Abstract
A spectral deferred correction method is presented for the initial value problems of fractional differential equations (FDEs) with Caputo derivative. This method is constructed based on the residual function and the error equation deduced from Volterra integral equations equivalent to the FDEs. The proposed method allows that one can use a relatively few nodes to obtain the high accuracy numerical solutions of FDEs without the penalty of a huge computational cost due to the nonlocality of Caputo derivative. Finally, preliminary numerical experiments are given to verify the efficiency and accuracy of this method.
Highlights
Fractional calculus, almost as old as the familiar integer-order calculus, has recently gained considerable popularity and importance due to its attractive applications in widespread fields of science and engineering (e.g., [1,2,3,4,5,6]), such as control theory [5], viscoelasticity [7], image processing [8], electromagnetism [9], anomalous diffusion [10, 11], and hydrology [12, 13]
A spectral deferred correction method for classical ordinary differential equations [31, 32] is extended and reconstructed to solve fractional differential equations (FDEs) based on accelerating the convergence of lower-order schemes
We list the number of iterations denoted by “Iter.,” the CPU time in second “Time[s]” and the errors in the maximum norm for different nodes n and fractional order index α. It demonstrates that when the new spectral deferred correction method is used to solve Example 1, only a relatively few nodes are used, and the high-order accuracy numerical solution is obtained with small computation complexity and computing time
Summary
A spectral deferred correction method is presented for the initial value problems of fractional differential equations (FDEs) with Caputo derivative. This method is constructed based on the residual function and the error equation deduced from Volterra integral equations equivalent to the FDEs. The proposed method allows that one can use a relatively few nodes to obtain the high accuracy numerical solutions of FDEs without the penalty of a huge computational cost due to the nonlocality of Caputo derivative. Preliminary numerical experiments are given to verify the efficiency and accuracy of this method
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