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

Abstract

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 physical equations. Here in part I we elucidate the form for many static, steady, and quasistatic equations.

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

“…These manipulations are similar to the manipulations of Cherkaev and Gibiansky [13] and the subsequent manipulations in [27] that led to (4.4), (5.2), and (9.2) in Part I [32], and which were generalized in [38] to include source terms. Now there is a close relation between J 0 and E 0 and they need not be real.…”

“…In Parts I, II, III, and IV [32][33][34][35] we established that an avalanche of equations in science can be rewritten in the form J(x, t) = L(x, t)E(x, t) − s(x, t),…”

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

“…GWM thanks the National Science Foundation for support through grant DMS-1814854, and Christian Kern for helpful comments on the manuscript. The work was largely based on the books [71,91] and in the context of the latter book many thanks go to the friends and colleagues mentioned in the acknowledgements of Part I [76]. The author also thanks Vladimir Shalaev and Sergei Tretyakov for references to early metamaterial related work.…”

“…As in Part I [76], we are interested in giving examples of linear science equations that can be expressed in the form…”

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

“…As in the previous parts [12][13][14][15][16][17] we are interested in the plethora of linear physics equations expressible in the form encountered in the extended abstract theory of composites:…”