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## Formulations of Initial‐Boundary Value Problems

*Józef Ignaczak and Martin Ostoja‐Starzewski*

### in Thermoelasticity with Finite Wave Speeds

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199541645
- eISBN:
- 9780191716164
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199541645.003.0002
- Subject:
- Mathematics, Applied Mathematics, Mathematical Physics

This chapter first presents a conventional and non‐conventional characterization of a thermoelastic process, giving four mixed initial‐boundary value problems (displacement‐temperature and ... More

## Elliptic Boundary Value Problems

*Karsten Urban*

### in Wavelet Methods for Elliptic Partial Differential Equations

- Published in print:
- 2008
- Published Online:
- May 2009
- ISBN:
- 9780198526056
- eISBN:
- 9780191712340
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526056.003.0003
- Subject:
- Mathematics, Applied Mathematics, Mathematical Finance

A model problem is introduced, namely the univariate two-point boundary value problem, both with periodic boundary conditions and homogeneous Dirichlet boundary conditions. The chapter describes the ... 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

## Mathematical Models of Electromagnetism

*Peter Monk*

### in Finite Element Methods for Maxwell's Equations

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198508885
- eISBN:
- 9780191708633
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198508885.003.0001
- Subject:
- Mathematics, Numerical Analysis

This chapter presents a summary of Maxwell’s equations. The time-harmonic Maxwell equations considered throughout this book are derived from the time domain system, and several linear constitutive ... More

## On the Origin of Truth Values

*Tim Maudlin*

### in Truth and Paradox: Solving the Riddles

- Published in print:
- 2004
- Published Online:
- January 2005
- ISBN:
- 9780199247295
- eISBN:
- 9780191601781
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199247293.003.0002
- Subject:
- Philosophy, Philosophy of Language

The formal language developed in ch. 1 is modelled as a directed graph with a boundary, and the problem of assigning truth values to sentences assimilated to a boundary‐value problem. A trivalent ... More

## INTEGRAL PROBLEMS

*Andrea Braides*

### in Gamma-Convergence for Beginners

- Published in print:
- 2002
- Published Online:
- September 2007
- ISBN:
- 9780198507840
- eISBN:
- 9780191709890
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198507840.003.0003
- Subject:
- Mathematics, Applied Mathematics

This chapter deals with integral problems. Using examples, it illustrates the need to use weak topologies to obtain compactness and coerciveness conditions. The sufficiency and necessity of the ... More

## Linear elastostatics

*Lallit Anand and Sanjay Govindjee*

### in Continuum Mechanics of Solids

- Published in print:
- 2020
- Published Online:
- September 2020
- ISBN:
- 9780198864721
- eISBN:
- 9780191896767
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198864721.003.0008
- Subject:
- Physics, Condensed Matter Physics / Materials

In this chapter the basic field equations in terms of displacement, strain, and stress, and typical boundary conditions which are necessary to formulate a complete three-dimensional ... More

## Variational formulation of boundary-value problems

*Lallit Anand and Sanjay Govindjee*

### in Continuum Mechanics of Solids

- Published in print:
- 2020
- Published Online:
- September 2020
- ISBN:
- 9780198864721
- eISBN:
- 9780191896767
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198864721.003.0010
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter discusses a variational formulation of boundary value problems in small deformation solid mechanics. It begins by introducing the important principle of virtual power, and shows that it ... 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

## Boundary Value Problems in Dimensions 7, 4 and 3 Related to Exceptional Holonomy

*Simon Donaldson*

### in Geometry and Physics: Volume I: A Festschrift in honour of Nigel Hitchin

- Published in print:
- 2018
- Published Online:
- December 2018
- ISBN:
- 9780198802013
- eISBN:
- 9780191840500
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198802013.003.0006
- Subject:
- Mathematics, Geometry / Topology

The variational point of view on exceptional structures in dimensions 6, 7 and 8 is one of Nigel Hitchin’s seminal contributions. One feature of this point of view is that it motivates the study of ... More

## Fundamentals of Transport Equation Formulation for Two-Phase Flow in Homogeneous and Heterogeneous Porous Media

*Michel Quintard and Stephen Whitaker*

### in Vadose Zone Hydrology: Cutting Across Disciplines

- Published in print:
- 1999
- Published Online:
- November 2020
- ISBN:
- 9780195109900
- eISBN:
- 9780197561058
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195109900.003.0005
- Subject:
- Earth Sciences and Geography, Oceanography and Hydrology

Most porous media of practical importance are hierarchical in nature; that is, they are characterized by more than one length-scale. When these length-scales are disparate, the hierarchical ... More

## Continuum dynamics

*Joseph F. Boudreau and Eric S. Swanson*

### in Applied Computational Physics

- Published in print:
- 2017
- Published Online:
- February 2018
- ISBN:
- 9780198708636
- eISBN:
- 9780191858598
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198708636.003.0019
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

The theory and application of a variety of methods to solve partial differential equations are introduced in this chapter. These methods rely on representing continuous quantities with discrete ... More

## Generalized Dirichlet and Neumann Boundary-Value Problems

*D. E. Edmunds and W. D. Evans*

### in Spectral Theory and Differential Operators

- Published in print:
- 2018
- Published Online:
- September 2018
- ISBN:
- 9780198812050
- eISBN:
- 9780191861130
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198812050.003.0006
- Subject:
- Mathematics, Pure Mathematics

In this chapter, the generalized or weak interpretation of the Dirichlet and Neumann problems for general elliptic expressions is motivated and then the Lax–Milgram Theorem is used to set the ... More

## Optimal Stopping Theory and American Options

*Tomas Björk*

### in Arbitrage Theory in Continuous Time

- Published in print:
- 2019
- Published Online:
- February 2020
- ISBN:
- 9780198851615
- eISBN:
- 9780191886218
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198851615.003.0028
- Subject:
- Economics and Finance, Econometrics

In this chapter we present the dynamic programming approach to optimal stopping problems. We start by presenting the discrete time theory, deriving the relevant Bellman equation. We present the Snell ... More

## Elasticity Equations with High Contrast Parameters

*Habib Ammari, Elie Bretin, Josselin Garnier, Hyeonbae Kang, Hyundae Lee, and Abdul Wahab*

### in Mathematical Methods in Elasticity Imaging

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691165318
- eISBN:
- 9781400866625
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691165318.003.0003
- Subject:
- Mathematics, Applied Mathematics

This chapter presents some recent results on the elasticity equations with high contrast coefficients. It first sets up the problems for finite and extreme moduli before discussing the incompressible ... More

## Optimal Control

*David J. N. Limebeer and Matteo Massaro*

### in Dynamics and Optimal Control of Road Vehicles

- Published in print:
- 2018
- Published Online:
- April 2019
- ISBN:
- 9780198825715
- eISBN:
- 9780191864636
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198825715.003.0008
- Subject:
- Physics, Condensed Matter Physics / Materials

Chapter 8 focuses on nonlinear optimal control and its applications. The chapter begins by introducing the fundamentals of optimal control and prototypical problem formulations. This is followed by ... More

## Boundary Value Problems for the Lorentzian Dirac Operator

*Christian Bär and Sebastian Hannes*

### in Geometry and Physics: Volume I: A Festschrift in honour of Nigel Hitchin

- Published in print:
- 2018
- Published Online:
- December 2018
- ISBN:
- 9780198802013
- eISBN:
- 9780191840500
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198802013.003.0001
- Subject:
- Mathematics, Geometry / Topology

On a compact globally hyperbolic Lorentzian spin manifold with smooth space-like Cauchy boundary, the (hyperbolic) Dirac operator is known to be Fredholm when Atiyah–Patodi–Singer boundary conditions ... More

## The Green’s function in linear elasticity

*Adrian P. Sutton*

### in Physics of Elasticity and Crystal Defects

- Published in print:
- 2020
- Published Online:
- August 2020
- ISBN:
- 9780198860785
- eISBN:
- 9780191893001
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198860785.003.0004
- Subject:
- Physics, Condensed Matter Physics / Materials, Crystallography: Physics

The elastostatic Green’s tensor function is the solution of a differential equation for the displacement field created by a unit point force in an infinite continuum. Its symmetry is derived using ... More

## Tsunamis

*John A. Adam*

### in Rays, Waves, and Scattering: Topics in Classical Mathematical Physics

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691148373
- eISBN:
- 9781400885404
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691148373.003.0013
- Subject:
- Mathematics, Applied Mathematics

This chapter describes a mathematical model of tsunami propagation (transient waves). A tsunami is a series of ocean waves triggered by large-scale disturbances of the ocean, including earthquakes, ... More

## Magnetic Sans of Nanoparticles and Complex Systems

*Andreas Michels*

### in Magnetic Small-Angle Neutron Scattering: A Probe for Mesoscale Magnetism Analysis

- Published in print:
- 2021
- Published Online:
- May 2021
- ISBN:
- 9780198855170
- eISBN:
- 9780191889547
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198855170.003.0005
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter provides an overview on the magnetic SANS of nanoparticles and complex systems, which include ferrofluids, magnetic steels, and spin glasses and amorphous magnets. The underlying ... More

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