RESOLUTION OF RICCATI EQUATION BY THE METHOD DECOMPOSITIONAL OF ADOMIAN

In this work, we will use the method of decompositional for Adomian solve the Riccati equation in the form:


INTRODUCTION
The Riccati equation it is named in honor of Jacopo Francesco Riccati (1676-1754) and his son Vincenzo Riccati (1707-1775). In general equation (1) is not solvable by quadratures, if he knows a particular solution up, the Riccati equation (1) reduces to a Bernoulli equation. And in the 80 G. Adomian proposed a new method to solve differential equations of different types. This method is to look for the solution in the form of a series, and decompose the non-linear operator in a series of function (polynomials Adomian) [4,5] K .Abbaoui and Y.Cherrault, place assumptions on the convergence of series of Adomian to the exact solution [1,2,3,6]. This work mainly concerns the resolution of the Riccati equation by the Adomian method, with application examples.

ADOMIAN METHOD
We consider the following problem: as N is a nonlinear operator, and L the invertible linear portion of F. Equation (2) it gives: (4) with Ai are called Adomian polynomials. and the terms of the standard solution defined by:

RICCATI DIFERENTIAL EQUATION
is an ordinary differential equation of first order of the form [7]: ′ = ( ) + ( ) + ( ) 2 (6) or a , b and c are continuous functions defined on an open interval I of R. In general there is no solution by quadrature, but if he knows a particular solution, a Riccati equation is reduced by substitution in a Bernoulli equation.

RESOLUTION KNOWING A PARTICULAR SOLUTION
If it is possible to find a particular solution u p , So the general solution is of the form: = + (7) By replacing u by + in equation (

RESOLUTION BY THE METHOD ADOMIAN
Consider the following problem: ′ = ( ) + ( ) + ( ) 2 (13) (0) = 0 the Adomian method is used to solve the Riccati equation in the problem (13). We have: and we ask: With : polynomials Adomian of the function 2 [4,5]. 2 Using the D,L in the neighborhood of 0 of functions a, b and c: and we obtain: and the solution given by:

EXAMPLE1
We consider the following problem (Riccati equation): with the particular solution = 1

DIRECT RESOLUTION
We have the following equation according to (7), (9)

CONCLUSION
Despite generally, there is no Resolution of Riccati equation, but the method of decomposition Adomian always given an approximate solution in the form of a convergent series.