Question

A metal rod of length a cm has initial temperature function f(x) = 2 sin 3x and its two ends are held at temperature zero for all time t>O The heat equation is given as: ди au 4 for 0 < x < it and t > 0 at @x? Boundary conditions: u(0,t) = u(1,t)=0, Initial conditions: u(x,0) = 2 sin 3x By using the method of separation of variables, calculate the general temperature u(x,t) for all cases, k = 22, k = 0, and k = -22.

Answer 1

We have to Solne su 2²4 st for ocnak d2² and to Boundary amelitring: 110,t) - uLt, t) = 0 initial andlitum uln,0) = 2 sinsu For By using the method of separation of vanastes. let ulu, t) = Xx) T(*) Then su st 4 2²4 duz XT! U 4X"T => TI 4T X х - K = ) x"-KX = 0 T-4KT = 0 and Also, 410, t) X(9) Tt) = 0 | X10) = 0 similarly LXcajo have to solve Now, nee X" - kx = 0 ; XO) = 0 = X(T) ko it let cose ka 1² 1

x" - X=0 Xcno = A en fa + Bé where AR Bare constit. Sinee X(0) = 0 => A+B=9 IBE-A] Also, X() = ST A é t Be B é 1h -> Alett +*) -> LÃ - o - Bio] Henee =) ulu,t) = 0 T =0 j.e it ko the general temperature is zen. ose-2 it k=0 X"=0 =) X(2) An & B where A & B are con steint.

Then By -*) -36t Az sin(3n) e ulnt) It init) -36t = 2 sin 13n) e