Weak initial regularity occurs quite commonly in nonlinear Caputo evolution equations even if the coefficient functions and boundary conditions are sufficiently smooth. The lack of regularity of the solution at the beginning is an essential numerical challenge since the error generated near the starting time propagates through the hereditary integration operation. The algebraic time transformation $t=s^{\eta /\alpha }$ with $\eta \in N^{+}$ is employed to derive product-integration schemes for nonlinear models having singular terms $t^{j\alpha }$ with $0<\alpha <1$. In particular, the power of the Caputo derivatives can be transformed into a smoother function of the transformed time, which allows local interpolation of the nonlinearity. The weakly singular memory kernel is incorporated directly within the weights of the product integration. In the case of $\eta =1$, piecewise constant interpolation yields first-order implicit and delayed-state rectangle methods. If $\eta =2$, then linear interpolation results in implicit, Newton-linearized, and extrapolated trapezoidal schemes with order two temporal accuracy. The proof of convergence consists of a careful study of the transformed quadrature weights along with a weakly singular discrete Gronwall lemma. The rectangle methods show first-order convergence while the trapezoidal schemes achieve second-order temporal accuracy for $\alpha =0.4,0.6,0.8$ for both the test examples considered, which are the exact solution $u\left(t\right)=t+t^{\alpha }$ for a nonlinear scalar Caputo reaction equation and the exact solution $u(x,t)=(t+t^{\alpha })\sin x$ of a time-fractional Allen--Cahn equation. It turns out that for the Allen--Cahn model, the transformed scheme performs effectively not only for the pure nonlinear Caputo problem but also when combined with the diffusion and cubic reaction. Moreover, the finite differences are second-order accurate in space for this equation.

Caputo derivative,product integration,initial-layer singularity,nonlinear fractional equation,Volterra integral equation,fractional Allen--Cahn equation,discrete Gronwall inequality,memory-dependent materials modelling