Composition constants for raising the orders of unconventional schemes for ordinary differential equations

Many models of physical and chemical processes give rise to ordinary differential equations with special structural properties that go unexploited by general-purpose software designed to solve numerically a wide range of differential equations. If those properties are to be exploited fully for the sake of better numerical stability, accuracy and/or speed, the differential equations may have to be solved by unconventional methods. This short paper is to publish composition constants obtained by the authors to increase efficiency of a family of mostly unconventional methods, called reflexive.

     

Investigating the arenyl-diene double [3 + 2] photocycloaddition reaction

The double [3 + 2] photocycloaddition reaction involving arenyl-dienes has been used to assemble seven separate [5.5.5.5] fenestrane structures that include ether and aza variants. The primary photolysis step was a meta photocycloaddition reaction, while a secondary photocycloaddition step formed the fenestrane structure. Investigations involving the insertion of an additional methylene group into the basic arenyl-diene skeleton failed to afford the desired [5.5.5.6] fenestrane structure. The presence of an oxime moiety in the aromatic photosubstrate allowed the primary photolysis step to take place; however, an attempted secondary photocycloaddition reaction involving the oxime did not provide the intended polyheterocyclic fenestrane. An alternative strategy to form various "criss-cross" double meta photocycloadducts was investigated and led to the discovery of a Paterno-Büchi cycloaddition reaction between acetone and an angular meta photocycloadduct. Other novel thermally and photochemically mediated skeletal rearrangement reactions were also recorded.