NUMERICAL SOLUTION OF FRACTIONAL DIFFERENTIAL EQUATION

ABSTRACT. We study a product-integration discretization for initial-value problems involvingthe ψ-Caputo derivative of order 0 < α < 1. After transforming the independent variable byx = ψ(t)- ψ(a), the equation takes a Volterra integral form with the standard weakly singularkernel. We define the numerical weights by integrating piecewise-linear hat functions againstthis kernel, including the full history and an implicit endpoint value. Under a global Lipschitzcondition, smoothness of the exact integrand, and an explicit contraction condition, we provea conditional global error bound of order O(h2). This result applies to the scheme specified inthis paper and does not assert unconditional stability. For a linear manufactured benchmark, thecorrected scheme is tested at four fractional orders; for a nonlinear manufactured benchmark, itis tested at α = 0.5. The observed orders approach two in both cases. Other proposed nonlinearexamples and stability analysis remain open.