A Quasilinearization–Fractional Spline Collocation Method for Nonlinear Fractional Volterra Integro-Differential Equations with Logarithmic Weakly Singular Kernels

Abstract

We construct a collocation scheme for nonlinear fractional Volterra integro-differential equations whose Volterra term contains a logarithmic weak singularity. Quasilinearization is used to obtain a sequence of linear fractional problems, and each iterate is represented by a six-parameter fractional non-polynomial spline on every mesh interval. The discrete formulation retains the full Caputo history and closes the spline coefficients globally through continuity, the initial value, and five interior collocation conditions per interval; no auxiliary endpoint derivative data are introduced. The logarithmic integral is evaluated by product integration. For sufficiently small mesh widths, the scaled algebraic system is shown to be nonsingular; conditional stability, consistency, and fixed-iteration convergence are then established under the stated assumptions. A manufactured test and two nonlinear benchmarks are used for verification. In the y=x8/5 benchmark, the empirical refinement rate increases toward two, while the benchmark with a logarithmic endpoint factor converges more slowly. The maximum error decreases on every reported mesh, and both nonlinear tests satisfy the stopping criterion after five quasilinearization iterations.