Open AccessXI. On the series of Strum and Liouville , as derived from a pair of fundamental integral equations instead of a differential equation
Author(s) -
Alfred Cardew Dixon
Publication year - 1912
Publication title -
philosophical transactions of the royal society of london. series a, containing papers of a mathematical or physical character
Language(s) - English
Resource type - Journals
eISSN - 2053-9258
pISSN - 0264-3952
DOI - 10.1098/rsta.1912.0011
Subject(s) - series (stratigraphy) , mathematics , differential equation , mathematical analysis , interval (graph theory) , convergence (economics) , universal differential equation , function (biology) , first order partial differential equation , exact differential equation , differential (mechanical device) , physics , thermodynamics , combinatorics , paleontology , evolutionary biology , economics , biology , economic growth
The series of Liouville and Strum are generally treated by means of approximate solutions of the fundamental differential equation, these approximations being valid when certain functions involved in the differential equation have differential coefficients. The object of the present paper is to relax this restriction, and for this purpose integral equations are used in place of a differential equation, and an approximation is investigated (§§ 4—11) depending on a function which is constant throughout each of a system of sub-intervals. In §§ 15-18 the results are applied, by help of Hobson's general convergence theorem, to that one of the Liouville series which is usually valid at the two ends of the fundamental interval, and in §§ 19-22 to the more general series discussed by me in ‘Proc. L. M. S.,’ ser. 2, vol. 3, pp. 83-103.