A fractional Adams–Simpson-type method for nonlinear fractional ordinary differential equations with non-smooth data
Tóm tắt
We propose a fractional Adams–Simpson-type method for nonlinear fractional ordinary differential equations with fractional order
$$\alpha \in (0,1)$$
. In our method, a nonuniform mesh is used so that the optimal convergence order can be recovered for non-smooth data. By developing a modified fractional Grönwall inequality, we prove that the method is unconditionally convergent under the local Lipschitz condition of the nonlinear term, and show that with a proper mesh parameter, the method can achieve the optimal convergence order
$$3+\alpha $$
even if the given data is not smooth. Under very mild conditions, the nonlinear stability of the method is analyzed by using a perturbation technique. The extensions of the method to multi-term nonlinear fractional ordinary differential equations and multi-order nonlinear fractional ordinary differential systems are also discussed. Numerical results confirm the theoretical analysis results and demonstrate the effectiveness of the method for non-smooth data.
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