Question

b) Solve the following problem using Laplace transformation: u = 104, +x.p?, 0<x< too, :>0, (0,1) = 1, u(x,0) = 0 Tim u(x,1)=0 *-* Xvx

can someone solve this question in good hand writing with explination of steps

Answer 1

(6) Som where, solve the following broblem using Laplace transferam u = 10 usex txt olacas to ulotl= , u(x,o =o lim u(x, t) =oy хохх we know that L (u) = sū - u(x, o) = su here. L{u(x, t)} = ū (2,8) & L (Uxx) - ffpu} = given problem is u = 10 uxx + x + 3/2 on taking Laplace (u) = L (louxx + x + 3/2) - L(it) = 1o Luna) + x L (1/2) - sū = 10 do + x ₂ / ₂ 1/2 dx? » so = 10 do + x, 3 x + dx2 48512 >> 10 - BE - EVE

48572 (108²-8) 0 = -x (350) - CF - CF = C, eta kot ce etiola P2 48/2 10028 = true love orya - svet] ! + 109° + D"))* - B JF (x +o) 48 7/2 PI = 350 x 487/2 C = CF & PI NTS/10)» ($9/10 x + 3VIE .r_0 + 3 487/2 - (4) Boundry condition are

u (o,t) = 1 Applysing Laplace ū (0,8)=L(11 & lim u(x,the nyo xa transforms sim Pū (x, 8) = L(0) - ayoo xx =) u (os) = 1/8 tim ū (x,x) = 0 nyo an from ū = c. e vello to è solo + 3 x 40 87/2 but x=0 => V = , +62 put x - lim asoo x x u x elio = x cen lim 3% + en 5/10 e 3 G lim nro xas 2 O clim - ) + - 0 + 0 ato 22 / do to c=0 , because him Hon from 0 then from 4 C2 = 18

= ū = t és + 3 x 4,772 » L {u(K, 2)} = + e + 355 x 15th on Applying & L > 1) = L", * V10] + syt Luca I years ? - este represe - error function we know that u(x, t) = erfa 35 x 5 5/2 Tala 512 اگ 512 E of (a) * Bavit a ulx,l) = este ) + sexy where exfc (8) = 2 jo e-heap lot where. 22 1