By Michael Christ, Carlos E. Kenig, Cora Sadosky
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Additional resources for Harmonic analysis and partial differential equations: in honor of Calderon
5. This definition suggests several other points that will be developed in the sections to follow, but are important enough to comment on now. 1) has index ^, then its solutions also satisfy an ODE where G is a function of partial derivatives of F. 4) are solutions of the original DAE. 4) is useful both theoretically and in some general numerical procedures. This ODE also sughhhgests that in general the solution of a DAE can involve derivatives of both input functions and system coefficients. This characteristic will turn out to have several ramifications for numerical procedures.
The fc-step (constantstepsize) BDF consists of replacing y' by the derivative of the polynomial which interpolates the computed solution at k + 1 times £ n , < n _ i , . . ,/ n _A;, evaluated at tn. This yields 41 42 CHAPTER 3. MULTISTEP METHODS where pyn = ]T^=o <*iyn_t- and a n * — 0 , 1 , . . , f c are the coefficients of the BDF method. The fc-step BDF method is stable for ODE's for k < 7. An introduction to the properties of multistep methods, and in particular BDF, for ODE's can be found in [114,137,159].
LINEAR TIME VARYING DAE'S 23 A rigorous discussion of the index is somewhat technical in nature and will be done at the end of this section after we have developed a better understanding of linear time varying DAE's. 1) by the implicit Euler method starting at time to with constant stepsize h. Let tn — to + nh, xn be the estimate for x(tn), and cn = c(tn) for c = /, A, B. 2) to uniquely determine xn given z n _i, we need A(tn) + hB(tn) to be nonsingular for small h. Thus we need regularity of the pencil \A(t) + B(t) for each t € 1.
Harmonic analysis and partial differential equations: in honor of Calderon by Michael Christ, Carlos E. Kenig, Cora Sadosky