Question

Problem 1 (Submit): Solve the following inhomogeneous initial boundary value problem for the wave equation: qu=cu, te+cos (31), 0<=<5, t>0 13(0,t) = 0 and u(t)=t, t>0 u(1,0) = cos(I), 24(1,0) = 1 + cos(51), 0<I

Answer 1

Soin: Given inhomogeneous initial boundary value problem for the wave loi : Una curet et cos (3x), Oxx<, txo Uzco,t)= 0 & u( $t)= + , to ul2,0)=cost, 44(%,0) = 1 + cos(53), 0245 Now, un, t)vcn, t) + wcn, ) → we take the w the function, Win,t) = Then, the function has to be the sol" of following problem, Vr = c? Vont el cos (an), o<n<y, tyo 1960,t) = 0 & VCA, 1) = t, tro vcn,o) . cogn, viinio)It cos on), Ocne => The eign-values & the corres ponding eigen function for the assoc Sturm-Liouville problem x"+920, ouncta xCo): ACK), 0. are given by an = n² & xg (n) = cos (nn). >> Van, t) - E Th Ct)cos on * Vt - Vonn = a {th (t) + n(t) / Cosniva - é'cos 32 which implies that D%C++ n TACH) . So that undh n> :,, n73, 121 Tocou as peon-my] cosando

2 m+ 3, Then In ct). In cově n? If n = 3, then, -90 973 (1): yct)e-96-976)e'+91 CH) +9Act) è at - A'CH)e-at=et so that, ack): est & ACHE AC) + s'esods - 109 +Cest) Hence, al venot) . E To ly) Cosmos į Test,je 't E Tn colē Eoson & nal uCot) = ***5.374, Tacosetcosm n>