MATH 251 Lecture Notes - Partial Differential Equation, Initial Value Problem, Ordinary Differential Equation
Document Summary
One-dimensional heat conduction equation revisited; temperature distribution of a bar with insulated ends; nonhomogeneous boundary conditions; temperature distribution of a bar with ends kept at arbitrary temperatures; steady-state solution. Previously, we have learned that the general solution of a partial differential equation is dependent of boundary conditions. The same equation will have different general solutions under different sets of boundary conditions. We shall witness this fact, by examining additional examples of heat conduction problems with new sets of boundary conditions. Keep in mind that, throughout this section, we will be solving the same partial differential equation, the homogeneous one-dimensional heat conduction equation: 2 uxx = ut where u(x, t) is the temperature distribution function of a thin bar, which has length l, and the positive constant 2 the bar. The equation will now be paired up with new sets of boundary conditions. is the thermo diffusivity constant of. The first step is the separation of variables.
