Consider the following wave equation for u(t, x) with boundary and initial conditions defined for 0...

Consider the following wave equation for u(t, x) with boundary and initial conditions defined for 0 ≤ x ≤ 2 and t ≥ 0.

∂ 2u ∂t2 = 0.01 ∂ 2u ∂x2 (0 ≤ x ≤ 2, t ≥ 0) (1)

∂u ∂x(t, 0) = 0 and ∂u ∂x(t, 2) = 0 (2)

u(0, x) = f(x) = x if 0 ≤ x ≤ 1 1 if 1 ≤ x ≤ 2.

(a) Compute the coefficients a0, a1, a2, . . . of the Fourier cosine series of f(x) in the interval 0 ≤ x ≤ 2. For your calculation, recall that sin(πn) = 0 for all integers n.

(b) Find all the possible values of λ, µ ≥ 0 such that the function v(t, x) = cos(λx) cos(µt) solves the wave equation (1) and the boundary condition (2).

(c) Use the superposition principle, the Fourier series coefficeints a0, a1, . . . from part (a) and the functions v(t, x) from part (b) to write down an expression for the solution u(t, x) of the boundary and initial condition problem (1)–(3).

Expert Solution

Colby Messinger answered 1 year ago

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