Consider the wave equation ∂^2u/∂x^2u/∂t^2, −∞ < x < ∞, t > 0, with the initial conditions u(x, 0) = f(x), ut(x, 0) = 0.

Question

Consider the wave equation ∂²u/∂x² = ∂²u/∂t², −∞ < x < ∞, t > 0, with the initial conditions u(x, 0) = f(x), ut(x, 0) = 0. 

a. Find the equation satisfied by L(x, s), where L(x, s) ≡ ∫dte^−stu(x, t) for t ∈ [0 ∞]. 

b. Assuming that both f(x) and L(x, s) have Fourier transforms, find L(x, s) in the form of a Fourier integral. (You are allowed to differentiate a Fourier integral by differentiating its integrand.) 

c. Find u(x, t). Note: the Laplace transform of u''(t) is equal to s²L(s)−u'(0)−su(0), where L(s) is the Laplace trandform of u(t).

Answer 1

a. Let the Laplace transform of u(x, t) be L(s, x).We have, by multiplying the equation above with e −st and integrate with respect to t from 0 to ∞,