Implicit-explicit numerical integration scheme for fractional differential equations (FDEs)
This project contains the first and second-order IMEX time-integration approaches for stiff/nonlinear FDEs presented in:
Zhou, Y., Suzuki, J.L., Zhang, C., Zayernouri, M. "Implicit-explicit time integration of nonlinear fractional differential equations". Applied Numerical Mathematics 156, 2020.
The main features of the presented IMEX approaches are:
- First- and second-order implicit-explicit (IMEX) solver for time-integration of stiff/nonlinear FDEs equations with fractional order α in (0,1), with proven convergence and linear stability
- The methods are based on a linear multi-step fractional Adams-Moulton method (FAMM), followed by the extrapolation of the nonlinear force terms
- The singularities nearby the initial time are addressed through Lubich-like corrections
- A fast inversion scheme is employed to achieve a computational complexity of O(N log N), where N denotes the number of time-steps
- Currently, the implemented methods support the solution of systems of FDEs with single fractional orders nonlinear force terms
- The example structures involve only a few steps on domain definition, correction terms, the main solver call, and the right-hand-side definition see src/Ex3_Case3.m
Users can access the examples/ folder for the following working examples:
- Ex2.m - Solution of a stiff system of FDEs from a fabricated multi power-law type solution
- Ex3_Case2.m - Solution of a nonlinear FDE from a fabricated polynomial solution
- Ex3_Case3.m - Solution of a nonlinear FDE with a nonlinear/harmonic RHS
The implemented solvers and auxiliary functions can be found in the src/ folder, which contains the following files:
- IMEX_I.m - First-order IMEX solver for single nonlinear FDE problems
- IMEX_II.m - Second-order IMEX solver for single nonlinear FDE problems
- IMEX_I_A.m - First-order IMEX solver for problems involving a nonlinear system of FDEs
- IMEX_II_A.m - Second-order IMEX solver for problems involving a nonlinear system of FDEs
Auxiliary files:
- sptoeplitz.m Sparse Toeplitz allocation: Toby Driscoll (2021). Sparse Toeplitz matrix construction (https://www.mathworks.com/matlabcentral/fileexchange/13353-sparse-toeplitz-matrix-construction), MATLAB Central File Exchange. Retrieved November 27, 2021.
- gjquadreal2f1.m Gauss-Jacobi quadrature for 2F1 Gauss hypergeometric functions from Pearson, J. W. (2009). Computation of hypergeometric functions (Doctoral dissertation, University of Oxford).
- qrule.m Gauss-Jacobi quadrature from https://www.mathworks.com/matlabcentral/mlc-downloads/downloads/submissions/32/versions/1/previews/qrule.m/index.html
Feel free to contact Jorge Suzuki at suzukijo@msu.edu if you have any questions.