In this paper, we propose an efficient method for constructing numerical algorithms for solving the fractional initial value problem by using the Pade approximation of fractional derivative operators. We regard the Grunwald–Letnikov fractional derivative as a kind of Taylor series and get the approximation equation of the Taylor series by Pade approximation. Based on the approximation equation, we construct the corresponding numerical algorithms for the fractional initial value problem. Finally, we use some examples to illustrate the applicability and efficiency of the proposed technique.

In the past decades, fractional differential equations were successfully applied to many problems in engineering, physics, chemistry, biology, economics, control theory, biophysics, and so on [

Among all the numerical methods, the direct numerical method [

In this paper, we propose an efficient method for constructing numerical algorithms for solving the fractional initial value problem by using the Pade approximation of fractional derivative operators. The advantage of the proposed technique is that efficient numerical methods can be constructed without calculation of long historical terms of the fractional derivative.

Consider the following homogeneous fractional initial problem:

For the following linear nonhomogeneous fractional initial problem,

Based on the

In numerical mathematics, Pade approximation [

Given a function

Equivalently, if

There are many definitions for the fractional derivative. The following equation is called the reverse Grunwald–Letnikov derivative:

For any given

Therefore, we can denote

So, we have

Let

Therefore,

Replacing

Noticing (

So, we get

Then, we get the following approximation to fractional derivative operators:

Consider fractional initial value problem (

Denote

From (

Due to the linearity and commutativity of the operators

Noticing

Then, we get the following numerical algorithm:

This is a multiple step algorithm for problem (

Let

Then, from (

In the same way, from the following

In the same way, the

So, we get the following numerical algorithm:

We can also get the following algorithm by using the [3, 3] Pade approximant to

The exact solution to this problem is

Computational errors for algorithms (1) and (2).

1 h | 2 h | 3 h | 4 h | 5 h | 6 h | 7 h | 8 h | 9 h | 10 h | |
---|---|---|---|---|---|---|---|---|---|---|

Algorithm (1) | 0.000 | 0.000 | 0.0155 | 0.0318 | 0.0455 | 0.0575 | 0.0716 | 0.0992 | 0.0783 | 0.0069 |

Algorithm (2) | 0.000 | 0.000 | 0.000 | 0.0011 | 0.0028 | 0.0043 | 0.0049 | 0.0042 | 0.0211 | 0.0263 |

Comparison between algorithms (1) and (2) with the exact solution.

We also make the comparison with the corresponding

Computational error comparison with the

1 h | 2 h | 3 h | 4 h | 5 h | 6 h | 7 h | 8 h | 9 h | 10 h | |
---|---|---|---|---|---|---|---|---|---|---|

Algorithm (1) | 0.000 | 0.000 | 0.0155 | 0.0318 | 0.0455 | 0.0575 | 0.0716 | 0.0992 | 0.0783 | 0.0069 |

Algorithm (2) | 0.000 | 0.000 | 0.000 | 0.0011 | 0.0028 | 0.0043 | 0.0049 | 0.0042 | 0.0211 | 0.0263 |

0.1232 | 0.1782 | 0.1642 | 0.2589 | 0.2529 | 0.2762 | 0.3856 | 0.3842 | 0.5485 | 0.5478 |

The exact solution to this problem is

Computational errors for algorithms (3) and (4).

1 h | 2 h | 3 h | 4 h | 5 h | 6 h | 7 h | 8 h | 9 h | 10 h | |
---|---|---|---|---|---|---|---|---|---|---|

Algorithm (3) | 0.000 | 0.000 | 0.0111 | 0.0123 | 0.0245 | 0.0267 | 0.0343 | 0.0355 | 0.0455 | 0.0058 |

Algorithm (4) | 0.000 | 0.000 | 0.0000 | 0.0009 | 0.0019 | 0.0038 | 0.0037 | 0.0042 | 0.0111 | 0.0165 |

The comparison with the corresponding

Comparison with the

1 h | 2 h | 3 h | 4 h | 5 h | 6 h | 7 h | 8 h | 9 h | 10 h | |
---|---|---|---|---|---|---|---|---|---|---|

Algorithm (1) | 0.000 | 0.000 | 0.0155 | 0.0318 | 0.0455 | 0.0575 | 0.0716 | 0.0992 | 0.0783 | 0.0069 |

Algorithm (2) | 0.000 | 0.000 | 0.000 | 0.0011 | 0.0028 | 0.0043 | 0.0049 | 0.0042 | 0.0211 | 0.0263 |

G-algm | 0.1255 | 0.1351 | 0.1385 | 0.1395 | 0.2452 | 0.2522 | 0.3836 | 0.3911 | 0.4263 | 0.5485 |

The exact solution is

Comparison between algorithm (5) and the exact solution.

We can construct numerical algorithms for solving the fractional initial value problem based on the Pade approximation of fractional derivative operators. Numerical tests show that this method is more efficient than the corresponding

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.