DETERMINATION OF THE FRACTIONAL ORDER OF OSCILLATORY SYSTEMS USING MASS DISPLACEMENT COORDINATES

Abstract

Summary. The pre­sent pa­per add­res­ses the prob­lem of de­ter­mi­ning the frac­tio­nal or­der in an os­cil­la­tory pro­cess oc­cur­ring wit­hin a New­to­ni­an flu­id. Ini­ti­ally, the go­ver­ning equa­ti­on is trans­for­med in­to a Vol­ter­ra in­teg­ral equa­ti­on of the se­cond kind with res­pect to the disp­la­ce­ment of mass co­or­di­na­tes. Sub­se­qu­ently, the so­lu­ti­on of this in­teg­ral equa­ti­on is rep­re­sen­ted in the form of a Neu­mann se­ri­es. Ba­sed on avai­lab­le sta­tis­ti­cal da­ta, a qu­ad­ra­tic func­tio­nal cha­rac­te­ri­zing the de­via­ti­on of mass disp­la­ce­ment co­or­di­na­tes is const­ruc­ted, and the unk­nown frac­tio­nal or­der is iden­ti­fi­ed using the le­ast squa­res met­hod. An ef­fi­ci­ent com­pu­ta­tio­nal al­go­rithm for de­ter­mi­ning the frac­tio­nal or­der is then pro­po­sed, and il­lust­ra­ti­ve re­sults are pre­sen­ted in ta­bu­lar form. The outco­mes of the com­pu­ta­tio­nal pro­ce­du­re de­monst­ra­te that the iden­ti­fi­ed frac­tio­nal or­der sa­tis­fi­es the con­di­ti­on of va­nis­hing first va­ria­ti­on of the qu­ad­ra­tic func­tio­nal with an ac­cu­racy of 10⁻⁸.

Key­words: Os­cil­la­tory system, frac­tio­nal de­ri­va­ti­ve, Vol­ter­ra in­teg­ral equa­ti­on of the se­cond kind, the le­ast squa­res met­hod