Numerical solution of the two-dimensional Helmholtz equation with variable coefficients by the radial integration boundary integral and integro-differential equation methods

Abstract

Abstract. This paper presents new formulations of the boundary-domain integral equation (BDIE) and the boundary-domain integro-differential equation (BDIDE) methods for the numerical solution of the two-dimensional Helmholtz equation with variable coefficients. When the material parameters are variable (with constant or variable wave number), a parametrix is adopted to reduce the Helmholtz equation to a BDIE or BDIDE. However, when material parameters are constant (with variable wave number), the standard fundamental solution for the Laplace equation is used in the formulation. The radial integration method is then employed to convert the domain integrals arising in both BDIE and BDIDE methods into equivalent boundary integrals. The resulting formulations lead to pure boundary integral and integrodifferential equations with no domain integrals. Numerical examples are presented for several simple problems, for which exact solutions are available, to demonstrate the efficiency of the proposed methods.

Implementation of a generalized exponential basis functions method for linear and non‐linear problems

“…In this example, we consider a non‐homogeneous Helmholtz equation with variable coefficients in a square domain, namely where a ( x ) and k ( x ) are known variable material coefficient and known variable wave number, respectively. Taking these material parameters as the analytical solution of this problem available in will be The problem domain and the defined boundary conditions for this example are illustrated in Figure . The source term f and the boundary conditions are determined from the analytical solution.…”

“…The convergence of the solution is shown in Figure . For a quantitative comparison, we used the results from various boundary element methods in . Table demonstrates the computed values of u along the middle line of the domain using boundary‐domain integro‐differential equation, radial integration boundary integro‐differential equation, boundary‐domain integral equation, radial integration boundary integral equation, analytical solution and present method.…”

Recent Progress in the Radial Integration Boundary Integral and IntegroDifferential Equation Methods for Numerical Solution of Problems with Variable Coefficients

“…This allows a reduction of the mathematical problem to a Boundary-Domain Integral or Integro-Differential Equation (BDIE or BDIDE) [14,[17][18][19][20]. AL-Jawary and Wrobel [14][15][16] have successfully implemented BDIE and BDIDE formulations for stationary heat transfer in isotropic materials with variable coefficients associated with Dirichlet, Neumann and mixed boundary conditions by using domain integrals.…”

“…A new type of boundary-only integral equation technique is developed for the treatment of the two-dimensional (2D) Helmholtz equation when the material parameters and wave number vary within the medium and non-homogeneous transient heat conduction problems with variable coefficients based on the use of a parametrix [17,22]. The RIM is used to convert the domain integrals appearing in both BDIE and BDIDE to equivalent boundary integrals.…”

“…The numerical results show that high rates of convergence are obtained with mesh refinement. A pure BEM using the RIM has been derived and implemented in [17,18] to solve the two-dimensional Helmholtz equation and steady state heat conduction with variable coefficients, respectively.…”