11.1 Introduction

The Adomian Decomposition Method (ADM) was first introduced by Adomian in the early 1980s [1,2]. It is an efficient semi‐analytical technique used for solving linear and nonlinear differential equations. It permits us to handle both nonlinear initial value problems (IVPs) and boundary value problems (BVPs).

The solution technique of this method [3,4] is mainly based on decomposing the solution of nonlinear operator equation to a series of functions. Each presented term of the obtained series is developed from a polynomial generated in the expansion of an analytic function into a power series. Generally, the abstract formulation of this technique is very simple, but the actual difficulty arises while calculating the required polynomials and also in proving the convergence of the series of functions. In this chapter, we present procedures for solving linear as well as nonlinear ordinary/partial differential equations by the ADM along with example problems for clear understanding.

The following section addresses the ADM for handling ordinary differential equations (ODEs).

11.2 ADM for ODEs

The ADM mainly depends on decomposing the governing differential equation 5,6] viz. F(x, p(x), p′(x)) = 0 into two components

(11.1)

where and N are the linear and nonlinear parts of F, respectively. The operator L_x is assumed as an invertible operator. Solving for L_x(p(x)), Eq. (11.1) leads to

(11.2)

Applying the inverse operator on both sides of Eq. (11.2) yields

(11.3)

where p(0) = φ(x) is the constant of integration which satisfies the given initial conditions. Now assume that the solution “p(x)” can be represented as an infinite series of the form

(11.4)

Furthermore, from Eq. (11.4) the nonlinear term N(p(x)) is written as infinite series [ 1 , 2 ] in terms of the Adomian polynomials A_n(x) of the form

(11.5)

where A_n Adomian polynomials of p₀, p₁, p₂, …, p_n are given by the following formula [ 3 , 4 ]:

(11.6)

Then, substituting Eqs. ( 11.4 ) and (11.5) in Eq. (11.3) gives

(11.7)

From Eq. (11.6), one may find the Adomian polynomials as below [ 4 , 5 ]

(11.8)

In order to get the solution , the process is longer but in practice all the terms of the series may not be required and the solution may be approximated by the truncated series (by using the convergence of the series).

Next, we present linear and nonlinear IVPs for clear understanding of the ADM for ODEs.

11.3 Solving System of ODEs by ADM

A system of first‐order ordinary differential equations can be considered as [7,8],

(11.21)

We can present the system of equations 11.21) as

(11.22)

where L_x is the linear operator with inverse linear operator, . Applying the inverse operator on Eq. (11.22), we get the following canonical form which is suitable for applying the ADM.

(11.23)

where u_i(0) are the initial conditions of Eq. ( 11.22 ).

As mentioned earlier, the solution of Eq. ( 11.22 ) using the ADM is considered as the sum of a series

(11.24)

and the integrand in Eq. (11.23) reduces to

(11.25)

where A_{i, j}(f_{i, 0}, f_{i, 1}, …, f_{i, n}) are the Adomian polynomials. Substituting Eqs. (11.24) and (11.25) in Eq. ( 11.23 ), we get

(11.26)

From Eq. (11.26), we have the following recursive relations as

11.27

By solving Eq. ( 11.26 ), we obtain f_{i, j} and .

11.4 ADM for Solving Partial Differential Equations

Let us consider a general nonlinear partial differential equations (PDEs) in the following form [9]:

(11.29)

The linear term of nonlinear PDE is represented as Lu + Ru, while the nonlinear term is represented by Nu. Here, L is the invertible linear operator and R is the remaining linear part. By applying inverse operator , Eq. (11.29) can be written as

(11.30)

where φ(x) is the constant of integration which satisfies the given initial conditions.

Then, the solution may be represented by the decomposition method given in Eq. (11.30) by the following infinite series:

(11.31)

The nonlinear operator Nu is then decomposed as

(11.32)

where A_n^'s are Adomian polynomials defined as

(11.33)

By substituting Eqs. (11.31) and (11.32) in Eq. ( 11.30 ), we have

(11.34)

Consequently, Eq. (11.34) can be written in terms of recursive relations as

(11.35)

Hence, all the terms of “u” may be calculated using the above recursive relations and the general solution obtained according to the ADM .

11.5 ADM for System of PDEs

ADM transforms system of PDEs into a set of recursive relation that can easily be handled [ 9 ,10]. To understand the method, we now consider the following system of linear PDEs:

(11.38)

subject to the initial conditions, p(x, 0) = g₁(x); q(x, 0) = g₂(x).

The system of linear PDEs, given in Eq. (11.38), can be further written as

(11.39)

where L_t and L_x are first‐order partial operators, and f₁ and f₂ are homogeneous terms. Now, by applying the inverse operator to Eq. (11.39) and then using the initial conditions we get

(11.40)

(11.41)

The ADM suggests that the linear terms p(x, t) and q(x, t) are decomposed by infinite series,

(11.42)

and

(11.43)

By substituting Eqs. (11.42) and (11.43) in Eqs. (11.40) and (11.41), respectively, we obtain

(11.44)

and

(11.45)

The system is then transformed into a set of following recursive relations:

and

Exercise

1. 1 Solve the following IVPs by using the ADM
   
   
2. 2 Solve the linear system of PDEs
   

   with the initial data

   
3. 3 Solve the problem
   

   with the initial condition

   

References

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2. 2 Adomian, G. (1990). A review of the decomposition method and some recent results for nonlinear equations. Mathematical and Computer Modelling 13 (7): 17–43.
3. 3 Momani, S. and Odibat, Z. (2006). Analytical solution of a time‐fractional Navier–Stokes equation by Adomian decomposition method. Applied Mathematics and Computation 177 (2): 488–494.
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8. 8 Jafari, H. and Daftardar‐Gejji, V. (2006). Revised Adomian decomposition method for solving a system of nonlinear equations. Applied Mathematics and Computation 175 (1): 1–7.
9. 9 Bildik, N. and Konuralp, A. (2006). The use of variational iteration method, differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations. International Journal of Nonlinear Sciences and Numerical Simulation 7 (1): 65–70.
10. 10 Wazwaz, A.M. (1999). A reliable modification of Adomian decomposition method. Applied Mathematics and Computation 102 (1): 77–86.