On the numerical solution of a hyperbolic initial boundary value problem by hypersingular boundary integral equations

Abstract: In this study, we consider the numerical solution of the Neumann initial boundary value problem for the wave equation in 2D domains. Employing the Laguerre transform with respect to the temporal variable, we effectively transform this problem into a series of Neumann elliptic problems. The development of a fundamental sequence for these elliptic equations provides us with the means to introduce modified double layer potentials. Consequently, we are able to derive a sequence of boundary hypersingular integral equations as a result of this transformation. To discretize the system of equations, we apply the Maue transform and implement the Nystr\"om method with trigonometric quadrature techniques. To demonstrate the practical utility of our approach, we provide numerical examples.