Question

get the coefficents of the fourier series in terms of an intigral

mts) Separate variables and find Xn(x),Tn (t) for Uz = 3uxx ux(0,t) = 0, u(1,t) = 0 u(x,0) = x(1 - x) the Fourier series in terms of an integral.

Answer 1

solution - Given entiel valle problem is = 30xx . and boundary conditions & Co, t)=0, U(1,t) = 0 and u(x,0) = x (1-2) let u(xt) = X(X) (t) be the solution of quation then = X(1) T'ct) and Wax- XNT(t) Now substitute me rate & Ut and an into equation 0 and we get XT= 3XT - Now use the separation & variable to solve above еччя him 2 х I': 3x" - 1 (lit) X Now I'- 31T=0 – and X-4x20 - Now solution of equation ( and & depends upon the value of t. There are three posibilities of it id=0 (ii) do cil) d = -4² (negative) since first two posibilities are gives the trivial solution so we consider third posibility = -1² to get solutia

and Now put t= -4² into equation Tt 34T =0 - 6 and x" + 4 x =0 - 6 Now solution - 34²7 - 342€ 7the c, et- and solution of equation & it XH) = C Cop Mx + C₃ Hinux - Now solution of given intial problem is uca,t) = Gesut (op 4 x 4 Cz 14x) - Now from boundary condition 0,0,1) = 0 + 1 ) = and ull,t) =0 cop (4x1) =0 u= 7 Hence using the superposition principle, we have 0 - 3 T² U1X,t) = Ese+ ( Cop NTTA) - © How using intial condition u(x,0) = a(1-2), we get (12) 5 Cọ tơ (23) which ip half-range pourier cosine perjes n=1 |

therefore courier coefficient en 10 en = d | 21-4) top (1TX) dz @ Hence from equation 0 and ③ we get the solution - 3n37²2 went = $ tej 41a) top (972) de] 2 Powered an