These last years, a lot of relatively new techniques as Adomian decomposition method (ADM), perturbation method, homotopy perturbation method, SOME BLAISE ABBO (SBA) method, variational iteration method etc. are used to solving a linear and nonlinear partial differential equations. Many problems are governed by partial differential equations, or by systems of partial differential equations. It is difficult to find their exact solutions. In this paper, we use the Adomian decomposition method (ADM)     to find the exact solution of the singular fourth-order partial differential equation. This equation has been studied in  , one used the homotopy perturbation to get the solution of the singular fourth-order partial differential equation in two space variables.
2. About the ADM Method
Suppose that we need to solve the following equation
in a real Hilbert space H, where is a linear or a nonlinear operator, and u is the unknown function. The principle of the ADM is based on the decomposition of the operator A in the following form:
where L + R is linear, N nonlinear, L invertible with L−1 as inverse. Using that decomposition, equation (1) is equivalent to
where verifies . (3) is called the Adomian’s fundamental equation or Adomian’s canonical form.We look for the solution of (1) in a series expansion
form and we consider where are special polynomials
of variables called Adomian polynomials and defined by     :
where is a parameter used by “convenience”. Thus (3) can be rewritted as follllows:
We suppose that the series and are convergent, and obtain by identification the following Adomian algorithm:
In practice it is often difficult to calculate all the terms of an Adomian series, so we approach the series solution by the truncated series:
where the choice of n depends on error requirements. If this series converges, the solution of (1) is:
3. Resolution of the Singular Fourth-Order Parabolic Partial Differential Equation in m Space Variables (m ≥ 2)
3.1. The Singular Fourth-Order Parabolic Partial Differential Equation in Three Space Variables
We consider the following singular fourth-order parabolic partial differential equation in three space variables:
with the initial conditions
From (8), we have:
We suppose that the solution of (8) has the following form:
From (10) and (11) we have:
From (12), we obtain the following Adomian algorithm:
From (13), we obtain:
Remark: In the case of the singular fourth-order parabolic partial differential equation in two space variables, we have:
and we recover the examined case in  , where ,
3.2. Main Result
The exact solution of the following singular fourth-order parabolic partial differential equation in m space variables , ( ):
with the following initial conditions
If t = 0, from (20), we have .
From (20), we get:
and if t = 0, we have:
From (22) and (25), we obtain:
Through this example, we showed again the usefulness of the Adomian decomposition method, in the search of an approximate solution of a linear or nonlinear equation; and this method gives us the exact solution.
 Abbaoui, K. (1995) Les fondements de la méthode décompositionnelle d'Adomian et application à la résolution de problèmes issus de la biologie et de la médécine. Thèse de doctorat de l'Université Paris VI.