The system of equations for mixed boundary value problem of partial differential equation with constant coefficient

Nur Syaza Mohd Yusop

Abstract

The aim of this research is to produce the system of equations for three different mixed Boundary Value Problems (BVPs). The potential problem which involves the Laplace’s equation on a square shape domain was considered, where the boundary is divided into four sets of linear boundary elements. The Boundary Element Method (BEM) was used to approximate the solutions for BVP. The mixed BVPs were reduced to Boundary Integral Equation (BIEs) by using direct method which were related with Green’s second identity representation formula. Then, linear interpolation was used on the discretized elements. The results showed that, there are three system of equations which were obtained. For some cases of mixed BVPs which involves discontinuous fluxes problems yields underdetermined systems. Out of the three problems that being considered, one of three BVPs leads to the underdetermined system of equations. Therefore, the transformation for the underdetermined system to the standard form is necessary for the numerical purposes. The gradient approach method which is widely applies to the Dirichlet problem was considered. This gradient approach method is extended to the underdetermined system of equations obtained from the mixed BVP which subsequently transformed to the standard system. In conclusion, the mixed BVP that involve discontinuous fluxes problem will yield to the underdetermined system of equations that prohibits in solving the system numerically. However, by the gradient approach method, the underdetermined system can be transformed to the standard form and can be solved the system numerically. The study implicates that the procedure used in this studies can be extended to higher dimensional mixed BVPs which involved the discontinuous fluxes problems.

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