Question

For (1) – (3), the model is with regards to a rod of length L with thermal diffusivity k coinciding along the interval (0, L) on the z-axis. Set up the boundary-value problem for the temperature u(x,t). (1) The left end is insulated and the right end is held at a temperature of 0°. The initial temperature is 1° throughout. (2) The left end is at a temperature of 50e-t, the right end if held at zero, and there is heat transfer from the lateral surface of the rod into the surrounding medium held at temperature zero. The initial temperature is 100° throughout. (3) The right end is insulated and the left end is held at a temperature of 0°. The initial temperature is 1°

Answer 1

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olour Date: / / Page no: © General relation t * heat wave equation is given by X=0 231 Heat equation = e² Una = ut - ė let ² x"T H = XT XT' using equatice XI_on both sides let equate both side divide by -12 x let L. Hos T c²T 2 x" y? let RH.S I'tc²T4²=0 x"'tdx=0 General Salution of this equation. far = (- c²²dt = lnt = -c²²t T = e-c²1²7 ulot = XT= (a cosche + Bsindn)e-c-it Now, applying Boundary conditione 0 @ Uyg X=0 Cchange in temperature is zexs) at ull, t) = 0 at ze=L @ on applying on equation 0 स Un Cat) = e (- Asindu td + Blosaard) :. Uplant) so en Corpasing we get Beol : using in equation o Note Book

Date: / / Page no: e-Reht to 2 do t = te-ci. @ lucou t) = _Acosdx_cottat o= a cosit e ori²c24 Azo Cosdl Cos (n+L (n=0,1,2 inda (ntal) 1 n = 0, 1, 2, 3 using in equaties @ (n+4)*x? + ula, t) = Ap cos (n+1) A e 1 initial temperature = °C M+4)2x2+ is į An Cos (m + 1 = [ An cos (n an -max*t/u2 noodd A = 1 this is true only for n = 0g 2, 4 Otherwise A=0 mar24/412 ulaut) = © @ ula=o) 500-t u (n =) = 0 and ini tilal temperature =100°c on applying on equation @ @ = 50 e e Ż 2 no . An Cos (m + 2) I те neo e 2L n = even. i cos .NIN e N=enell 2L i. 3 Note Book

Cours Date: Page no: 0 = (Acosde + Bsindu) ete's i. Acosch = B Sindh B = -A lot dL 0 ning equation @ and ☺ in equation mbete soet (cosdx-cot dk Sindue) - 6 Now intial temperature = 100°C t=07 Cosette -Cot de Sindn 100 = 50 e o Cosda cot dl sindas 72 we get using in equation 6 ulnot) = 50e-t x2 loo ett 3 FO ull) = UA (L) = o insulated end. 63 ulugo) = using in equation Pulau Acosdx + 8sinduentzat O = A in U14) = Bsinda e un = d Bloody e col met Cat =b; 4x=o) cosdl = 0 = cos (n+1 and²c²t AL n+ 1 / 2 da na 1 L 1+2) 0,4 2,2- :. u.CN, t) = 2 Bn Sin (n+1Le muje sčat (n+1) 1 L2 n=0 Thanks Note Book

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