Computation of Neumann Localised Boundary Domain Integral Equations

Abstract

Most integrals in Localised Boundary Domain Integral Equations (LBDIEs) comprise singularities. This paper aims to produce numerical solutions of the LBDIEs for the Partial Differential Equations with variable coefficients. The singularities of the boundary integrals in LBDIEs will be handled by using a semi-analytic for logarithmic singularity and a semi-quadratic analytic method for r−2 singularity. Whereas the singular domain integrals are handled by using the Duffy transformation. The LBDIEs that we consider are associated with the Neumann problem, which can be solved with a condition. If it can be solved, the solution is, however, unique up to an additive constant. We add a perturbation operator to the LBDIEs to convert the LBDIE to a uniquely solvable equation. The perturbed integral operator leads the perturbed LBDIEs to a dense matrix system that disable the use of methods in solving sparse matrix system. We solve the system of linear equations by Lower-Upper (LU) decomposition method. The numerical results indicate that high accuracy results can be attained. It gives the impression that the methods we use in this numerical experiment are reliable in handling the boundary and domain singular integrals.