Power-Series Collocation for Linear and Nonlinear Fractional Volterra Integro-Differential Equations: Two Verified Polynomial Benchmarks

Abstract

A power-series collocation procedure is developed for a class of multi-term fractional Volterra integro-differential equations containing Caputo derivatives, variable coefficients, and linear or nonlinear memory terms. The equation is first converted to an equivalent fractional integral equation; a polynomial trial function is then inserted and the resulting residual is collocated at uniformly distributed points, with the initial conditions used as algebraic rows. Continuity and uniqueness follow from a Banach’s contraction argument under an explicit Lipschitz bound, while the computable residual gives an a posteriori error estimate whenever the contraction constant is below one. Two complementary examples are considered: a nonlinear equation with a cubic Volterra term and a linear equation with an exponential kernel. The reported degree-four and degree-three polynomials reproduce the stated quadratic and cubic reference solutions, respectively, to approximately eleven decimal places at the tested nodes. A coefficient-level audit is included to separate the actual polynomial errors from zeros caused by twelvedecimal tabulation. The study demonstrates the economy of the collocation construction for polynomial solutions and records the source-data consistency conditions needed for reproducible use of the two benchmarks.