LIL

Implicit quadratic LIL solver for the Caputo FDE

The code, a faster algebraic implementation based on precomputed closed-form weights, implements the assembled implicit quadratic LIL solver for Caputo FDEs.

Starting from the Volterra integral formulation, we derived a full-memory discretization based on

backward quadratic Lagrange interpolation, implemented through assembled subinterval weights. Under suitable smoothness assumptions, the LIL method was shown to be globally third-order convergent on uniform

meshes. More precisely, the proved \(O(h^3)\) convergence result concerns the quadratic

assembled core of the discretization.

The method retains the intrinsic memory effect of Caputo-type fractional models:

although the interpolation on each subinterval is local, the resulting numerical

scheme depends on the entire past history through the accumulated convolution

weights. In this sense, the LIL method may be viewed as a two-step

backward extension of the classical ABM approach,

since LIL uses interpolation over \(t_{k+1}, t_k, t_{k-1}\), whereas ABM is

based on one-step backward linear interpolation.

The experiments confirm that the LIL method is consistently more accurate than the ABM PECE discretization on uniform meshes.

For smooth data it realizes the expected third-order behavior in the linear

polynomial test and remains clearly superior in the nonlinear smooth test. For

mildly weakly singular data, both methods lose accuracy order on a uniform mesh, but the LIL method still preserves a substantially better convergence rate than ABM.

The proposed LIL discretization provides an accurate and

effective full-memory approach for Caputo fractional differential equations.

The analysis presented in [1] establishes third-order accuracy for the quadratic assembled core on

uniform meshes under suitable smoothness assumptions, while the linear startup

treatment may limit the asymptotic order of the complete implementation.

Nevertheless, the numerical results show that the method consistently outperforms

the classical ABM scheme on the tested meshes and offers a favorable balance of

accuracy and computational cost, particularly in the smooth-data regime.