2024, Vol. 9, Issue 4, Part A

The singularity of the kernel in a weakly singular integral equation typically leads to the singularity of the derivatives of the solution at the border of the domain. As you are aware, first-kind Volterra integral equations are more ill-posed compared to the second-kind equations. The degree of ill-posedness may be quantified by the v-smoothing of the integral operator. Here, a highly effective method is presented for solving weakly singular first-kind Volterra integral equations. This method is based on the use of polynomials that satisfy interpolation criteria. A novel category of highly implicit multi-step collocation methods is developed to get a higher order approach for numerically solving this problem. Additionally, we utilize a one-step approach to solve Volterra integral equations of the first class with weak singularity. We then compare the numerical results to demonstrate the effectiveness and precision of the suggested method.

AMS subject classifications: Volterra integral equations of the first-kind; Collocation methods; Multi-step numerical method, weakly singular kernel.