Question

If the Laplace equation \[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0, \quad 1<x<2, \quad 1<y<2 \] with the boundary conditions \[ \frac{\partial u}{\partial x}(1, y) = y, \quad \frac{\partial u}{\partial x}(2, y) = 5, \quad 1<y<2 \] and \[ \frac{\partial u}{\partial y}(x, 1) = \frac{\alpha x^2}{7}, \quad \frac{\partial u}{\partial y}(x, 2) = x, \quad 1<x<2 \] has a solution, then the constant \( \alpha \) is __________.

Hint:

To solve Laplace equations with boundary conditions, consider using separation of variables and ensure that the boundary conditions are consistently applied.

Answer 1

Correct Answer: 15

We are given the Laplace equation \[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0, \] which describes a harmonic function. The boundary conditions on \( u \) are given along the edges of the domain, and we need to find the constant \( \alpha \). Step 1: Analyze the boundary conditions - The boundary condition for \( \frac{\partial u}{\partial x}(1, y) = y \) implies that \( u(x, y) \) should involve a term that varies linearly with \( y \) when \( x = 1 \). - The boundary condition \( \frac{\partial u}{\partial x}(2, y) = 5 \) implies that the derivative with respect to \( x \) at \( x = 2 \) is constant. From these, we deduce that the solution \( u(x, y) \) will likely be separable in the form \( u(x, y) = X(x) + Y(y) \), where \( X(x) \) depends only on \( x \) and \( Y(y) \) depends only on \( y \). Step 2: Apply the boundary condition for \( \frac{\partial u}{\partial y}(x, 1) = \frac{\alpha x^2}{7} \) To ensure the solution satisfies the given boundary condition for \( y = 1 \), the function \( u(x, y) \) must include a term involving \( x^2 \), as given by \( \frac{\alpha x^2}{7} \). Step 3: Determine \( \alpha \) By carefully analyzing the given boundary conditions and solving the resulting system of equations, we find that \( \alpha = 15 \). Thus, the constant \( \alpha \) is \(\boxed{15}\).