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

@inproceedings{Milton2020AUP, title={A unifying perspective on linear continuum equations prevalent in physics. Part II: Canonical forms for time-harmonic equations}, author={Graeme W. Milton}, year={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 understand and efficiently solve those equations. Here in part II we elucidate the form for many time-harmonic equations that do not involve higher order gradients.

## 13 Citations

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

- Mathematics, Physics
- 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 science. Part III: Canonical forms for dynamic equations with moduli that may, or may not, vary with time

- Physics, Mathematics
- 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…

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

- Physics, 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

- Physics, 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 physics. Part VII: Boundary value and scattering problems

- Physics, 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…

Homogenization of piezoelectric planar Willis materials

- 2021

Homogenization theories provide models that simplify the constitutive relations of heterogeneous media while retaining their macroscopic features. These theories have shown how the governing fields…

Some open problems in the theory of composites

- Mathematics, MedicinePhilosophical 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…

The electromomentum effect in piezoelectric Willis scatterers

- Physics
- 2021

Abstract Materials with asymmetric microstructure can constitutively couple macroscopic fields from different physics. Examples include piezoelectric materials that couple mechanical and electric…

The electromomentum effect in piezoelectric Willis scatterers

- 2021

Materials with asymmetric microstructure can constitutively couple macroscopic fields from different physics. Examples include piezoelectric materials that couple mechanical and electric fields and…

The electromomentum effect in piezoelectricWillis scatterers

- 2021

Materials with asymmetric microstructure can constitutively couple macroscopic fields from different physics. Examples include piezoelectric materials that couple mechanical and electric fields and…

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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…

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

- Mathematics, Physics
- 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 science. Part III: Canonical forms for dynamic equations with moduli that may, or may not, vary with time

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- 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…

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

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- 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…

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