Question

k2 Solve the following partial differential equation by Laplace transform: д?у ду dx2 at , with the initial and boundary conditions: t = 0, y = A x = 0, y = B[u(t) – uſt - to)] x = 0, y = 1 5 Where, k, A, B and to are constants

solve

Answer 1

Solution: Given K² zzy oy da² at t=0 4= 1 OC=0 ,ya B[uct) – ult-to)] OLEDO y=1 at hence y = constant which is equal to = B [ul- ult-to] yllo) B[ult) - ult-to) ] & ylo) = Now: take laplace transform of equation K? [ 52 yus) - Ylo) - ylos de (593) - yoool. k?[ se yls) – Sefult)-ult-to)] ] = SYS) A ( S- S) 913 K’s Blult) - ult-to)] - A = ys) Ko Sul1)- 44-to) 1-A s( K²S-1) A y(5) IdB (414) - ult-to) ] (K2S-1) 5 (125-1) KB Cult)- ult-to) ] ? 15 - 11/22) A/K2 s(5-

Y(S) = Blu 44 - ult-to)] BR) A/K? 5(5-) A K² K2 + Y(S) = b[ult) - ultito] ] S 12 is s-1/2) A Y(s) = + B[ult) - 017-to) ] ismes up (5 - 11/22 By formula at =e 1 ( S-a hence Yl) = ['y(s) = Blult) - ult-tol J. ok? Raft B tA - A.