ASYMPTOTICAL METHOD FOR DETERMINING THE FRACTIONAL ORDER IN OSCILLATORY SYSTEMS BY MOVING MASS COORDINATES

Abstract

No­te that in an os­cil­la­tory system, when the mass of oil lif­ted by the plun­ger is suf­fi­ci­ently lar­ge, an asymp­to­ti­cal met­hod is int­ro­du­ced to sol­ve the gi­ven prob­lem, ta­king the in­ver­se of the mass as a small pa­ra­me­ter. At this ti­me, the mo­ti­on of the ob­ject is re­du­ced to the Vol­ter­ra in­teg­ral equa­ti­on of the se­cond kind with res­pect to the disp­la­ce­ment of mass co­or­di­na­tes. Ba­sed on sta­tis­ti­cal da­ta, a qu­ad­ra­tic func­tio­nal is const­ruc­ted. Using the le­ast squa­res met­hod for de­ter­mi­ning the frac­tio­nal or­der an ef­fec­ti­ve al­go­rithm is pro­po­sed and the tab­le with cer­ta­in frac­tio­nal or­ders is pre­sen­ted in the examp­le. The re­sult of the com­pu­ta­tio­nal al­go­rithm shows that the first va­ria­ti­on of the qu­ad­ra­tic func­tio­nal with res­pect to the frac­tio­nal or­der is equ­al to ze­ro with an ac­cu­racy of 10^-16. This frac­tio­nal or­der is the most ef­fec­ti­ve or­der. The ob­tai­ned frac­tio­nal or­der sa­tis­fi­es the first va­ria­ti­on of the qu­ad­ra­tic func­tio­nal with an ac­cu­racy of 10^-16.

Key­words: os­cil­la­tory system, frac­tio­nal or­der dif­fe­ren­ti­al equa­ti­on, Vol­ter­ra in­teg­ral equa­ti­on, the le­ast squa­res met­hod