The complex‐valued fundamental solution ln(z) can be decomposed into the radial basis function (RBF) and the angular basis function (ABF). In this paper, not only the RBF but also the ABF are employed to solve the problem of interaction between a screw dislocation and an elliptical hole or rigid inclusion. The problem is decomposed into a free field with a screw dislocation and a boundary value problem subject to a specific boundary condition. The boundary value problem is solved by using the RBF and the boundary integral equation. Since the geometric shape is an ellipse, the degenerate kernel is expanded to a series form under the elliptic coordinates, while the unknown boundary densities are expanded to Fourier series. By combining the degenerate kernel and the null‐field integral equation, the boundary value problem can be easily solved. Finally, two examples are demonstrated to verify the validity of the present approach.

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<p style='text-indent:20px;'>Thanks to the fundamental solution, both BIEs and BEM are effective approaches for solving boundary value problems. But it may result in rank deficiency of the influence matrix in some situations such as fictitious frequency, spurious eigenvalue and degenerate scale. First, the nonequivalence between direct and indirect method is analytically studied by using the degenerate kernel and examined by using the linear algebraic system. The influence of contaminated boundary density on the field response is also discussed. It's well known that the CHIEF method and the Burton and Miller approach can solve the unique solution for exterior acoustics for any wave number. In this paper, we extend a similar idea to avoid the degenerate scale for the interior two-dimensional Laplace problem. One is the external source similar to the null-field BIE in the CHIEF method. The other is the Burton and Miller approach. Two analytical examples, circle and ellipse, were analytically studied. Numerical tests for general cases were also done. It is found that both two approaches can yield an unique solution for any size.</p>

<abstract> <p>The electrostatics of two cylinders charged to the symmetrical or anti-symmetrical potential is investigated by using the null-field boundary integral equation (BIE) in conjunction with the degenerate kernel of the bipolar coordinates. The undetermined coefficient is obtained according to the Fredholm alternative theorem. The uniqueness of solution, infinite solution, and no solution are examined therein. A single cylinder (circle or ellipse) is also provided for comparison. The link to the general solution space is also done. The condition at infinity is also correspondingly examined. The flux equilibrium along circular boundaries and the infinite boundary is also checked as well as the contribution of the boundary integral (single and double layer potential) at infinity in the BIE is addressed. Ordinary and degenerate scales in the BIE are both discussed. Furthermore, the solution space represented by the BIE is explained after comparing it with the general solution. The present finding is compared to those of Darevski ^{[<xref ref-type="bibr" rid="b2">2</xref>]} and Lekner ^{[<xref ref-type="bibr" rid="b4">4</xref>]} for identity.</p> </abstract>