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.

翻译：本研究考虑二维区域中波动方程Neumann初边值问题的数值求解。通过引入关于时间变量的拉盖尔变换，我们成功将该问题转化为一系列Neumann椭圆问题。针对这些椭圆方程的基本解序列的构建，使我们能够引入修正的双层势函数，进而推导出一组边界双奇异积分方程。为离散化该方程组，我们应用Maue变换并结合三角求积技术实施Nyström方法。最后通过数值算例验证了该方法在实际应用中的有效性。