Question

Consider the nonhomogeneous 1D wave problem utt​+ut​=4uxx​+f,00u(0,t)=h(t),ut​(L,t)=r(t)u(x,0)=g(x),ut​(x,0)=k(x)​ where the functions f,r,h,g,k are all continuous. x.) (0.5pt) Write the problem satisfied by w=u1​−u2​, where u1​ and u2​ are two solutions of the problem. b.) (2.5 pts) Prove that w(x,t)=0, for all x∈[0,L] and t≥0.

Answer 1

To find the problem satisfied by w=u1−u2, substitute w into the wave equation utt+ut=4uxx+f,00u(0,t)=h(t),ut​(L,t)=r(t)u(x,0)=g(x),ut​(x,0)=k(x)​ and simplify. To prove that w(x,t)=0 for all x∈[0,L] and t≥0, assume that u1 and u2 satisfy the wave equation and substitute w into the equation to show that w=0.

Step-by-step explanation:

In the given 1D wave problem, consider two solutions u1 and u2 of the problem. We can find the problem satisfied by w=u1−u2 as follows:

Step 1: Substitute w into the wave equation utt+ut=4uxx+f,00u(0,t)=h(t),ut​(L,t)=r(t)u(x,0)=g(x),ut​(x,0)=k(x)​.

Step 2: Simplify the equation to solve for w(x,t).

b.) To prove that w(x,t)=0 for all x∈[0,L] and t≥0, we can assume that u1 and u2 satisfy the wave equation. Since w=u1−u2, it will also satisfy the wave equation. By substituting w into the equation, we can show that w=0.