Answer to Question #40555 in Math for nate ruiz

Question

Partial Differential Equations:

Please assume that the improper integrals ∫_(-∞)^∞▒u_0 (x)dx,∫_(-∞)^∞▒〖u_1 (x)dx〗,and ∫_(-∞)^∞▒f(x,t)dx ∀t are convergent.
Prove that if u(x,t) is the solution of
u_tt-c^2 u_xx=f(x,t) x∈R,t≥0
u(x,0)=u_0 (x)
u_t (x,0)=u_1 (x)
then:
∫_(-∞)^∞▒u(x,t)dx=∫_(-∞)^∞▒u_0 (x)dx+t∫_(-∞)^∞▒〖u_1 (x)dx〗+∫_0^t▒〖(t-τ〗)∫_(-∞)^∞▒f(x,τ)dx dτ
Hint: Use the D’Alembert’s Formula for u(x,t) and change the order of integration dx and ds in the 2nd and 3rd terms.
Under what conditions on u_0,u_1,f is it true that ∫_(-∞)^∞▒〖u(x,t)dt=constant?〗

Answer 1

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