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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 fasshauer@iit.edu MATH461–Chapter2 4 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
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