Heat Equation

Heat Equation The temperature variation u = u(a:,t) in a rod of length 1r. initially at temperature given by f(x). then positioned with one end in ice and the other end insulated. can be modeled by the heat equation c'3;u=6§u, O<x<1r, t>O, with boundary conditions (BCs) u(O, t) = O, 6Iu(1r,t) = 0, and initial condition (IC) u(a:,0) = f(x). (3) Show that the general solution satisfying the heat equation and the BCs is given by cc 2 2 + 1 2 + 1 u(a:,t) = 2A,, s1n 27) exp [- tj nro (b) Show that. when fitting the IC, the unknown coefficients An can be determined from f(:r:) via 2 7" . 2n+ 1 An - 1-r/0 f(a:) s1n dx. You may use the result “sin flat sin flat dz: 0’ nsém n,mEN. 0 2 2 1r/2, n = m [10 marks] - Laplace's Equation for an annulus Find a solution 12 = v(r,9) to the following Dirichlet problem for Laplace's equation in polar coordinates. 1 1 83v+;8rv+r-28§v=0, 1<r<2,-rrgélgrr, v(1, 9) = sin 46 - cos 9, v(2, 0) = -sin3 0 + 3(cos2 0)(sin 0) PLACE THIS ORDER OR A SIMILAR ORDER WITH US TODAY AND GET AN AMAZING DISCOUNT :)