No abstract

Âîññòàíîâëåíèå ïîëèíîìèàëüíîãî ïîòåíöèàëà â çàäà÷å ØòóðìàËèóâèëëÿ c ⃝ À. Ì. Àõòÿìîâ 1 , È. Ì. Óòÿøåâ 2 Àííîòàöèÿ. Ðàññìîòðåíà çàäà÷à èäåíòèôèêàöèè ïîëèíîìèàëüíîãî êîýôôèöèåíòà óïðóãîñòè ñðåäû ïî ñîáñòâåííûì ÷àñòîòàì êîëåáëþùåéñÿ â ýòîé ñðåäå ñòðóíû. Ïðèâåäåí ìåòîä ðåøåíèÿ çàäà÷è, îñíîâàííûé íà ïðåäñòàâëåíèè ëèíåéíî íåçàâèñèìûõ ðåøåíèé äèôôåðåíöèàëüíîãî óðàâíåíèÿ â âèäå ðÿäîâ Òåéëîðà ïî ïåðåìåííûì x è λ . Ðàçðàáîòàí òàêaeå ìåòîä, êîòîðûé ïîçâîëÿåò äîêàçûâàòü íå îäíîãî èëè ìíîãèõ âîññòàíîâëåííîãî ïîëèíîìèàëüíûõ êîýôôèöèåíòîâ óïðóãîñòè ñðåäû ïî êîíå÷íîìó ÷èñëó ñîáñòâåííûõ ÷àñòîò êîëåáàíèé ñòðóíû. Äàííûé ìåòîä îñíîâàí íà ìåòîäå âàðèàöèè ïðîèçâîëüíîé ïîñòîÿííîé. Ïðèâåäåíû ïðèìåðû ðåøåíèÿ çàäà÷è è îöåíêà ïîãðåøíîñòè ðåçóëüòàòà.  ðàáîòå ïîêàçàíî, ÷òî äëÿ îäíîçíà÷íîé èäåíòèôèêàöèè n + 1 êîýôôèöèåíòîâ ïîëèíîìà ñòåïåíè n , ÿâëÿþùèìñÿ ïîòåíöèàëîì â çà-äà÷å Øòóðìà-Ëèóâèëëÿ, äîñòàòî÷íî èñïîëüçîâàòü n + 1 ñîáñòâåííîå çíà÷åíèå. Ïðè ýòîì ñîáñòâåííûå çíà÷åíèÿ áåðóòñÿ èç äâóõ ðàçíûõ êðàåâûõ çàäà÷, îòëè÷àþùèõñÿ îäíèì èç êðàåâûõ óñëîâèé. Êîëè÷åñòâî ñîáñòâåííûõ çíà÷åíèé â êàaeäîé çàäà÷å áåðåòñÿ ïî ïîëîâèíå. Åñëè ýòî ÷èñëî ÿâëÿåòñÿ íå÷åòíûì, òî êîëè÷åñòâî ñîáñòâåííûõ çíà÷åíèé èç ñïåêòðà îäíîé èç çàäà÷ áóäåò íà åäèíèöó áîëüøèì. Ïðèâåäåí êîíòðïðèìåð, èç êîòîðîãî ñëåäóåò, ÷òî èñïîëüçîâàíèå ñîáñòâåííûõ ÷àñòîò òîëüêî èç îäíîãî ñïåêòðà íå ïîçâîëÿåò íàéòè åäèíñòâåííîå ðåøåíèå. Ïî ñóòè, ïðèâåäåííûå ðåçóëüòàòû óòî÷íÿþò èçâåñòíóþ òåîðåìó Áîðãà íà ñëó÷àé, êîãäà ïîòåíöèàë ÿâëÿåòñÿ ïîëèíîìîì. Êðîìå ýòîãî, ìåòîä, ïîçâîëÿþùèé âûÿâèòü êëàññ èçîñïåêòðàëüíûõ çàäà÷, äëÿ êîòîðûõ ñïåêòð ñîáñòâåííûõ ÷àñòîò ñîâïàäàåò.Êëþ÷åâûå ñëîâà:ñïåêòðàëüíàÿ çàäà÷à, èäåíòèôèêàöèÿ ïîòåíöèàëà, ñòðóíà, îáðàòíàÿ çàäà÷à, ñîáñòâåííûå çíà÷åíèÿ, ïîëèíîìèàëüíûé ïîòåíöèàë, çàäà÷à Øòóðìà-Ëèóâèëëÿ,âîññòàíîâëåíèå ïîòåíöèàëà.

The paper considers the problem of determining the local inhomogeneity of the medium from the natural frequencies of string oscillation. The inhomogeneity is modeled in three sections: in the first and third sections medium is homogeneous, and on average section the elastic characteristics are modeled by a quadratic function. This model is implemented using the conjugation conditions at boundary between media. It is shown that to identify the center of an inhomogeneity and determine its size, two natural frequencies are enough, and in the case of rigid fixing of both ends of the string, the solution of the problem is dual. The problem is solved by expanding the fundamental system of solutions into a power series in the variables x and λ. The estimates of the error of the method are given.

There are situations when there is a breakthrough of a pipeline with oil products under water. As a result, oil spills on the surface, polluting the environment. Underwater currents and wind can carry an oil stain away from the point of leakage. Therefore, it is not always possible to visually determine the location of a pipeline break by a spot of oil on the surface.To solve such problems, it is proposed to install strain sensors along the pipeline, which take the values of the displacement derivative ∂u(x,t)/∂x (deformations) at different instants of time, and use the simplest model of a pipeline based on the equations of longitudinal oscillations of a homogeneous rod. Formulas for determining the moment and location of the pipeline rupture were obtained from the strain sensors data and a scheme of interaction with GLONASS was proposed that allows instantly detecting leaks and damages of pipelines laid under water and timely eliminating the consequences of the accident. The application of the proposed scheme minimizes the consequences of the accident for the environment and financial costs.