Triple reciprocity method for unknown function's domain integral in boundary integral equation

In this paper, triple reciprocity method (TRM) is employed to transform the unknown function's domain integral appearing in boundary integral equation (BIE) to boundary integrals. The three-dimensional interior Helmholtz equation is represented as a Poisson equation with an unknown function in the right-hand side. The fundamental solution of Laplace equation is used to derive the BIE and there is a domain integral containing unknown field function in the BIE. The triple reciprocity method (TRM) is used to transform this domain integral to boundary integrals. The improved triple reciprocity approximation (TRA) is used to approximate the unknown domain function and solve the high-order derivatives of it virtually. Employing the reciprocity theory, the domain integral is transferred into the boundary integrals containing the unknown field function. Finally, the BIE can be solved without internal cells. The proposed formulas can be employed to treat the domain integral containing time derivatives of unknown variable, both known and unknown functions and other similar types. Three numerical examples are presented to demonstrate the efficiency and accuracy of the proposed formulas. Results show that TRM can solve the BIE containing unknown function's domain integral efficiently and accurately.