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Numerical Methods for Delay Differential Equations

Alfredo Bellen and Marino Zennaro

Published in print:
2003
Published Online:
September 2007
ISBN:
9780198506546
eISBN:
9780191709609
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198506546.001.0001
Subject:
Mathematics, Numerical Analysis

The main purpose of the book is to introduce the numerical integration of the Cauchy problem for delay differential equations (DDEs) and of the neutral type. Comparisons between DDEs and ordinary ... More


The Standard Approach via Continuous ODE Methods

Alfredo Bellen and Marino Zennaro

in Numerical Methods for Delay Differential Equations

Published in print:
2003
Published Online:
September 2007
ISBN:
9780198506546
eISBN:
9780191709609
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198506546.003.0004
Subject:
Mathematics, Numerical Analysis

This chapter focuses on the error and convergence analysis of the technique based on the use of a discrete method for ODEs endowed with some continuous extension, which is called the standard ... More


LINEAR CAUCHY PROBLEM WITH CONSTANT COEFFICIENTS

Sylvie Benzoni-Gavage and Denis Serre

in Multi-dimensional hyperbolic partial differential equations: First-order systems and applications

Published in print:
2006
Published Online:
September 2007
ISBN:
9780199211234
eISBN:
9780191705700
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199211234.003.0001
Subject:
Mathematics, Applied Mathematics

For first-order systems, the well-posedness of the Cauchy problem is equivalent to an algebraic condition on the symbol of the operator: the hyperbolicity. The hyperbolic Cauchy problem has the ... More


INTIAL BOUNDARY VALUE PROBLEM IN A HALF-SPACE WITH CONSTANT COEEFICIENTS

Sylvie Benzoni-Gavage and Denis Serre

in Multi-dimensional hyperbolic partial differential equations: First-order systems and applications

Published in print:
2006
Published Online:
September 2007
ISBN:
9780199211234
eISBN:
9780191705700
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199211234.003.0004
Subject:
Mathematics, Applied Mathematics

This chapter drops the assumption of symmetry, or at least it does not assume the dissipativity in a classical sense. The search for a necessary condition for maximal estimates (strong well-posedness ... More


A CLASSIFICATION OF LINEAR IBVPs

Sylvie Benzoni-Gavage and Denis Serre

in Multi-dimensional hyperbolic partial differential equations: First-order systems and applications

Published in print:
2006
Published Online:
September 2007
ISBN:
9780199211234
eISBN:
9780191705700
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199211234.003.0008
Subject:
Mathematics, Applied Mathematics

Besides strongly well-posed (with a uniform K.-L. condition) and strongly ill-posed (where K.-L. fails) IBVPs, there are interesting problems that are only C∞ -well posed, but ... More


VARIABLE-COEFFICIENTS INITIAL BOUNDARY VALUE PROBLEMS

Sylvie Benzoni-Gavage and Denis Serre

in Multi-dimensional hyperbolic partial differential equations: First-order systems and applications

Published in print:
2006
Published Online:
September 2007
ISBN:
9780199211234
eISBN:
9780191705700
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199211234.003.0009
Subject:
Mathematics, Applied Mathematics

This chapter turns to linear IBVPs with variable coefficients. The techniques mix those of Chapters 2 and 4. Dissipative boundary symmetrizers have now variable coefficients, and are viewed as ... More


PERSISTENCE OF MULTIDIMENSIONAL SHOCKS

Sylvie Benzoni-Gavage and Denis Serre

in Multi-dimensional hyperbolic partial differential equations: First-order systems and applications

Published in print:
2006
Published Online:
September 2007
ISBN:
9780199211234
eISBN:
9780191705700
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199211234.003.0012
Subject:
Mathematics, Applied Mathematics

This chapter is partly influenced by Majda's work. Viewing shock fronts as free boundaries, it addresses their stability and existence by fixing front locations through a solution-dependent change of ... More


Well-posed hybrid systems and their properties

Rafal Goebel, Ricardo G. Sanfelice, and Andrew R. Teel

in Hybrid Dynamical Systems: Modeling, Stability, and Robustness

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691153896
eISBN:
9781400842636
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691153896.003.0006
Subject:
Mathematics, Applied Mathematics

This chapter defines nominally well-posed hybrid systems and well-posed hybrid systems to be those hybrid systems, vaguely speaking, for which graphical limits of graphically convergent sequences of ... More


Asymptotic stability, an in-depth treatment

Rafal Goebel, Ricardo G. Sanfelice, and Andrew R. Teel

in Hybrid Dynamical Systems: Modeling, Stability, and Robustness

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691153896
eISBN:
9781400842636
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691153896.003.0007
Subject:
Mathematics, Applied Mathematics

This chapter defines local pre-asymptotic stability for a compact (closed and bounded) set and studies its properties for systems that are nominally well-posed or well-posed. While Chapter 3 dealt ... More


HYPERBOLIC REDUCTIONS OF THE FIELD DQUATIONS

Miguel Alcubierre

in Introduction to 3+1 Numerical Relativity

Published in print:
2008
Published Online:
September 2008
ISBN:
9780199205677
eISBN:
9780191709371
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199205677.003.0005
Subject:
Physics, Theoretical, Computational, and Statistical Physics

From the early 1990s, a large number of alternative formulations of the 3+1 evolution equations have been proposed. There are currently more formulations than there are numerical groups capable of ... More


Well Posedness

G. F. Roach, I. G. Stratis, and A. N. Yannacopoulos

in Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691142173
eISBN:
9781400842650
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691142173.003.0004
Subject:
Mathematics, Applied Mathematics

This chapter presents rigorous mathematical results concerning the solvability and well posedness of time-harmonic problems for complex electromagnetic media, with a special emphasis on chiral media. ... More


Well Posedness

G. F. Roach, I. G. Stratis, and A. N. Yannacopoulos

in Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691142173
eISBN:
9781400842650
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691142173.003.0007
Subject:
Mathematics, Applied Mathematics

This chapter deals with electromagnetic fields in complex media in the time domain. First, the chapter discusses the Maxwell equations in complex media in the time domain and shows that they can be ... More


Nonlinear Problems

G. F. Roach, I. G. Stratis, and A. N. Yannacopoulos

in Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691142173
eISBN:
9781400842650
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691142173.003.0011
Subject:
Mathematics, Applied Mathematics

This chapter studies nonlinearity in electromagnetic media. Nonlinearity appears in a number of cases in which the dispersion relations or equivalently the coefficients of the constitutive relations ... More


Well Posedness

G. F. Roach, I. G. Stratis, and A. N. Yannacopoulos

in Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691142173
eISBN:
9781400842650
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691142173.003.0012
Subject:
Mathematics, Applied Mathematics

This chapter first presents a model for the Maxwell equations for complex random media in terms of stochastic integrodifferential equations. It then provides the appropriate functional framework that ... More


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