*Mauro Fabrizio and Morro Angelo*

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198527008
- eISBN:
- 9780191713316
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198527008.003.0007
- Subject:
- Mathematics, Applied Mathematics

This chapter provides a wide class of mathematical problems which originates from the application of Maxwell’s differential equations to electromagnetic systems. Static, stationary, and transient ...
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This chapter provides a wide class of mathematical problems which originates from the application of Maxwell’s differential equations to electromagnetic systems. Static, stationary, and transient problems are investigated for dielectrics or conductors with instantaneous response or with memory, subject to various boundary conditions. The main concern is to establish existence, uniqueness, and stability of the solution. This objective is realized by applying different mathematical techniques both to adhere to the features of the model and to offer a profitable set of approaches.Less

This chapter provides a wide class of mathematical problems which originates from the application of Maxwell’s differential equations to electromagnetic systems. Static, stationary, and transient problems are investigated for dielectrics or conductors with instantaneous response or with memory, subject to various boundary conditions. The main concern is to establish existence, uniqueness, and stability of the solution. This objective is realized by applying different mathematical techniques both to adhere to the features of the model and to offer a profitable set of approaches.

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

- 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 ...
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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 expressed as integrodifferential equations. It then provides a convenient functional setting that allows the Maxwell equation to be treated as an integrodifferential evolution equation, and also provides some solvability and well posedness results for this problem based on a semigroup approach. Two alternative approaches to the solvability of evolution problems in the time domain, namely, the evolution family approach and an approach using finite-dimensional approximations (Faedo-Galerkin) approach, are then discussed. Finally, the chapter presents some extensions related to evolution problems.Less

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 expressed as integrodifferential equations. It then provides a convenient functional setting that allows the Maxwell equation to be treated as an integrodifferential evolution equation, and also provides some solvability and well posedness results for this problem based on a semigroup approach. Two alternative approaches to the solvability of evolution problems in the time domain, namely, the evolution family approach and an approach using finite-dimensional approximations (Faedo-Galerkin) approach, are then discussed. Finally, the chapter presents some extensions related to evolution problems.

*Luis Caffarelli and Luis Silvestre*

*Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, Stephen Wainger, Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, and Stephen Wainger (eds)*

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691159416
- eISBN:
- 9781400848935
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159416.003.0004
- Subject:
- Mathematics, Numerical Analysis

This chapter studies evolution problems that are related to continuous time random walks (CTRW), having a discontinuous path for which both the jumps and the time elapsed in between them are random. ...
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This chapter studies evolution problems that are related to continuous time random walks (CTRW), having a discontinuous path for which both the jumps and the time elapsed in between them are random. These processes are governed by a generalized master equation which is nonlocal both in space and time. To illustrate, the chapter considers kernels K(t, x, s, y) in a particular function. Here, studying correlated kernels provides a more flexible framework where more interesting physical phenomena can be observed, and more subtle mathematical questions appear. The regularity estimates are in fact more interesting (harder mathematically) when the jumps in space and the waiting times are strongly correlated.Less

This chapter studies evolution problems that are related to continuous time random walks (CTRW), having a discontinuous path for which both the jumps and the time elapsed in between them are random. These processes are governed by a generalized master equation which is nonlocal both in space and time. To illustrate, the chapter considers kernels *K*(*t*, *x*, *s*, *y*) in a particular function. Here, studying correlated kernels provides a more flexible framework where more interesting physical phenomena can be observed, and more subtle mathematical questions appear. The regularity estimates are in fact more interesting (harder mathematically) when the jumps in space and the waiting times are strongly correlated.