Mathematical Methods for Physics and Engineering - Matematica.NET

PDES: SEPARATION OF VARIABLES AND OTHER METHODSf(x)−L0Lx−HL/kFigure 21.4 The appropriate continuation for a Fourier series containingonly sine terms.in order to satisfy ∂v(L, t)/∂x =0werequirecosλL =0,andsoλ is restricted to thevaluesλ = nπ2L ,where n is an odd non-negative integer, i.e. n =1, 3, 5,... .Thus, to satisfy the boundary condition v(x, 0) = −Hx/k, we must have∑ ( nπx)B n sin = − Hx2L k ,n oddin the range x =0tox = L. In this case we must be more careful about the continuationof the function −Hx/k, for which the Fourier sine series is required. We want a series thatis odd in x (sine terms only) and continuous as x =0andx = L (no discontinuities, sincethe series must converge at the end-points). This leads to a continuation of the functionas shown in figure 21.4, with a period of L ′ =4L. Following the discussion of section 12.3,since this continuation is odd about x = 0 and even about x = L ′ /4=L it can indeed beexpressed as a Fourier sine series containing only odd-numbered terms.The corresponding Fourier series coefficients are found to beB n = −8HL (−1) (n−1)/2kπ 2 n 2and thus the final formula for u(x, t) isu(x, t) = Hxk− 8HLkπ 2∑n odd(−1) (n−1)/2n 2sinfor n odd,( nπx) ( )exp − kn2 π 2 t,2L 4L 2 sρgiving the temperature for all positions 0 ≤ x ≤ L and for all times t ≥ 0. ◭We note that in all the above examples the boundary conditions restricted theseparation constant(s) to an infinite number of discrete values, usually integers.If, however, the boundary conditions allow the separation constant(s) λ to takea continuum of values then the summation in (21.16) is replaced by an integralover λ. This is discussed further in connection with integral transform methodsin section 21.4.724