Find the steady-state temperature distribution in a rectangular plate covering the area \(0

The Laplace equation is a fundamental partial differential equation (PDE) in mathematics, often written as \[ \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0\]. It's crucial in problems involving steady-state heat distribution and other areas such as electrostatics and fluid flow. When we say 'steady-state,' we mean that the temperature at each point in the system doesn’t change over time. Understanding this equation is key to solving problems like the rectangular plate heat problem. Here, you have no dependence on time, which simplifies the problem significantly by focusing only on the spatial variables x and y. Use the Laplace equation to set up the problem before applying the boundary conditions.

Boundary conditions are the constraints required to solve differential equations such as the Laplace equation. In the rectangular plate problem, we have specific temperature values at the edges:

• T=0 when x=0, x=1, and y=2
• T=1-x for y=0

These conditions allow us to determine the unique solution for the temperature distribution. Essentially, these constraints enclose our problem within a defined 'boundary,' guiding us toward the final solution.
Each boundary condition translates into mathematical expressions that shape our approach and solution process. For steady-state problems, these conditions are typically set by physical constraints or empirical observations.

Separation of variables is a method used to solve PDEs like the Laplace equation. It involves assuming that the solution can be written as the product of two functions, each depending on a single coordinate. In this case, let \[ T(x, y) = X(x)Y(y) \]. When we substitute this form into the Laplace equation, we get\[ X''(x)Y(y) + X(x)Y''(y) = 0 \]. Dividing by \[ X(x)Y(y) \] lets us separate the equation into two ordinary differential equations (ODEs): one for \[ X(x) \]and one for \[ Y(y) \]. This technique simplifies solving the PDE since each ODE can be tackled independently before recombining the solutions.

Ordinary Differential Equations (ODEs) are equations involving a function of one variable and its derivatives. In the process of separation of variables, we break the Laplace equation into two ODEs. For the rectangular plate problem, these ODEs are:

• \[ X''(x) + \lambda^2 X(x) = 0 \]
• \[ Y''(y) - \lambda^2 Y(y) = 0 \]

Solving these for given boundary conditions provides functions for X(x) and Y(y). The solution typically involves standard functions like sines, cosines, exponentials, or hyperbolic functions. These ODE solutions will help us form the overall temperature distribution solution. They turn a complex PDE problem into more manageable parts.

The rectangular plate heat problem involves finding the temperature distribution in a rectangular area with given boundary conditions. This problem is framed by the given boundaries and conditions, which in this case, are:

• 0 < x < 1
• 0 < y < 2
• T=0 for x=0, x=1, y=2
• T=1-x for y=0

Following the steps:

• Set up the Laplace equation.
• Use separation of variables to propose a solution.
• Solve resulting ODEs for \[ X(x) \]and \[ Y(y) \].
• Combine these findings to form an overall solution.
• Apply boundary conditions to determine specific coefficients.

By following these steps, we get a precise mathematical model describing how heat distributes across the plate in a steady state, resolving the given boundary conditions.