Solving weakly singular integral equations utilizing the meshless local discrete collocation technique

• Main Author: Pouria Assari
• Format: Article
• Language: English
• Published: Elsevier 2018-12-01
• Series: Alexandria Engineering Journal
• Online Access: http://www.sciencedirect.com/science/article/pii/S1110016817303009

The current work presents a computational scheme to solve weakly singular integral equations of the second kind. The discrete collocation method in addition to the moving least squares (MLS) technique established on scattered points is utilized to estimate the solution of integral equations. The MLS scheme approximates a function without mesh refinement over the domain that includes a locally weighted least squares polynomial fitting. The discrete collocation technique for the approximate solution of integral equations results from the numerical integration of all integrals in the method. We utilize an accurate quadrature formula based on the use of non-uniform composite Gauss-Legendre integration rule and employ it to compute the singular integrals appeared in the approach. The proposed scheme does not require any meshes, so it can be called as the meshless local discrete collocation (MLDC) method. Error analysis is also given for the method. Illustrative examples are shown clearly the reliability and efficiency of the new scheme and confirm the theoretical error estimates. Keywords: Discrete collocation method, Weakly singular integral equation, Meshless method, Moving least squares (MLS), Error analysis