changes over time, reflecting equilibrium.From our earlier steps, using separation of variables and applying boundary conditions, we establish that:\[ u(x, y) = -\sin(\pi x) e^{-\pi y} + 3\sin(2\pi x) e^{-2\pi y} \]This expression combines the spatial and exponential decay elements from solving the equations for \( X(x) \) and \( Y(y) \). It tells how the temperature behaves not only at different points along the plate but also as you move up along the \( y \)-direction away from the bottom boundary. Each term in the expression corresponds to a mode of heat conduction across the plate, quickly reaching a steady state. What this achieves is an insightful model of how the physical system dissipates thermal energy in a structured way, with the exponential terms ensuring that the influence of any temperature on the boundary reduces as we move further away along \( y \).