![]() a useful strategy in developing an analytical solution when designing practical extended surfaces with suitable geom-etry for temperature response.Solve Nonhomogeneous 1-D Heat Equation Example: In nite Bar Objective: Solve the initial value problem for a nonhomogeneous heat equation with zero initial condition: ( ) ˆ ut kuxx = p(x t) 1 0 u(x 0) = f(x) 1 If we substitute X (x)T t) for u in the heat equation u t = ku xx we get: X dT dt = k d2X dx2 T: Divide both sides by kXT and get 1 kT dT dt = 1 X d2X dx2: D. ![]() A more fruitful strategy is to look for separated solutions of the heat equation, in other words, solutions of the form u(x t) = X(x)T(t).(II) there is an ODE for the density with analytic solutions for any n BUT more important the ODE for the velocity field: For n =1/2 and n= 3/2 HeunT functions tooo elaborate BUT for n=2 we have the Whitakker functions The solution for the non-compressible case There is no kappa 0 limit to compare. The compressible Newtonian Navier-Stokes eq.2D Heat Equation in Cartesian Coordinates Two-Dimensional Wave Equation 2D Heat Equation in Polar Coordinates Exercises 226 226 227 231 231 231 233 234 239 239 244 Finite Element Method 249 10.1 General Framework 250 10.2 ID Elliptical Example 252 10.2.1 Reformulations 252 10.2.2 Equivalence in Forms 253 10.2.3 Finite Element Solution 255
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