Question

Problem 1. Find the general solution of an ID heat equation: Tt(x,t) = 4Txx(x,t) with the boundary conditions T(0,t) = T(2,t) = 0. Note that T(x,t) denotes the temperature profile along x of a uniform rod of length 2. Problem 2. Solve the following ID wave equation: Ott(x,t) = 0xx(x,t) with the boundary conditions 0 (0,t) = 0;(1,t) = 0, where 0(x,t) refers to the twist angle of a uniform rod of unit length. Problem 3. Show that the solution of the partial differential equation (Laplace equation), Wxx(x, y) + Wyy(x, y) = 0, with the four boundary conditions: w(x,0) = 0, w(x, 1) = 0, w(0,y) = 0 and w(1, y) = 24 sin ny, can be obtained as w(x,y) – 2 sinh nx · sin ny.

Answer 1

solution- given that Boundary condition probelm & Tt (x, t) = 4 Tox(xit) treger) TCO,t) = T(21t) = 0 let 4 TOC,t) = x (T()t> equation ® Tt = XT', Tococ = X'T → put in equation ③ XT = "X" T > XdT = 4 d²T T T dx² I di T dt dt 4 d²T Х doc? -dc-ve constant term) 2 + ---, y ч "Т X doc? - T I dt 2 -> d²T - dT dt d X 4 2 > IdT -IT dIt td x=0 dt dx? finding complementary functiona T=cent X = Acasd x t B sinda çete 4 TC, t) = X(X) T(t) (Acos dx + Bsinda) absorb in this - det T(x, t) = e Cacos dx + Bond 0c) > equation 4

using boundary conditions Tot) = TC2it) = 0 taking boundary conditions in equation ③ -*t Tot)=0 V (Acasd.co) + B Sindo) 4 it T 877?t never be zero 4 4 A=0 put in equation ③ T(x, t) = е -it (Acer qx + Bsindx) ( Bsind x) Tcxit)= e -dt → equation another taking Boundary conditions 4 B never be zero T (ait)=0 -24 ( Bsin b sindxx) Bsind=0 B sinA= sina B= */ put in eq - ( 7 sind x) uniform rod length De=2 At T(x, t) = T Sir sin Ax?) Txt) I sin equation V T(x,t)= eft 4 th 7.

The equation & becomes temperature profile along sc of a uniform rod of length @ -dt Text) = e (77 * sind Answer