LIL

Implicit quadratic LIL solver for the Caputo FDE

LIL is a MATLAB solver for systems of commensurate Caputo fractional differential equations with a common fractional order (0<\alpha<1). The method is based on an implicit quadratic Lagrange Interpolation at the Last step (LIL) predictor-corrector scheme derived from the Volterra integral formulation of the problem.

The first interval is treated by linear startup interpolation, while all subsequent intervals use backward quadratic Lagrange interpolation. The resulting implicit corrector retains the full Volterra memory and is solved by fixed-point iteration. A fractional Adams--Bashforth predictor is used as the initial guess for the nonlinear iteration.

The routine returns the numerical solution and, optionally, the number of fixed-point iterations required at each time step. The implementation terminates if the prescribed nonlinear convergence tolerance is not achieved within the maximum number of iterations, so that an unconverged implicit iterate is not propagated.

The theoretical formulation, convergence analysis, numerical experiments, and implementation details are presented in [1]

For systems with different fractional orders (\alpha_i), see the companion LIL_nc implementation on MATLAB Central File Exchange.