Meshless method and shifted Chebyshev polynomials for solving a class of VIEs of the third kind

Abstract

The main goal of this study is to solve a class of linear and nonlinear Volterra integral equations (VIEs) of the third kind. The proposed technique estimates the solution of these equations using a technique based on MLS approximation and shifted Chebyshev polynomials. The approach is independent of the geometry of the domain and does not require any background meshes; therefore, it can be considered a meshless approach. The new method is effective and more flexible for many types of VIEs of the third kind, and its algorithm can be simply implemented on computers. The construction of a new technique for the proposed equations has been introduced. The convergence analysis is also studied. The convergence precision of the new approach is checked on classes of VIEs of the third kind, and the results validate the theoretical error estimates. Finally, numerical tests are provided to illustrate the accuracy and reliability of this approach.