Reproducing kernel method for solving Fredholm integro-differential equations with weakly singularity

Singular integral equations (SIEs) are often encountered in certain contact and fracture problems in solid mechanics. Numerical methods for solving SIEs have been the focus of much research, including reproducing kernel methods. However, there are no reports on reproducing kernel methods for solving differential-integral equations with weakly singular kernels. We developed a reproducing kernel method for solving Fredholm integro-differential equations with weakly singular kernels in reproducing kernel Hilbert space. This involves changing a weakly singular kernel to a logarithm kernel to a Kalman kernel. Weak singularity is removed by applying a smooth transformation to the Kalman kernel. Solution representations are obtained in reproducing kernel Hilbert space. Numerical experiments show that our reproducing kernel method is efficient.