advertisement

MA2332 Tutorial Sheet: due on 1st April, 2016 1 1. Use separation of variables to solve the two-dimensional Laplace equation for φ(x, y) with boundary conditions given by φ(0, y) = 0, ∂ 1 φ(0, y) = sin(ny), (n ∈ Z). ∂x n called the Hadamard conditions. 2. Using separation of variables to solve the two-dimensional Laplace equation ∇2 V (x, y) = 0, subject to the boundary conditions: V (0, y) = V (d, y) = 0 and V (x, 0) = V0 sin 2πx , d with d and V0 ∈ R . 3. Use separation of variables to find a solution to the 1-dimensional wave equation 2 ∂ 2φ 2∂ φ = c ∂t2 ∂x2 ∂φ (x, 0) = g(x) ∂t Note: you may assume that the formula for the coefficients of a Fourier sine series for a function h(x) defined on 0 ≤ x ≤ L is with the boundary conditions: φ(x, 0) = f (x), φ(0, t) = 0, φ(L, t) = 0, 2 bn = L 1 Z L h(x) sin nπx 0 L dx, n = 1, 2, 3, . . . SineĢad Ryan, see http://www.maths.tcd.ie/˜ryan/231.html 1