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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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