Nonlinear differential algebraic equations

We consider a system of nonlinear ordinary differential equations that are not solved with respect to the derivative of the unknown vector function and degenerate identically in the domain of definition. We obtain conditions for the existence of an operator transforming the original system to the normal form and prove a general theorem on the solvability of the Cauchy problem.

[1]  Alberto L. Sangiovanni-Vincentelli,et al.  The Waveform Relaxation Method for Time-Domain Analysis of Large Scale Integrated Circuits , 1982, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[2]  Mark Horowitz,et al.  Signal Delay in RC Tree Networks , 1983, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[3]  E. Coddington,et al.  Theory of Ordinary Differential Equations , 1955 .

[4]  Zdzislaw Jackiewicz,et al.  Waveform Relaxation Methods for Functional Differential Systems of Neutral Type , 1997 .

[6]  D. Bainov,et al.  Integral Inequalities and Applications , 1992 .

[7]  O. Nevanlinna,et al.  Convergence of dynamic iteration methods for initial value problems , 1987 .

[8]  Jr. J. Wyatt.  Monotone sensitivity of nonlinear uniform RC transmission lines, with application to timing analysis of digital MOS integrated circuits , 1985 .

[9]  Zdzislaw Jackiewicz,et al.  Convergence of Waveform Relaxation Methods for Differential-Algebraic Systems , 1996 .

[10]  E. Griepentrog,et al.  Differential-algebraic equations and their numerical treatment , 1986 .

[11]  W. Rheinboldt,et al.  Theoretical and numerical analysis of differential-algebraic equations , 2002 .

[12]  R. Fletcher.  A Nonlinear Programming Problem in Statistics (Educational Testing) , 1981 .

[13]  V. Lakshmikantham,et al.  Monotone iterative techniques for nonlinear differential equations , 1985 .

[14]  Stephen L. Campbell,et al.  Uniqueness of completions for linear time varying differential algebraic equations , 1992 .

[15]  Stefan Vandewalle,et al.  SOR WAVEFORM , 1997 .

[16]  Wai-Shing Luk,et al.  Convergence-Theoretics of Classical and Krylov Waveform Relaxation Methods for Differential-Algebraic Equations (Special Section on VLSI Design and CAD Algorithms) , 1997 .

[17]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[18]  W. Walter.  Differential and Integral Inequalities , 1970 .

[19]  Stefan Vandewalle,et al.  Multigrid waveform relaxation on spatial finite element meshes , 1996 .

[20]  Yaolin Jiang,et al.  Convergence conditions of waveform relaxation methods for circuit simulation , 1998, ISCAS '98. Proceedings of the 1998 IEEE International Symposium on Circuits and Systems (Cat. No.98CH36187).

[21]  Stephen L. Campbell,et al.  Solvability of General Differential Algebraic Equations , 1995, SIAM J. Sci. Comput..

[22]  A necessary and sufficient condition for the convergence of iterative procedures for solving equations of nonlinear monotone resistive networks , 1997, Proceedings of 1997 IEEE International Symposium on Circuits and Systems. Circuits and Systems in the Information Age ISCAS '97.

[23]  Benedict Leimkuhler,et al.  Rapid convergence of waveform relaxation , 1993 .

[24]  Andrew Lumsdaine,et al.  Spectra and Pseudospectra of Waveform Relaxation Operators , 1997, SIAM J. Sci. Comput..

[25]  Caren Tischendorf,et al.  Recent Results in Solving Index-2 Differential-Algebraic Equations in Circuit Simulation , 1997, SIAM J. Sci. Comput..

[26]  Linda R. Petzold,et al.  Numerical solution of initial-value problems in differential-algebraic equations , 1996, Classics in applied mathematics.

[27]  R. März.  Differential Algebraic Systems with Properly Stated Leading Term and MNA Equations , 2003 .

[28]  Volker Mehrmann,et al.  Regular solutions of nonlinear differential-algebraic equations and their numerical determination , 1998 .

[29]  Charles A. Zukowski,et al.  Relaxing Bounds for Linear RC Mesh Circuits , 1986, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.