On the Numerical Solution of the Initial-Boundary Value Problem with Neumann Condition for the Wave Equation by the Use of the Laguerre Transform and Boundary Elements Method

Abstract

We consider a numerical solution of the initial-boundary value problem for the homogeneous wave equation with the Neumann condition using the retarded double layer potential. For solving an equivalent time-dependent integral equation we combine the Laguerre transform (LT) in the time domain with the boundary elements method. After LT we obtain a sequence of boundary integral equations with the same integral operator and functions in right-hand side which are determined recurrently. An error analysis for the numerical solution in accordance with the parameter of boundary discretization is performed. The proposed approach is demonstrated on the numerical solution of the model problem in unbounded three-dimensional spatial domain.

Система Multimathframework Для Розв’язування Початково-Крайових Задач З Використанням Методу Граничних Елементів Та Перетворення Лаґерра

“…3) çàäà÷à Íåéìàíà ç âèêîðèñòàííÿì ïîòåíöiàëó ïîäâiéíîãî øàðó [13]; 8) çàäà÷à Äiðiõëå ó ïiâïðîñòîði ç âêëþ÷åííÿìè [10];…”

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“…Äëÿ çíàõîäaeåííÿ ðîçâ'ÿçêiâ çãàäàíèõ âèùå ÏÊÇ i âiäïîâiäíèõ ×ÃIÐ ó äîñèòü øèðîêèõ ôóíêöiéíèõ ïðîñòîðàõ îäèí ç åôåêòèâíèõ ïiäõîäiâ ïîëÿã๠ó ïî¹äíàííi ïåðåòâîðåííÿ Ëà åððà (ÏË) ç ìåòîäîì ãðàíè÷íèõ åëåìåíòiâ (ÌÃÅ). Çîêðåìà, ÿêùî ó âèïàäêó çàäà÷i Äiðiõëå ïîäàò訨ðîçâ'ÿçîê ÷åðåç ïîòåíöiàë iç ÷àñîâèì çàïiçíåííÿì ïðîñòîãî øàðó, à â çàäà÷i Íåéìàíà ïîäâiéíîãî, òî çà äîïîìîãîþ ÏË ç âiäïîâiäíèõ ×ÃIÐ ìîaeíà îòðèìàòè ïîñëiäîâíîñòi ÃIÐ ç åëiïòè÷íèìè ãðàíè÷íèìè îïåðàòîðàìè [14,15].…”

On the discretization and application of two space–time boundary integral equations for 3D wave propagation problems in unbounded domains

“…In the last decades, space-time Boundary Integral Equations (BIE) have been successfully applied to wave propagation problems defined in the exterior of a bounded domain (see, for example, [9], [24], [3], [19], [26], [15], [10], [2], [27], [16], [5], [4], [25], [18], [17], [22], [12]).…”