Question

1. Use the Laplace transform to solve the following BVP au au əx2 = ət, x > 0,t > 0 u(0, 1) = tuo, lim u(x, t) = u1, t> 0, u(x,0) = U1, x > 0.

Answer 1

Sony Given: = - 2 du at let's apply Laplace transform on both sides We know how ) = shul- ufro ) duz let L(u) = U and we know u(0,0) = U, So, 1104) - SU-U Then ear * will be: 2 = SU-u =) [a²v - su=-41 To solve this D.E let's first find out the complementary function , ie by substituting wro arv - SU=0 - 0 We know the general solution of this egn, which is : V(x,S) = Als) e S x + Bljeta (Hint: To obtain this solh, just suppose u = Aeta and substitute in 1 to obtain → Now we have to find the particular solution which is U (x,s) = Tu So net solution we obtain is : U (2,5) = Als) e Jon + Bisless x + . Now, We are given lim ulat) = u to. TERROR

infinito As x approaches So Brs) = 0 for So, we have infinity , and tum approches uln,s) to be finite. -JSX U(X,s) - Alsle + un - 09 Also, we have ulo,t) = no Taking Laplace of this we get Equating this to egr 2 for x=0 uo = A(s) + u so, A() = (uo-ul) Then, (as) = 1 Uo-ui) e Is x tui To obtained the solution we have to use laplace transform S inverse (ulet) - woll-as (re) + w of)