Open AccessDENSENESS OF SETS OF CAUCHY PROBLEMS WITHHOUT SOLUTIONS AND WITH NONUNIQUE SOLUTIONS IN THE SET OF ALL CAUCHY PROBLEMS
Author(s) -
V. Yu. Slyusarchuk
Publication year - 2020
Publication title -
bukovinsʹkij matematičnij žurnal
Language(s) - English
Resource type - Journals
ISSN - 2309-4001
DOI - 10.31861/bmj2020.02.11
Subject(s) - mathematics , banach space , cauchy problem , cauchy distribution , initial value problem , cauchy's convergence test , set (abstract data type) , space (punctuation) , cauchy's integral theorem , mathematical analysis , pure mathematics , cauchy's integral formula , cauchy boundary condition , computer science , boundary value problem , programming language , free boundary problem , operating system
When nding solutions of dierential equations it is necessary to take into account thetheorems on innovation and unity of solutions of equations. In case of non-fulllment of theconditions of these theorems, the methods of nding solutions of the studied equations used incomputational mathematics may give erroneous results. It should also be borne in mind thatthe Cauchy problem for dierential equations may have no solutions or have an innite numberof solutions.The author presents two statements obtained by the author about the denseness of setsof the Cauchy problem without solutions (in the case of innite-dimensional Banach space)and with many solutions (in the case of an arbitrary Banach space) in the set of all Cauchyproblems.Using two examples of the Cauchy problem for dierential equations, the imperfection ofsome methods of computational mathematics for nding solutions of the studied equations isshown.