Question

4.[10] Find the solution to given initial-boundary value problem: 4uxx = U, 0 < x <TT, t> 0 u(0,t) = 5, u(t, t) = 10, t> 0 u(x,0) = = sin 3x - sin 5x, 0<x<

Answer 1

Sol 4 x x = U t> o ulo, t) = 5 ULT, t) = 10 tro u(xo) = Sin 3a - Sin 50 О< x < ТТ Steady state utao Junx = so au an 2 А x = A U = AX + B Qex, t) = = AX+B Steady sol? when X=0 9 y (0,t) 25=B B = 5 when a=T, U (T ) = 10 = A.TT + 5 A = 5 UX, t) = A x + B = U(X+)- 16%) - Us (x, t) U(x, t) - Ult) = - 15x+5) when x=0 Ulogt) a ulo,t) - Uslot) ulo,t) 5 - 5 ulo,t) = 0 New Left hand side B C

T(t) & Co e-4 n2T when 2-T UCT, t) = 17,t) - Us ( TT, t) 10- 10 UlTT,+) New Right hand side BC when tro U12,0) = U( x 0) - Us (x, o) Ulxo) = sin3x-sin5x – 2 x - 5 Zero BCS ų = 4 Uxx 110,0) = 0 U ( TT, t) = 0 Ulx,0) = Sinza sin 5x - 5 - 5 - 5 Case I f Caset are ginen trivial sol K=-22 Case II x", 2x=0 xol=0 X (X) = A cosant Bsinan X(o)= A co X (x1 = B sinar X(T) = 0= BSinani & Sina0 non rivical sol аТЕ ПТ an n-1, 2, 3 - - X(X) = Bn Sinna T' = 4KT Kard2 T' = 4 (-22) T T'=-4 m2 T

Salue. 4 Uxx = Ut 0 < x < TT, to ulo,t) = 5 UCTT, t) =10 to ulx, o) - sin 3x - sin 5x 0 < x < TT a²u 4. a x2 - au at separation of variables! Let ulx, t) = X(X Tit) VO, t) = 5 = X(0) T(+) = 5 tt UITT,t) = 10 =) X CTT) T(+) = 10 tt au at =XT' 2 X(0) E X (TT) 2 =x"T ax2 4X"T=XT' T! X! 11 4T k is the const. T = K T'=4KT X!! X =) x" - KX=0 (2) x"_KX=0 Case 1 Ka da zo azo 9 X" – a 2x=0 X (2) i Aeda & Be-an XO) = A + B = * (TV) A+B = X(TI 2 XLT) =Ae -dT + Be

U(X,+) XCX T,t) 4024 ulx, t) = { An sinna e Nal 5 U (x,0) = f(x) = sin 3u - sin 58 - x – 5 휴가 Sin 3x- sin5n - 5 - - - - n-5= e An Sinna nal sinna sinma da i nem म nem s (sinan - Sin 5x – 2 2 - 5) sin nada s E An sinna s in mx da na Sinan Simnit - sin si sinna o cionis An ( + 0+0.-) T - 5 Sinan Sina 5 sinna 2 An T 1/2 - 2 Anh 2 1 here sinn a geet put the general sol? we and