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The aim of this paper is to apply Adomian decomposition method (ADM) for solving some classes of nonlinear delay differential equations (NDDEs) with accelerated Adomian polynomial called El-kalla polynomial proposed by El-kalla [1]. The main advantages of El-kalla polynomials can be summarized in the following main three points: 1) El-kalla polynomials are recursive and do not have derivative terms so, El-kalla formula is easy in programming and save much time on the same processor compared with the traditional Adomian polynomials formula; 2) Solution using El-Kalla polynomials converges faster than the traditional Adomian polynomials; 3) El-Kalla polynomials used directly in estimating the maximum absolute truncated error of the series solution. Some convergence remarks are studied and some numerical examples are solved using the Adomian decomposition method using the two polynomials (Adomian polynomial and El-kalla polynomial). In all applied cases, we obtained an excellent performance that may lead to a promising approach for many applications.

Delay differential equations are frequently used to model real life phenomena in various fields such as mechanics, computer science, biology and chemistry. Some of the recent studies involving delay differential equations include topics as varied as epidemic models that describe the fraction of a population infected by a virus [

L u ( x ) = h ( x ) + f ( x , u ( x ) , g ( u ( x ) ) ) , 0 ≤ x ≤ 1 (1)

where L = d n d x n is the highest derivative respect to the variable x,

f ( x , u ( x ) , g ( u ( x ) ) ) is the nonlinear term and h ( x ) is any other terms. We will separate the highest derivative on the left side of the equation. Then we take L − 1 to both sides, where L − 1 is integration from 0 to x. After integration the nonlinear term will be the Adomian polynomial term or El-kalla polynomial such that:

u ( x ) = u ( 0 ) + L − 1 h ( x ) + L − 1 ( A n ) . (2)

Then Adomian solution assumes that:

u ( x ) = ∑ n = 0 ∞ u n = u 0 + u 1 + u 2 + u 3 + ⋯ (3)

u 0 = u ( 0 ) + L − 1 h ( x ) u n + 1 = L − 1 ( A n ) where n = 0 , 1 , 2 , 3 , ⋯ ⋮ (4)

After making the integration we get the solution, where A 0 , A 1 , A 2 , ⋯ are Adomian polynomials, or we can use the new El-kalla polynomials A ¯ 0 , A ¯ 1 , A ¯ 2 , ⋯

A n = 1 n ! ( d n d ƛ n [ N ( ∑ i = 0 n ƛ i u i ) ] ) ƛ = 0 , (5)

where N ( u i ) is the nonlinear term.

A ¯ n = f ( S n ) − ∑ i = 0 n − 1 A ¯ i , (6)

where A ¯ n , are El-kalla polynomials, A ¯ 0 , A ¯ 1 , A ¯ 2 , ⋯ , f ( S n ) is the substitution of the summation of dependent variable in the nonlinear term.

For example the Adomian polynomials & El-Kalla polynomials of the nonlinear term y 2 are showen in

Also the nonlinear term y 3 the Adomian polynomials & El-kalla polynomials are shown in

Many authors discussed Convergence of the Adomian decomposition method. For example, K. Abbaoui and Y. Cherruault [

Adomian polynomials of y 2 | El-Kalla polynomials of y 2 |
---|---|

A 0 = y 0 2 | A ¯ 0 = y 0 2 |

A 1 = 2 y 0 y 1 | A ¯ 1 = 2 y 0 y 1 + y 1 2 |

A 2 = y 1 2 + 2 y 0 y 2 | A ¯ 2 = 2 y 0 y 2 + 2 y 1 y 2 + y 2 2 |

A 3 = 2 y 1 y 2 + 2 y 0 y 3 | A ¯ 3 = 2 y 0 y 3 + 2 y 1 y 3 + 2 y 2 y 3 + y 3 2 |

A 4 = y 2 2 + 2 y 1 y 3 + 2 y 0 y 4 | A ¯ 4 = 2 y 0 y 4 + 2 y 1 y 4 + 2 y 2 y 4 + 2 y 3 y 4 + y 4 2 |

Adomian polynomials of y 3 | El-kalla polynomials of y 3 |
---|---|

A 0 = y 0 3 | A ¯ 0 = y 0 3 |

A 1 = 3 y 1 y 0 2 | A ¯ 1 = 3 y 1 y 0 2 + 3 y 0 y 1 2 + y 1 3 |

A 2 = 3 y 0 y 1 2 + 3 y 2 y 0 2 | A ¯ 2 = 3 y 2 y 0 2 + 6 y 0 y 1 y 2 + 3 y 0 y 2 2 + 3 y 2 y 1 2 + 3 y 1 y 2 2 + y 2 3 |

A 3 = y 1 3 + 3 y 3 y 0 2 + 6 y 0 y 1 y 2 | A ¯ 3 = 3 y 3 y 0 2 + 6 y 0 y 1 y 3 + 6 y 0 y 2 y 3 + 3 y 0 y 3 2 + 3 y 3 y 1 2 + 6 y 1 y 2 y 3 + 3 y 1 y 3 2 + 3 y 3 y 2 2 + 3 y 2 y 3 2 + y 3 3 |

Consider the nonlinear delay differential equation:

d y ( x ) d x = 1 − 2 [ y ( x 2 ) ] 2 , y ( 0 ) = 0 , 0 ≤ x ≤ 1 (7)

We will solve this problem using Adomian decomposition method using Adomian polynomials and El-kalla polynomials.

Let, the solution

y = ∑ i = 0 ∞ y i = y 0 + y 1 + y 2 + ⋯ , (8)

d y ( x ) d x = 1 − 2 [ y ( x 2 ) ] 2 (9)

Make integration of both sides from 0 to x, we get:

y ( x ) = x − ∫ 0 x 2 [ y ( x 2 ) ] 2 d x (10)

y ( x ) 0 + y ( x ) 1 + y ( x ) 2 + ⋯ = x − 2 ∫ 0 x [ A ( x 2 ) 0 + A ( x 2 ) 1 + A ( x 2 ) 2 + ⋯ ] d x (11)

y ( x ) 0 = x

y ( x ) 1 = − 2 ∫ 0 x A ( x 2 ) 0 d x

y ( x ) 2 = − 2 ∫ 0 x A ( x 2 ) 1 d x

y ( x ) 3 = − 2 ∫ 0 x A ( x 2 ) 2 d x (12)

where the nonlinear term is [ y ( x 2 ) ] 2 , we calculate A ( x 2 ) 0 , A ( x 2 ) 1 , A ( x 2 ) 2 , ⋯ from the Equation (5),

A ( x 2 ) 0 = ( x 2 ) 2 A ( x 2 ) 1 = − x 4 48 A ( x 2 ) 2 = x 6 1440 ⋮ (13)

y ( x ) 0 = x y ( x ) 1 = − x 3 6 y ( x ) 2 = x 5 120 y ( x ) 3 = − x 7 5040 (14)

So, the Solution is

y = ∑ i = 0 ∞ y i = y 0 + y 1 + y 2 + ⋯ (15)

y = x − x 3 6 + x 5 120 − x 7 5040 + ⋯ , (16)

which leads to the closed form solution:

y ( x ) = sin ( x ) , (17)

which equal to the exact solution.

The solution is the same as before in Equations (8)-(12) except when we calculate El-kalla polynomials we use Equation (6) as follow.

A ¯ ( x 2 ) 0 = ( x 2 ) 2 A ¯ ( x 2 ) 1 = ( y ( x 2 ) 0 + y ( x 2 ) 1 ) 2 − A ¯ ( x 2 ) 0 A ¯ ( x 2 ) 2 = ( y ( x 2 ) 0 + y ( x 2 ) 1 + y ( x 2 ) 2 ) 2 − A ¯ ( x 2 ) 0 − A ¯ ( x 2 ) 1 ⋮ (18)

y ( x ) 0 = x y ( x ) 1 = − 2 ∫ 0 x A ¯ ( x 2 ) 0 d x = − x 3 6 y ( x ) 2 = − 2 ∫ 0 x A ¯ ( x 2 ) 1 d x = − x 5 ∗ ( 5 ∗ x 2 − 336 ) 40320 y ( x ) 3 = − 2 ∫ 0 x A ¯ ( x 2 ) 2 d x = − x 7 ∗ ( 715 ∗ x 8 − 443520 ∗ x 6 + 112379904 ∗ x 4 ) 5713316492083200 + x 7 ∗ ( − 15006351360 ∗ x 2 + 425097953280 ) 5713316492083200 (19)

The solution is

y = ∑ i = 0 ∞ y i = y 0 + y 1 + y 2 + ⋯ (20)

In

The time elapsed of the program that calculates the solution of Example 1 in Matlab R2014a:

Using Adomian polynomials = 6.3183 seconds;

Using El-kalla polynomials = 5.6548 seconds;

This data calculated by taking six terms of the series solution y = y 0 + y 1 + y 2 + y 3 + y 4 + y 5 in Example 1 (Figures 1-3).

Consider the nonlinear delay differential equation

d 3 y ( x ) d x 3 = − 1 + 2 [ y ( x 2 ) ] 2 , y ( 0 ) = 0 , d y d x ( 0 ) = 1 , d 2 y d x 2 ( 0 ) = 0 , 0 ≤ x ≤ 1 (21)

We will solve this problem using Adomian decomposition method using Adomian polynomials and El-kalla polynomials.

Let, the solution

y = ∑ i = 0 ∞ y i = y 0 + y 1 + y 2 + ⋯ , (22)

d 3 y ( x ) d x 3 = − 1 + 2 [ y ( x 2 ) ] 2 (23)

Make integration of both sides from 0 to x, we get

d 2 y d x 2 = − x + 2 ∫ 0 x [ y ( x 2 ) ] 2 d x (24)

Make integration of both sides from 0 to x, we get

d y d x = 1 − x 2 2 + 2 ∫ 0 x ∫ 0 x [ y ( x 2 ) ] 2 d x d x (25)

x | (ARE) of solution using Adomian polynomials | (ARE) of solution using El-kalla polynomials |
---|---|---|

0.1 | 1.60582*10^{−}^{23} | 3.44432*10^{−}^{28} |

0.2 | 1.31530*10^{−}^{19} | 2.81772*10^{−}^{24} |

0.3 | 2.55923*10^{−}^{17} | 5.47133*10^{−}^{22} |

0.4 | 1.07688*10^{−}^{15} | 2.29565*10^{−}^{20} |

0.5 | 1.958*10^{−}^{14} | 4.15860*10^{−}^{19} |

0.6 | 2.09383*10^{−}^{13} | 4.42705*10^{−}^{18} |

0.7 | 1.55232*10^{−}^{12} | 3.26462*10^{−}^{17} |

0.8 | 8.80170*10^{−}^{12} | 1.83964*10^{−}^{16} |

0.9 | 4.06629*10^{−}^{11} | 8.43953*10^{−}^{16} |

1 | 1.598285*10^{−}^{10} | 3.29123*10^{−}^{15} |

Make integration of both sides from 0 to x, we get

where the nonlinear term is

So, the Solution is

which leads to the closed form solution

which equal to the exact solution.

The solution is the same as before in Equations (22)-(28) except when we calculate El-kalla polynomials as follow:

The solution is

In

The time elapsed of the program that calculate the solution of Example 2 in Matlab R2014a:

Using Adomian polynomials = 11.8629 seconds;

Using El-kalla polynomials = 6.5055 seconds;

This data calculated by taking three terms of the series solution

x | (ARE) of solution using Adomian polynomials | (ARE) of solution using El-kalla polynomials |
---|---|---|

0.1 | 8.13863*10^{−}^{24} | 2351693*10^{−}^{2}^{5} |

0.2 | 6.66554*10^{−}^{2}^{0} | 1924773*10^{−}^{2}^{1} |

0.3 | 1.29671*10^{−}^{16} | 3.74036*10^{−}^{19} |

0.4 | 5.45508*10^{−}^{16} | 1.5711*10^{−}^{17} |

0.5 | 9.91548*10^{−}^{15} | 2.8501*10^{−}^{16} |

0.6 | 1.05994*10^{−}^{13} | 3.03934*10^{−}^{15} |

0.7 | 7.85472*10^{−}^{13} | 2.24589*10^{−}^{14} |

0.8 | 4.45139*10^{−}^{12} | 1.26858*10^{−}^{3} |

0.9 | 2.05531*10^{−}^{11} | 5.83546*10^{−}^{13} |

1 | 8.07337*10^{−}^{11} | 2.28259*10^{−}^{12} |

From all the previous examples, we extract that the solution of NDDEs by using the Adomian decomposition method with the new polynomial, El-kalla polynomial, which is faster and more accurate than using it with the traditional polynomial called Adomian polynomial. Also the formula that calculates El-kalla polynomial is simple, but the formula that calculates Adomian polynomial has a derivative term that takes time in calculations. It is clear that the time elapsed in the program that calculates the solution using El-kalla polynomial is less than the time elapsed in the program that calculates the solution using Adomian polynomial that means saving time in Matlab R2014a. Also, the maximum absolute relative error between the solution using El-kalla polynomial and the exact solution is less than the maximum absolute relative error between the solution using Adomian polynomial and the exact solution. And thus El-kalla polynomial can be used in solving a wide range of a nonlinear differential equation in many applications.

The authors declare no conflicts of interest regarding the publication of this paper.

El-Kalla, I.L., Elgaber, K.M.A., Elmahdy, A.R. and Sayed, A.Y. (2019) Solution of a Nonlinear Delay Differential Equation Using Adomian Decomposition Method with Accelerated Formula of Adomian Polynomial. American Journal of Computational Mathematics, 9, 221-233. https://doi.org/10.4236/ajcm.2019.94017