Investigation of Relaxation Methods and Iterative Schemes Befitting Matrices of Bounded Linear Operators

Abstract

Following a newly established paradigm in precursor works at LMAM, diverging from widely recognised conventions, and inspired by an article on non-linear equations, we embark on the interdisciplinary mathematical mission to carry on the pursuit of numerically and theoretically discussing the approximation of solutions to the general Fredholm integral equation of the second kind defined on a large interval. Firstly, we show how efficient it is to truncate a Neumann's Series resulting in enhancing the outcomes and reducing numerical costs further all whilst manoeuvring the same well-constructed environment of the Banach spaces from previous works in order to approximate the solution of the equation cited hereinabove. Secondly, delving deeper into it, we demonstrate that with a shift in focus towards the Hilbert space L^2, new horizons emerge. The need for more generalisations of classically known algebraic iterative methods, with a particular care landing on generalising those of relaxation; namely, the Jacobi Over-relaxation (JOR) scheme, uncovers various theoretical corners, demonstrating that, with implicit analogies to R^n, we provide coherent and consistent findings as well as highlight the promising possibility of additional investigations despite the handful of limitations encountered. Our work concludes with enticing perspectives and inviting goals for richer and more comprehensive explorations.