MATH461: Fourier Series and Boundary Value
                                 Problems
                     Chapter II: Separation of Variables
                               Greg Fasshauer
                          Department of Applied Mathematics
                            Illinois Institute of Technology
                                  Fall 2015
        fasshauer@iit.edu        MATH461–Chapter2                          1
  Outline
   1  Model Problem
   2  Linearity
   3  Heat Equation for a Finite Rod with Zero End Temperature
   4  Other Boundary Value Problems
   5  Laplace’s Equation
        fasshauer@iit.edu        MATH461–Chapter2                            2
                             Model Problem
   For much of the following discussion we will use the following 1D heat
   equation with constant values of c,ρ,K0 as a model problem:
         ∂             ∂2           Q(x,t)
         ∂tu(x,t) = k∂x2u(x,t)+       cρ   ,    for 0 < x < L, t > 0
   with initial condition
                       u(x,0) = f(x)     for 0 < x < L
   and boundary conditions
                u(0,t) = T (t),   u(L,t) = T (t)     for t > 0
                           1                2
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                                   Linearity
   Linearity will play a very important role in our work.
   Definition
   Theoperator L is linear if
                     L(c u +c u ) = c L(u )+c L(u ),
                         1 1     2 2      1    1     2    2
   for any constants c ,c and functions u ,u .
                       1   2                 1  2
   Differentiation and integration are linear operations.
   Example
        Consider ordinary differentiation of a univariate function, i.e.,
        L= d. Then
             dx
                    d (c f +c f )(x) = c d f (x)+c d f (x).
                   dx    1 1    2 2         1dx 1         2dx 2
         fasshauer@iit.edu        MATH461–Chapter2                             6