In this article, we consider weakly singular Volterra integral equations (WSVIEs) in 2D with combined logarithmic kernel, \(\begin{aligned} {\texttt{f}}(\texttt{x}, \texttt{t})=\texttt{Au}(\texttt{x}, \texttt{t})+\int \limits _{0}^{\texttt{x}} \int \limits _{0}^{\texttt{t}}\frac{{\log }{{(\texttt{x}- \texttt{r})}}\texttt{G}({\mathtt {u(r, s)}})}{{{(\texttt{x}-\texttt{r})}}^{\alpha _1}{{(\texttt{t}- \texttt{s})}}^{\alpha _2}}\texttt{d}\texttt{r}\texttt{d}\texttt{s},~\texttt{x, t}{\in }[0,1], \end{aligned}\) where, \(\mathtt {u(x, t)}\) is a unknown function, \(\mathtt {f(x, t)}\) is a known function and the nonlinear term \(\texttt{G}({\mathtt {u(r, s)}})={\mathtt {u^{2}(r, s)}}\) . Equations of this kind, introduced by V. Volterra himself within the framework of the theory of functional compositions, are also directly related to the modern problems of fractional dynamics. We have constructed the scheme which is based on employing the operational matrix technique with the 2D shifted Legendre polynomials (2D-SLP) as a basis function. This approach reduces WSVIE to a system of nonlinear algebraic equations with coefficients and finds these coefficients which provide the approximate solution. We have provided some test functions highlighting such a workflow in addition to establishing the error bound, convergence analysis and stability analysis.
