Advancements in Numerical Analysis: Techniques for Solving Volterra and Fredholm Equations

This work concerns the construction of techniques to facilitate numerical analysis to solve basic computational mathematics problems of Volterra and Fredholm integral equations. In an attempt to merge contemporary machine learning techniques with conventional methods like the trapezoidal rule, the proposed methods seek enhanced accuracy, stability, and computing efficiency in applications involving synthetic and realistic kernel and forcing function data. With Python and computational libraries such as SciPy and NumPy, the algorithms are applied to actual physics and engineering problems and show greater precision and performance in various types of kernels, namely smooth, oscillatory, and weakly singular kernels. The methods for Volterra-type problems are highly stable, with iterative systems solving them effectively, and matrix-based methods solving Fredholm-type equations. This book contributes to numerical analysis through the presentation of new algorithms that enhance the practical solution of integral equations and have ramifications in computer science, engineering, and physics. The creation of hybrid numerical algorithms blending machine learning with conventional techniques bridges gaps in the literature and opens the door to more effective solutions to intricate integral equation problems.