Hoff equations on graphs

The Higher-Order Sobolev-Type Models

“…Èìåííî â ýòîì êîíòåêñòå áûëà ïîñòðîåíà òåîðèÿ òàêèõ çàäà÷ äëÿ ëèíåéíûõ óðàâíåíèé ñîáî-ëåâñêîãî òèïà ïåðâîãî ïîðÿäêà, è ðàçðàáîòàíû ïðèëîaeåíèÿ ýòîé òåîðèè. äàííîé ðàáîòå ýòè èäåè è ìåòîäû ðàñïðîñòðàíåíû íà ñëó÷àé óðàâíåíèé ñîáîëåâñêîãî òèïà âûñîêîãî ïîðÿäêà. Íåîáõîäèìî îòìåòèòü, ÷òî â íàñòîÿùåå âðåìÿ íà÷àëüíî-êîíå÷íûå çàäà÷è äëÿ íåêëàññè÷å-ñêèõ óðàâíåíèé ìàòåìàòè÷åñêîé ôèçèêè àêòèâíî èçó÷àþòñÿ, â òîì ÷èñëå è íà ìíîaeåñòâàõ ðàçëè÷íîé ãåîìåòðè÷åñêîé ñòðóêòóðû [26].…”

“…Îïåðàòîðû â ïðàâûõ ÷àñòÿõ (26), (27) ìîaeíî îòîaeäåñòâèòü ñ íåïðåðûâíûìè îïåðàòîðàìè, îïðåäåëåííûìè íà ïðîñòðàíñòâàõ U è F ñîîòâåòñòâåííî. Ïîýòîìó óðàâíåíèÿ (26), (27) óäîáíî ðàññìàòðèâàòü êàê êîíêðåòíûå èíòåðïðåòàöèè óðàâíåíèÿ…”

“…îáðàçîâàííûé ëó÷àìè, âûõîäÿùèìè èç íà÷àëà êîîðäèíàò ïîä óãëàìè θ è −θ, îïðåäåëÿþò ïðîïàãàòîðû îäíîðîäíîãî óðàâíåíèÿ (26) è (27) ñîîòâåòñòâåííî.…”

Numerical Solution for Non-Stationary Linearized Hoff Equation Defined on Geometrical Graph

“…For the first time, the methods of the Sobolev type equations theory [2] were used to study the equation (1) in [8]. Later, within the framework of this theory, the Hoff equation 1was investigated in various aspects (see, for example, [9][10][11][12][13][14]). In this paper we consider the non-stationary linearized Hoff equation of the form…”

“…Non-stationary Sobolev type equations were first considered in [15] and the proposed methods were applied to investigate various problems, for example, in [16][17][18]. The Sobolev type equations on graphs were first considered in the paper [19] and the proposed methods were applied to various models on graphs (see, for example, [10,11,13,[20][21][22][23][24][25]), including the Hoff model [10,11,13,23,24]. The Hoff equation, which is defined on a finite connected oriented graph G, models the dynamics of the structure consisting of I-beams.…”