Advancing from 2m-sets of triangular functions to 3m-sets of parabolic functions: Direct solutions for integral equations

Orthogonal functions have long been fundamental in mathematical modeling and analysis, particularly for their adaptability to computational environments. With the increasing complexity of digital technologies, piecewise orthogonal functions have gained prominence for their efficiency in simplifying differential and integral equations. This work introduces a novel class of 3m-sets of piecewise orthogonal parabolic (quadratic) functions (PFs). These functions decompose differential and integral equations into computationally manageable components, offering improvements in accuracy compared to the existing 2m-sets and 1m-set of piecewise orthogonal linear and constant functions, respectively. We investigate the mathematical properties of these PFs, develop operational matrices for definite integrals, and demonstrate their effectiveness in solving second-kind Volterra integral equations. As we have shown, the derived formula can be useful in cases where kernels with fractional memory are involved. This new methodology enables the transformation of equations into explicit algebraic forms, eliminating the need for traditional integration methods and offering substantial computational benefits. The approach enhances both the precision and applicability of numerical solutions across various fields of applied mathematics and engineering, expanding the toolkit available to researchers and practitioners.