New Adomian’s Polynomials Formulas for the Non-linear and Non-autonomous Ordinary Differential Equations

In this paper, Adomian decomposition method has been adopted to resolve the non-linear and non-autonomous ordinary differential equations. It has been proved that this technique permits to give new expressions for the Adomian’s polynomials (??) and (??). Citation: Zaouagui IN, Badredine T (2017) New Adomian’s Polynomials Formulas for the Non-linear and Non-autonomous Ordinary Differential Equations. J Appl Computat Math 6: 373. doi: 10.4172/2168-9679.1000373

Numerical examples treated in this work in order to compare the Adomian decomposition method with other classical methods such as Euler, Heun and the Runge Kutta method of the 4 th order.

Application of the Decomposition Method to Nonautonomous and Non-linear Differential Equations
Let us consider the following non-linear and non-autonomous differential equation: , du f t u dt (1) u(t 0 )=c (2) where f is the non-linear term.
By integrating both sides the equation (??) we obtain: The canonical form [11,14] of Cauchy's problem of the eqn. (3) is: The Adomian's method consists in setting the solution in a series form: and the non-linear term f(t,u) is written by the form: where the A n 's are terms dependent on t,u 0, u 1,…., u n and are called Adomian's polynomials Inserting eqns. (5) and (6) into eqn. (4) leads to: which implies:

Remark
In practice it's difficult to calculate all the terms of the series (5) that's why an approximation of the solution is used by truncating the series:   and f is a multivariate function, we note: k k nk

Adomian's Polynomials Theorem
The terms A n are given by the formulas:

Proof
For convenience we define two operators F and L as follows: From equation (??) we have: L(U)=F(U), so L -1 (L(U))= L -1 (F(U)) and it follows that: We obtain an equation of the form: According to Abbaoui et al. and Himoun et al. [11,14], the Adomian's polynomials A n associated to N(U) are given by: As a result of this: Considering the fact that the sequence (U n ) n∈N is defined by: U 0 =(t 0 ,u 0 ),U 1 =(t-t 0 ,u 1 ) and U n =(0,u n ) for n≥2, we obtain: which ends the proof by a simple identification with eqn. (8).

Lamma
Let f be a function of two variables u and v. The Adomian's polynomials associated with f are given by the relationship [3,4]:

Theorem
The n A 's are given by:

Proof
It's sufficient to apply the eqn (11) to f(t,u) by using the following decompositions: where t 1 =t-t 0 and t n =0 for n≥2, so we have:  is a decreasing sequence of non-negative integers then:

A n 's Expressions for n=5
The following table shows the solution of the system:

Numerical Example
Let us solve the following equation: The exact solution of this equation is the function: ( ) = ln u t t t − . By the Adomian decomposition method, we obtain: from where, it follows that: It's easy to see that φ 7 (t) is an approximation of the 6 th order of Taylor's expansion of in the neighborhood of 1.

Conclusion
In this paper we have generalized the application of the decomposition method to non-linear and non-autonomous differential equations. We have obtained new expressions for the Adomian polynomials (??) and (??). The case of autonomous nonlinear differential equation [2] becomes, of course, a particular case of this work. Moreover, we not that the Adomian method is more general than some classical methods because the terms of truncated series can be easily deduced in a recursive way. However, the solution coincides quite often with that of another classical method (in the example studied, the solution coincides with the Taylor series).