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The aim of this paper is to approximate the solution of system of fractional delay differential equations. Our technique relies on the use of suitable spline functions of polynomial form. We introduce the description of the proposed approximation method. The error analysis and stability of the method are theoretically investigated. Numerical example is given to illustrate the applicability, accuracy and stability of the proposed method.

Recently, the use of various types of spline function in the numerical treatment of ordinary differential equations [

is studied by Kia Dithelm and N. J. Ford [

where

with constant

An extension of the spline functions form defined in [

considered. The spline functions

nomial form as:

for

Ramadan, M. A. obtained in [

using the spline function of the polynomial form which defined as:

where

Ramadan, Z. in [

where

Consider the system of first order delay differential equations:

The function g is called the delay function and it is assumed to be continuous on the interval

Suppose that

and there exists a constant

with

Suppose also that

and there exists a constant

with

These conditions assure the existence of unique solution y and z of system (4).

Let

Define the new form of system of fractional spline function

where

with

Such that

To estimate the error of the approximate solution, we write the exact solution

where

Moreover, we denote to the estimated error of

and at

Define the modulus of continuity of

and

Next lemma gives an upper bound to the error.

Lemma 1

Let

holds for all

Proof

Using the Lipschitz condition, Taylor expansion, definition of error estimation and (15) we get, by dropping

where

Therefore,

Thus,

and

where

Similarly,

where the constant

where

In the same manner we can prove that

where

The lemma is proved.

For analyzing the stability properties of the given method, we make a small change of the starting values and study the changes in the numerical solution produced by the method.

Now, we define the spline approximating function

where

Lemma 2

Let

holds where

Proof

Using Lipsechitz condition and (9), (17), (19) and (20) we get, by dropping a:

but

where

Thus from (21) and (22) we obtain:

where,

In the same manner we can prove that

where

Consider the system of fractional ordinary delay differential equations

The exact solution is given by

The obtained numerical results are summarized in

Absolute diff. between the two Appr. solutions | Appr. solution for the perturbed problem | Absolute Error | Appr. solution for the problem | x | a |
---|---|---|---|---|---|

1.50275 × 10^{−8} 2.33452 × 10^{−8} 3.03307 × 10^{−8} 3.66017 × 10^{−8} 4.24062 × 10^{−8} | 0.000925771 0.00286047 0.00555184 0.00890311 0.0128568 | 8.2 × 10^{−4} 2.5 × 10^{−3} 4.7 × 10^{−3 } 7.3 × 10^{−3 } 1.0 × 10^{−2} | y = 0.000925756 y = 0.00286044 y = 0.00555181 y = 0.00890308 y = 0.0128567 | 0.01 0.02 0.03 0.04 0.05 | 0.1 |

9.30474 × 10^{−9} 1.54803 × 10^{−8} 2.09326 × 10^{−8} 2.59856 × 10^{−8} 3.07735 × 10^{−8} | 0.000540099 0.00178793 0.00361276 0.0059613 0.00880111 | 4.4 × 10^{−4} 1.4 × 10^{−3} 2.7 × 10^{−3 } 4.4 × 10^{−3 } 6.1 × 10^{−3} | y = 0.00054009 y = 0.00178791 y = 0.00361274 y = 0.00596128 y = 0.00880108 | 0.01 0.02 0.03 0.04 0.05 | 0.2 |

5.71685 × 10^{−9} 1.01865 × 10^{−8} 1.43368 × 10^{−8} 1.83092 × 10^{−8} 2.21638 × 10^{−8} | 0.000313707 0.00111263 0.00234066 0.00397413 0.0059986 | 2.1 × 10^{−4} 7.1 × 10^{−4} 1.4 × 10^{−3 } 2.4 × 10^{−3 } 3.5 × 10^{−3} | y = 0.000313701 y = 0.00111262 y = 0.00234065 y = 0.00397411 y = 0.00599858 | 0.01 0.02 0.03 0.04 0.05 | 0.3 |

3.4872 × 10^{−9} 6.65526 × 10^{−9} 9.74972 × 10^{−9} 1.28095 × 10^{−8} 1.58509 × 10^{−8} | 0.000181441 0.000689483 0.00151013 0.0026383 0.00407143 | 8.1 × 10^{−5} 2.9 × 10^{−4} 6.1 × 10^{−4 } 1.0 × 10^{−3 } 1.0 × 10^{−3} | y = 0.000181438 y = 0.000689476 y = 0.00151012 y = 0.00263828 y = 0.00407141 | 0.01 0.02 0.03 0.04 0.05 | 0.4 |

2.11285 × 10^{−9} 4.31917 × 10^{−9} 6.58633 × 10^{−9} 8.90272 × 10^{−9} 1.12616 × 10^{−8} | 0.000104516 0.000425537 0.000970365 0.00174444 0.00275232 | 4.5 × 10^{−6} 2.6 × 10^{−5} 7.0 × 10^{−5 } 1.4 × 10^{−4 } 2.5 × 10^{−4} | y = 0.000104514 y = 0.000425532 y = 0.000970359 y = 0.00174443 y = 0.00275231 | 0.01 0.02 0.03 0.04 0.05 | 0.5 |

Absolute diff. between the two Appr. solutions | Appr. solution for the perturbed problem | Absolute Error | Appr. Solution for the problem | X | a |
---|---|---|---|---|---|

1.50275 × 10^{−8} 2.33452 × 10^{−8} 3.03307 × 10^{−8} 3.66017 × 10^{−8} 4.24062 × 10^{−8} | 0.000925771 0.00286047 0.00555184 0.00890311 0.0128568 | 8.2 × 10^{−4} 2.5 × 10^{−3} 4.7 × 10^{−3 } 7.3 × 10^{−3 } 1.0 × 10^{−2} | z = 0.000925756 z = 0.00286044 z = 0.00555181 z = 0.00890308 z = 0.0128567 | 0.01 0.02 0.03 0.04 0.05 | 0.1 |

9.30474 × 10^{−9} 1.54803 × 10^{−8} 2.09326 × 10^{−8} 2.59856 × 10^{−8} 3.07735 × 10^{−8} | 0.000540099 0.00178793 0.00361276 0.0059613 0.00880111 | 4.4 × 10^{−4} 1.4 × 10^{−3} 2.7 × 10^{−3 } 4.4 × 10^{−3 } 6.1 × 10^{−3} | z = 0.00054009 z = 0.00178791 z = 0.00361274 z = 0.00596128 z = 0.00880108 | 0.01 0.02 0.03 0.04 0.05 | 0.2 |

5.71685 × 10^{−9} 1.01865 × 10^{−8} 1.43368 × 10^{−8} 1.83092 × 10^{−8} 2.21638 × 10^{−8} | 0.000313707 0.00111263 0.00234066 0.00397413 0.0059986 | 2.1 × 10^{−4} 7.1 × 10^{−4} 1.4 × 10^{−3 } 2.4 × 10^{−3 } 3.5 × 10^{−3} | z = 0.000313701 z = 0.00111262 z = 0.00234065 z = 0.00397411 z = 0.00599858 | 0.01 0.02 0.03 0.04 0.05 | 0.3 |

3.4872 × 10^{−9} 6.65526 × 10^{−9} 9.74972 × 10^{−9} 1.28095 × 10^{−8} 1.58509 × 10^{−8} | 0.000181441 0.000689483 0.00151013 0.0026383 0.00407143 | 8.1 × 10^{−5} 2.9 × 10^{−4} 6.1 × 10^{−4 } 1.0 × 10^{−3 } 1.0 × 10^{−3} | z = 0.000181438 z = 0.000689476 z = 0.00151012 z = 0.00263828 z = 0.00407141 | 0.01 0.02 0.03 0.04 0.05 | 0.4 |

2.11285 × 10^{−9} 4.31917 × 10^{−9} 6.58633 × 10^{−9} 8.90272 × 10^{−9} 1.12616 × 10^{−8} | 0.000104516 0.000425537 0.000970365 0.00174444 0.00275232 | 4.5 × 10^{−6} 2.6 × 10^{−5} 7.0 × 10^{−5 } 1.4 × 10^{−4 } 2.5 × 10^{−4} | z = 0.000104514 z = 0.000425532 z = 0.000970359 z = 0.00174443 z = 0.00275231 | 0.01 0.02 0.03 0.04 0.05 | 0.5 |

shown in the fifth column. To test the stability, the difference between the two approximate solutions is computed as shown in the six column

From the obtained results in

We adapt the spline functions with some additional assumptions and definitions for approximating the solution of system of ordinary delay differential equation with fractional order which studied in [

Mahmoud N. Sherif,1 1, (2016) Numerical Solution of System of Fractional Delay Differential Equations Using Polynomial Spline Functions. Applied Mathematics,07,518-526. doi: 10.4236/am.2016.76048