An Effective Interpolation Numerical Technique for Solutions of Fractional Volterra Integral Equations

In this sequel, we employ an approach that combines the modified version of the Lagrange polynomial approximation technique with the biconjugate gradient stabilized method (BiCGSTAB) to analyze numerical solutions of fractional Volterra integral equations (FVIEs). These equations emerge as indispensable tools in modern physics, enabling the modeling and analysis of intricate dynamical systems with memory effects and nonlocal behaviors. From fractal dynamics to quantum mechanics, electromagnetism, and biophysics, these equations provide a unified framework to capture the rich complexity inherent in diverse physical phenomena. The scheme for establishing the uniquely existence of a solution to the integral equation is elucidated in relation to the Banach contraction principle and the Bielecki norm. Furthermore, we rigorously prove several theorems concerning the method's convergence and error estimation. Notably, our proposed approach effectively handles potential singularities in the solution, a facet that, to the best of our knowledge, has remained unaddressed in existing literature. To showcase the reliability and efficacy of our method, we present illustrative examples along with comparative analyses. All numerical computations are conducted using MATLAB 23.2.0.2410171 (R2023b).