On the numerical treatments of classes of third-kind auto-convolution VIEs via moving least squares collocation scheme

Purpose This contribution investigates the numerical solution of Volterra integral equations with auto-convolution of the third kind (AVIE). The numerical method applied in this investigation employs a collocation method based on the moving least squares (MLS) approximation. The MLS approximation is an effective way of approximating an unknown function by taking a disordered dataset. This method is a mesh-free approach since it does not require background interpolation or approximation cells, and is independent of domain geometry. The proposed method reduces the solution of third-kind AVIEs to the solution of systems of algebraic equations. By employing the Gauss–Legendre integration formula, we can estimate all the integrals of these equations. The applicability and validity of this method are demonstrated by numerical experiments, and its efficiency and robustness are proven by comparison with existing methods. Design/methodology/approach The numerical method applied in this study uses a collocation method based on moving least squares (MLS) approximation. This method is a mesh-free approach since it requires no background interpolation or approximation cells and is independent of domain geometry. Using the Gauss–Legendre integration formula, we can estimate all the integrals of these equations. Findings The authors declare that they have no known competing financial interests. Originality/value The manuscript has not been copyrighted or published previously and is not under consideration for publication elsewhere.