Question

1. Let u be a solution of the wave equation u 0. Let the points A, B, C, D be the vertices of the paralleogram formed by the two pairs of characteristic lines r-ctC1,x- ct-2,+ ct- di,r +ct- d2 Show that u (A)+u (C)-u (B) + u (D Use this to find u satisfying For which (x, t) can you determine u (x, t) uniquely this way? 2. Suppose u satisfies the wave equation utt -curr0 in the strip 0 < z < L, t > 0 and the boundary conditions u (0,t) u(L,t) 0 for t > 0. Show that u (z, t+ )--u (L-x,t). Conclude that u has period 2 in t. (Hint: use the above problem.) 3. Solve the initial-boundary value problem uz (0,t)-ka(0, t) = 0, k E R, t > 0 Hint: Set v(x,t) - ur,t)- ku(x,t) and consider the initial-boundary value problem for v 4. Prove the Duhamel's principle: Let u,t, T) be the solution of the initial value problem -00 < x < oo, t>0 u(z,0,r) = f(x, τ), -oo < x < oo, t > 0 and set Prove that v is the solution of the problem Vt - kvxz-f(x,t), -00<0 (x,0)f(x), -00 < x < oo, t>0

Answer 1

According to rules and regegulations , we are able to do only one question at a time... So i do question 4th...

Any query in solution then comment beloq.. i will explain..  

2. As uut that