Question

An infinitely long copper wire coated insulting material on its surface is assumed perfectly insulated laterally. Thus heat in the wire flows in x-direction only.- In this case the temperature of the wire can be modeled by 1-D heat equation: au au ax at Assume that initially the temperature distribution of the wire is u(x,0)-T for x<a A. Please show that the solution of 1-D heat equation in this case is ulxt)-o [Acos(px)Bsin (px)]eptdP where A and B are the coefficients to be determined by the initial condition, c is the thermal diffusivity B. According to the given initial condition, please find A and B in (A). C. Please write down the entire solution in integral form the satisfying (B)

Answer 1

Let u (x, f) = X (x) T(t) therefore partial differential u/ partial differential f = e^2 partial differential^2 u/ partial differential x^2 gives X partial differential T/ partial differential t = e^2 T partial differential^2 X/V x^2 rightarrow 1/T partial differential T/ partial differential t = e^2 1/X partial differential^2 X/ partial differential x^2 in the above equation left hand rule is a form of f only merges right hand side is a fine of x only so, they must be consistant therefore 1/e^2 T = partial differential T/ partial differential t = 1/ X partial differential^2X/ partial differential x^2 = -P^2 therefore partial differential T/ partial differential t = -p^2 e^2 T rightarrow partial differential T/T = -e^2 p^2 partial differential t rightarrow ln T = -e^2 p^2 t + k rightarrow T = A e^-e^2 p^2 t \r\n l partial differential^2 X/ partial differential x^2 + p^2 X = 0