Numerical solution of locally loaded Volterra integral equations
Vladislav Byankin, Aleksandr Tynda, Denis Sidorov, Aliona Dreglea
TL;DR
This work addresses numerical solution of linear LVIEs with local loads (frozen-argument) by formulating the LVIE in operator form and analyzing existence/uniqueness via a resolvent-based approach and SLAE solvability. It introduces a direct numerical method using a piecewise linear discretization on a mesh, converting the problem into a SLAE with a mean-rectangle integral approximation, and proves an $O(h^2)$ convergence rate. Numerical experiments on two model problems confirm the method's accuracy and second-order convergence, implemented in C++ with Gauss-Jordan elimination. The study lays groundwork for future extension to nonlinear LVIE and Hammerstein-type problems and demonstrates a practical approach for LVIE with local loads. The results have implications for reliable numerical treatment of loaded integral equations in applied contexts.
Abstract
Volterra's integral equations with local and nonlocal loads represent the novel class of integral equations that have attracted considerable attention in recent years. These equations are a generalisation of the classic Volterra integral equations, which were first introduced by Vito Volterra in the late 19th century. The loaded Volterra integral equations are characterised by the presence of a load which complicates the process of their theoretical and numerical study. Sometimes these equation are called the equations with ``frozen'' argument. The present work is devoted to the study of Volterra equations with locally loaded integral operators. The existence and uniquness theorems are proved. Among the main contributions is the collocation method for approximate solution of such equations based on the piecewise linear approximation. To confirm the convergence of the method, a number of numerical results for solving model problems are provided.