# A unifying perspective on linear continuum equations prevalent in science. Part III: Canonical forms for dynamic equations with moduli that may, or may not, vary with time

@article{Milton2020AUP, title={A unifying perspective on linear continuum equations prevalent in science. Part III: Canonical forms for dynamic equations with moduli that may, or may not, vary with time}, author={Graeme W. Milton}, journal={arXiv: Mathematical Physics}, year={2020} }

Enlarging on Parts I and II we write more equations in the desired format of the extended abstract theory of composites. We focus on a multitude of full dynamic equations, including equations where the medium is moving or otherwise changing in time. The motivation is that results and methods in the theory of composites then extend to these equations.

## 8 Citations

A unifying perspective on linear continuum equations prevalent in science. Part IV: Canonical forms for equations involving higher order gradients

- Mathematics
- 2020

Enlarging on Parts I, II, and III we write more equations in the desired format of the extended abstract theory of composites. We focus on a multitude of equations involving higher order derivatives.…

A unifying perspective on linear continuum equations prevalent in physics. Part II: Canonical forms for time-harmonic equations

- Physics
- 2020

Following some past advances, we reformulate a large class of linear continuum science equations in the format of the extended abstract theory of composites so that we can apply this theory to better…

A unifying perspective on linear continuum equations prevalent in science. Part VI: rapidly converging series expansions for their solution

- Mathematics
- 2020

We obtain rapidly convergent series expansions of resolvents of operators taking the form ${\bf A}=\Gamma_1{\bf B}\Gamma_1$ where $\Gamma_1({\bf k})$ is a projection that acts locally in Fourier…

A unifying perspective on linear continuum equations prevalent in physics. Part V: resolvents; bounds on their spectrum; and their Stieltjes integral representations when the operator is not selfadjoint

- Mathematics
- 2020

We consider resolvents of operators taking the form ${\bf A}=\Gamma_1{\bf B}\Gamma_1$ where $\Gamma_1({\bf k})$ is a projection that acts locally in Fourier space and ${\bf B}({\bf x})$ is an…

A unifying perspective on linear continuum equations prevalent in science. Part I: Canonical forms for static, steady, and quasistatic equations

- Physics
- 2020

Following some past advances, we reformulate a large class of linear continuum science equations in the format of the extended abstract theory of composites so that we can apply this theory to better…

A unifying perspective on linear continuum equations prevalent in physics. Part VII: Boundary value and scattering problems

- Mathematics
- 2021

We consider simply connected bodies or regions of finite extent in space or space-time and write conservation laws associated with the equations in Parts I-IV. We review earlier work where, for…

Some open problems in the theory of composites

- EngineeringPhilosophical Transactions of the Royal Society A
- 2021

A selection of open problems in the theory of composites is presented. Particular attention is drawn to the question of whether two-dimensional, two-phase composites with general geometries have the…

An extremal problem arising in the dynamics of two-phase materials that directly reveals information about the internal geometry

- Mathematics
- 2020

Author(s): Mattei, Ornella; Milton, Graeme W; Putinar, Mihai | Abstract: In two phase materials, each phase having a non-local response in time, it has been found that for appropriate driving fields…

## References

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A unifying perspective on linear continuum equations prevalent in physics. Part II: Canonical forms for time-harmonic equations

- Physics
- 2020

Following some past advances, we reformulate a large class of linear continuum science equations in the format of the extended abstract theory of composites so that we can apply this theory to better…

A unifying perspective on linear continuum equations prevalent in physics. Part I: Canonical forms for static and quasistatic equations

- Physics
- 2020

Following some past advances, we reformulate a large class of linear continuum physical equations in the format of the extended abstract theory of composites so that we can apply this theory to…

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A procedure is described to generate fundamental solutions or Green’s functions for time harmonic point forces and sources. The linearity of the field equations permits the Green’s function to be…

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Field patterns, first proposed by the authors in Milton and Mattei (2017 Proc. R. Soc. A 473 20160819), are a new type of wave propagating along orderly patterns of characteristic lines which arise…