LIL_nc

Fast grouped implicit quadratic LIL solver for non-commensurate Caputo fractional differential systems

LIL_nc: Implicit quadratic solver for non-commensurate Caputo fractional differential equations

LIL_nc is a MATLAB solver for systems of non-commensurate Caputo fractional differential equations with fractional orders 0<αi<10<\alpha_i<1. The method is based on the same 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, the number of fixed-point iterations required at each time step, and information about the detected groups of equal or near-equal fractional orders. 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]

LIL_nc extends LIL to non-commensurate systems.