By T. Geveci (auth.), Dr. D. P. Laurie (eds.)

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Extra resources for Numerical Solution of Partial Differential Equations: Theory, Tools and Case Studies: Summer Seminar Series Held at CSIR, Pretoria, February 8–10, 1982

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4. 1 Let us consider the initial-boundary-value problem for the heat equation: au at(t,x) - v2u(t,x) u(t,x) = 0, t > 0, ° ,t u(O,x) = 'P(x), We may write this problem as an bert Space in various ways. 1) E an , x En. quatio~ in a Hil= The first alternative is to take L2 (n) to be the basic space H and consider the (unbounded) operator _v 2 on its domain D(A) * L2(n). For example, we can take D(A) = H~(n) n H2(n), and this choice corresponds to the semi-classical setting for the Dirichlet problem for _v 2• ((L 2 (rl),H 2 (rl))-well-posedness was discussed in the previous 34 section).

For a satisfactory discussion of the spectral properties the reader is again referred to [IJ, [10J, [14J. For future reference let us note the representation of the solution operators T for u = 0 on ,hI, and for -v 2 u + u = f in rl au an = 0 on arl, in terms of eigenfunctions {I/'j}. 51) E ]1 . (f ,If! \ 2(~) If!. j=l J J J , j=1,2, ... For the Neumann problem ~ dn = 0, A1 = 0, and the corresponding eigenspace is spanned by constants, so that for (f,l)L2(~) = 0, Tf= E ]1. ' j=2 J J J (T maps the orthogonal complement of the span of 1 into itself).

Am{~)} be the eigenvalues of A{-L{~)) - Define -L{~). = max me A.. 38) is L2 -well-posed [13]. The other important special case is the class of ~y~~em~. 6~~-okdet hypetbofie Consider au ) - d a d -rr{t,x I: A. 44) u{O,x) = 'P{x), x Em, where u is :Rm-valued, Aj'S and Bare mxm matrices with real entries. ~. J J (~E :Rd) {iAl{~), ... ,iAm{~)} and can be diagonali= 49 o o = iA(1;) , and sup {IIS(i;)II, I S-1 (I;)II } :s; i; M for some M. 42) and therefore L2 -well-posedness can be established [14J.

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