Preface
This section illustrates applications of the Adomian decomposition method (ADM for short) for numerical approximations of solutions to certain classes of nonlinear ordinary differential equations of first order. In layman's terms, as the name suggests, the decomposition method is an appropriate decomposition of the equation to be solved, and then it uses a special technique to further analyze and deal with the terms separately, and finally to give the approximate analytical solution of the arbitrary precision required. Therefore, the basic spirit of the decomposition method mainly includes three levels of meaning; the first is to decompose the true solution of an equation into the sum of several solution components, try to find the solution components of each order, and then let the sum of these solution components be a high accuracy approximation of the true solution. The second is to properly decompose the entire equation into several parts, mainly according to the operator into linear, nonlinear, and input parts; in principle, it can be arbitrarily decomposed, but you need to pay attention to assumptions, such as the deterministic linear operator selected being invertible (this will involve the difficulty of finding the Green's function), so it is easy to obtain the partial solution of the corresponding equation of the linear operator, and then use the unknown initial value or boundary value conditions to find the correlation between the rest of the solution and the partial solution from the set method. The most important thing is that the middle and high order solution components depend only on the loworder solution components, so that any higherorder solution components can be derived from the loworder according to a certain set of rules. The third is to propose a clever method for dealing with the most important nonlinear term (function) in the nonlinear equation. This term is replaced with a special polynomial that is called the Adomian polynomial described below. This polynomial is only determined by the preceding lowerorder solution components and the nonlinear function.
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Glossary
Adomian Decomposition Method
The Adomian decomposition method (ADM) is a wellknown systematic method for practical solution of nonlinear functional equations, including ordinary differential equations. The method was developed by the American mathematician and aerospace engineer of Armenian descent George Adomian (19221996), chair of the Center for Applied Mathematics at the University of Georgia. Since the introduction to the ADM was done previously, we emphasize the main steps in its applications to the initial value problems for the first order differential equations:
Let us review the main idea in the decomposition method. Before going into detail, it is convenient to reformulate the initial value problem in operator form:
Implementation of the Adomian decomposition method includes several steps. The first and most crucial step is the assumption that the true solution to the initial value problem (or boundary value problem) can be represented as the infinite convergent series
The next step consists of substituting the above infinite series into the given equation
Next, we substitute these two generating functions into the given differential equation assuming that the slope function is the sum: f(x,y) = N(x,y) + g(x),
It should be noted that when the Adomian decomposition method is applied to nonhomogeneous equations, it may miss some terms in the explicit formula or impose terms with wrong coefficients that are called the noise terms. Therefore, the ADM supplies only an approximation to the true solution, similar to Picard's scheme. These unwanted terms usually do not affect the approximation because they are rapidly damped out numerically. The noise terms appear only for inhomogeneous equations whereas noise terms do not appear for homogeneous equations. Several authors have proposed a variety of modifications to ADM to remove the noise terms.
One way to reduce the effect of noise terms or increase the region of convergence is to implement the nonstandard decomposition method. Unlike the standard method, the nonstandard approach distributes the nonhomogeneous term and initial conditions between other components of the Adomian series. Recall that in the standard Adomian method, only the initial component includes the nonhomogeneous term and initial condition. We redistribute the burden of the nonhomogeneous term between other components. This approach is based on representation of the forcing term g(x) or/and the initial value y_{0} as the sums:
Convergence of the Adomian decomposition method for initial value problems was discussed in many articles (see references in publications by Abdelrazec & Pelinovsky and Ray). In spite of the variety of publications on convergence, computational complexity, improvements, and applications of the ADM, no precise criterion of convergence has formulated in the literature. When the ADM is a contraction operation, we expect convergence of the Adomian series. The main progress was achieved by utilizing the CauchyKowalevskaya theorem, which has no practical applications. The following two hypotheses are necessary for proving convergence of Adomian's technique.
 The given problem has a series solution \( \displaystyle y(x) = \sum_{n\ge 0} u_n (x) \) such that \( \displaystyle \sum_{n\ge 0} \left( 1+ \varepsilon \right)^n \left\vert u_n (x) \right\vert < \infty ,\) where ε > 0 but may be very small.
 The nonlinear term f(x,y) in the equation can be developed in series according to the sequence {u_{n}}.
There are two main differences of the ADM compared to the Picard iteration. The first advantage of the ADM is that the initial term is a function taking into account external and initial or boundary inputs rather than the initial condition considered by Picard's procedure. This gives the solution method more flexibility than direct Taylor series expansion or Picard's iteration. The second advantage of the ADM is the decomposition of the problem under consideration into a sequence of similar (very simple) initial value auxiliary initial value problems. So the amount of labor required by the ADM remains almost the same at each iteration step subject that the Adomian polynomials are available. Therefore, the ADM represents the solutions as the (infinite) sum of components rather than the limit of the sequence avoiding repetition that slows down Picard's scheme because the latter becomes more and more cumbersome at each iteration.
Golberg (1999) showed that two methods, the Adomian and Picard, only have their meeting point in the linear differential equations case. However, this equivalence does not hold for nonlinear equations. Since the ADM deals with holomorphic functions (having all derivatives), the rate of convergence is expected to be better than Picard's one (such conclusion is based on numerous examples). Unlike Picard's method, the region of convergence for the ADM is unknown, which is understandable because utilizing the fixed point theorems is too restrictive for most physical and engineering applications to be verified in practice.
These two approaches, the ADM and Picard's iteration, suffer a similar drawback: it is difficult to implement them practically, in general. Picard's iteration often leads to the integration problem that can be overcome only for polynomial inputs. Although the Adomian decomposition method allows to treat much more complicated nonlinearities, it transfers all difficulties into evaluation of the Adomian polynomials. Since the latter requires differentiation of the composition of the slope function with infinite series, the labor involved in Adomian's polynomial evaluations grows exponentially. As a rule, the ADM has a fast convergence, but it is very hard to establish its domain of convergence.
In general, evaluation of Adomian's polynomials remains the main obstacle in computations because the number of terms in their representations grows exponentially. There are several known improvements in its evaluations and modifications, but the issue remains. Since, as a rule, the labor involved in evaluation of sequential Adomian polynomials grows as a snowball, the ADM is usually considered as a truncated series solution that only gives a good approximation to the actual solution in a small region. To overcome this drawback, two aftertreatment techniques have been proposed. One approach is known as multistage ADM, which divides the solution space into a finite number of small regions, similar to a onestep numerical procedure. Another aftertreatment technique is based on Padé approximants, Laplace transform and its inverse to deal with the truncated series solution obtained by the ADM.
To solve numerically the initial value problem
Instead of utilizing a general backdrop, we show how the Adomian's decomposition method works in a series of examples. We try to convince the reader of the credibility of the method by using connections with familiar techniques and plotting Adomian's approximations along with true solutions. There are several known algorithms to determine these polynomials. It turns out that the number of terms to calculate A_{n}, according to Uspensky's formula (1920), grows exponentially. Therefore, it is not practical to calculate many terms in the corresponding truncated series \( \sum_{n= 1}^M u_n (x) \) unless the slope function is quite simple. However, for numerical calculations we just need some small number of such terms to make one step, similar to the RungeKutta algorithms. Fortunately, the step size in the ADM is usually larger than in many other numerical methods.
Fortunately, the Adomian polynomials A_{n} can be determined recursively according to Rach's rule (discovered by Randolph Rach in his 1984 paper: A convenient computational form for the Adomian polynomials, Journal of Mathematical Analysis and Applications, Vol. 102, No. 2, September, 1984, pp. 415419):
Z[[1,1]]=Subscript[u,1];U=Table[Subscript[u,1]^l/l!,{l,0,n}];
For[m=2,m<=n,m++,Z[[m,1]]=Subscript[u,m];
For[k=2,k<=Floor[m/2],k++,
Z[[m,k]]=Expand[Take[U,k].(Table[Z[[mk,kl]],{l,0,k1}]/. Table[Subscript[u,i]>Subscript[u,i+1],{i,1,n}])]];
For[k=Floor[m/2]+1,k<=m,k++,
Z[[m,k]]=Integrate[Z[[m1,k1]],Subscript[u,1]]]]; Z];
dir=Table[D[f[Subscript[u,0]],{Subscript[u,0],k}],{k,1,n}];
For[m=1,m<=n,m++,Subscript[A,m]=Take[dir,m].Z[[m]]]];
Accelerated Adomian Polynomials
There are four known kinds of Adomian polynomials and it is understood that certain circumstances could benefit from a different choice; however for most problems the classical Adomian polynomials are satisfactory. A special class of them constitute the accelerated Adomian polynomials that were first introduced by George Adomian in his 1989 book, and were detailed and intensively used in 2008 Randolph Rach article. There is no standard notation for the accelerated Adomian polynomials, and we denote them by 𝑎A_{n}. For example, the accelerated Adomian polynomials would be good for exponential nonlinearities. They are defined as follow:
Classical Adomian’s polynomials  Accelerated Adomian’s polynomials 

A_{0} = u_{0}²  𝑎A_{0} = u_{0}² 
A_{1} = 2u_{0}u_{1}  𝑎A_{1} = u_{1}² + 2u_{0}u_{1} 
A_{2} = u_{1}² + 2u_{0}u_{2}  𝑎A_{2} = u_{2}² + 2u_{0}u_{2} + 2u_{1}u_{2} 
A_{3} = 2u_{1}u_{2} + 2u_{0}u_{3}  𝑎A_{3} = u_{3}² + 2u_{0}u_{3} + 2u_{1}u_{3} + 2u_{2}u_{3} 
A_{4} = u_{2}² + 2u_{1}u_{3} + 2u_{0}u_{4}  𝑎A_{4} = u_{4}² + 2u_{0}u_{4} + 2u_{1}u_{4} + 2u_{2}u_{4} + 2u_{3}u_{4} 
A_{5} = 2u_{1}u_{4} + 2u_{2}u_{3} + 2u_{0}u_{5}  𝑎A_{5} = u_{5}² + 2u_{0}u_{5} + 2u_{1}u_{5} + 2u_{2}u_{5} + 2u_{3}u_{5} + 2u_{4}u_{5} 
In general, the classical Adomian’s polynomials for square function are
Mathematica code for evaluating onevariable classical Adomian polynomials (JunSheng Duan and Randolph Rach, The degenerate form of the Adomian polynomials in the power series method for nonlinear ordinary differential equations, Journal of Mathematics and System Science, 2015, 5, 411428):
Table[c[n,k], {n,1,M}, {k,1,n}];
der = Table[D[f[Subscript[u, 0]], {Subscript[u, 0],k}],{k,1,M}];
A[0] = f[Subscript[u, 0]];
For[n=1, n<=M, n++, c[n,1] = Subscript[u, n];
For[k = 2, k<=n, k++, c[n,k] = Expand[1/n*Sum[(j+1)*Subscript[u, j+1]* c[n1j,k1], {j,0,nk}]]];
A[n] = Take[der,n].Table[c[n,k], {k,1,n}]];
Table[A[n], {n,0,M}]]
JJ[0] = f[Subscript[u, 0]];
For[n = 1, n <= M, n++, JJ[n] = Simplify[ f[Sum[Subscript[u, k], {k, 0, n}]]  f[Sum[Subscript[u, k], {k, 0, n  1}]]]];
Table[JJ[n], {n, 0, M}]]
Th [ m_ ] := Expand [ Sum [ D [ f [ u 0 ], { u 0 , k }]/ k ! ∗ Sum [ u i , { i , 1 , Which [ class == 2 , m − 1 , class == 3 , m − k ]}] ∧ k , { k , 0 , m − 1 }]];
A 0 = Th [ 1 ];
Do [ A i = Th [ i + 1 ] − Th [ i ], { i , 1 , n }];
Table [ A i , { i , 0 , n }]]
Example: Consider the initial value problem for the following autonomous equation
In general, since the Adomian decomposition method requires the slope function to be holomorphic in both variables, which is more restrictive than the Lipschitz condition that is needed fro Picard's iteration, we expect that the ADM converges faster than the Picard method. We demonstrate this method in many examples, and we start with a linear differential equations.
Example: We start with a simple initial value problem for linear equation:
Example: Our next example is also related to the linear differential equation, but now we consider variable coefficient nonhomogeneous equation:
Table[y[n, x], {n, 0, 6}]
Do[y[n, x_] =
f[x] /. First @ DSolve[{f'[x] == 2*x*y[n  1, x], f[1] == 0}, f, x], {n, 1, 5}]
phi[x_] := Simplify[Sum[y[n, x], {n, 0, 5}]]
Plot[{z[x], phi[x]}, {x, 1, 2}, PlotStyle > {{Thick, Blue}, {Thick, Orange}}, PlotLabels>{"exact","ADM approximation"}]
Now we turn our attention to initial value problems for nonlinear differential equations such as
Option 1: \( \displaystyle L = \texttt{D} = {\text d}/ {\text d}x \) and \( \displaystyle f(x,y) = g(x)  a(x)\,y(x)  b(x)\,N[y] . \)
The main idea of the Adomian decomposition method is to represent the nonlinear term as the sum of Adomian polynomials:
Option 2: \( \displaystyle L = \texttt{D} + a(x) \) and \( \displaystyle f(x,y) = g(x)  b(x)\,N[y] . \)
In this case, one have to solve a linear differential equation \( L\left[ y \right] = \varphi (x) \) at each step for some known function φ:Example: We apply the Adomian decomposition method to find an approximate solution to the initial value problem for logistic equation:
A[0, t] := 100
u[1, t_] = f[t] /. First@ DSolve[{f'[t] == (1/50)*u[0,t]  (1/50)*100, f[0] == 0}, f, t]
A[1, t] = 2*u[0,t]*u[1,t]
u[2, t_] = f[t] /. First@ DSolve[{f'[t] == (1/50)*u[1, t]  (1/50)*A[1, t], f[0] == 0}, f, t]
A[2,t_]=20*u[2,t] + (u[1,t])^2
u[3, t_] = f[t] /. First@ DSolve[{f'[t] == (1/50)*u[2, t]  (1/50)*A[2, t], f[0] == 0}, f, t]
A[3,t_]= 20*u[3,t] + 2*u[1,t]*u[2,t]
u[4, t_] = f[t] /. First@ DSolve[{f'[t] == (1/50)*u[3, t]  (1/50)*A[3, t], f[0] == 0}, f, t]
A[4,t_]= 20*u[4,t] + (u[2,t])^2 + 2*u[1,t]*u[3,t]
u[5, t_] = f[t] /. First@ DSolve[{f'[t] == (1/50)*u[4, t]  (1/50)*A[4, t], f[0] == 0}, f, t]
phi[t_] := 10  (9 t)/5 + (171 t^2)/500  (1623 t^3)/25000 + (61617 t^4)/ 5000000  (2924103 t^5)/1250000000
Plot[{z[t], phi[t]}, {t, 0, 3.0}, PlotStyle > {{Thick, Blue}, {Thick, Orange}}]
A[5, t_] = 4236099*t^5 /1250000000;
u[6, t_] = f[t] /. First@ DSolve[{f'[t] == (1/50)*u[5, t]  (1/50)*A[5, t], f[0] == 0}, f, t]
phi6[t] = phi[t]  u[6, t]
Plot[{z[t], phi[t], phi6[t]}, {t, 0, 8}, PlotStyle > {{Thick, Blue}, {Thick, Orange}, {Thick, Magenta}}]
Let us compare the obtained Adomian approximation with the true solution:
Finally, we apply the Adomian's decomposition method with accelerated polynomials to this initial value problem. We proceed exactly in the same manner as the ADM requires, but use accelerated polynomials aA_{n} instead of the classical Adomian polynomials A_{n}. We start with
Example: Consider the initial value problem for the logistic equation
Example: Consider the initial value problem for the integrodifferential equation
A[0, t_] = 1/100*50^2  1/10*50*Integrate[50, {x, 0, t}]
u[1, t_] = f[t] /. First@ DSolve[{f'[t] == 23*5 + A[0, t], f[0] == 0}, f, t]
Out[21]= 5 (18 t + 25 t^2)
u[2, t_] = f[t] /. First@ DSolve[{f'[t] == 23*u[1,t]/10 + A[1, t], f[0] == 0}, f, t]
Out[23]= 1/6 (351 t^2  1675 t^3 + 1250 t^4)
u[3, t_] = f[t] /. First@ DSolve[{f'[t] == 23*u[2,t]/10 + A[2, t], f[0] == 0}, f, t]
Out[25]= 1/720 (1188 t^3  167925 t^4 + 402750 t^5  212500 t^6)
u[4, t_] = f[t] /. First@ DSolve[{f'[t] == 23*u[3, t]/10 + A[3, t], f[0] == 0}, f, t]
Out[27]= (1353807 t^4 + 100065 t^5 + 134567125 t^6  258056250 t^7 + 115000000 t^8)/50400
u[5, t_] = f[t] /. First@ DSolve[{f'[t] == 23*u[4, t]/10 + A[4, t], f[0] == 0}, f, t]
Out[29]= (1/45360000)(600313518 t^5 + 6071613975 t^6 + 21100995000 t^7  149290171875 t^8 + 193617578125 t^9  69406250000 t^10)
Do[phi5[t] = phi5[t] + u[j, t], {j, 1, 5}]
A[5, t_] = Simplify[ u[5, t]*(1 + 50*t) + u[2, t]*u[3, t]/50 + u[4, t]*u[1, t]/50 + u[4, t]*Integrate[u[1, x], {x, 0, t}]/10 + u[3, t]*Integrate[u[2, x], {x, 0, t}]/10 + u[2, t]*Integrate[u[3, x], {x, 0, t}]/10 + u[1, t]*Integrate[u[4, x], {x, 0, t}]/10 + 50*Integrate[u[5, x], {x, 0, t}]]
u[6, t_] = f[t] /. First@ DSolve[{f'[t] == 23*u[5, t]/10 + A[5, t], f[0] == 0}, f, t]
phi6[t_] = phi5[t]+u[6,t]
Plot[Tooltip@{phi5[t], phi6[t]}, {t, 0, 0.8}, PlotStyle > {Red, Green}, PlotLegends > {"ADM(5)", "ADM(6)"}]
 Using ADM, solve the initial value problem: \( y' + \sqrt{y} = 2x , \quad y(0) =1 . \)
 Using ADM, solve the initial value problem: \( y' + \sqrt{y} = 1+x+x^2 , \quad y(0) =2 . \)
 Using ADM, solve the initial value problem: \( y' + \sqrt{1+y} = 0 , \quad y(0) =3 . \)
 Using ADM, solve the initial value problem: \( y' + 2x\, y = 1+ x^2 + y^2 , \quad y(0) =1 . \)
 Using ADM, solve the initial value problem: \( y' + 2x\, y = e^x + 2y\left( \ln y \right)^2 , \quad y(0) =1 . \)

Using ADM, solve the initial value problem:
\( y' = y^p , \quad y(0) =1 ; \) with p ≥ 1.
Its true solution is
\[ y^{1} (t) = \left( 1  (p1)t \right)^{1/(p1)} \] 
Using ADM, solve the initial value problem:
\( y' = 2y  y^2 , \quad y(0) =1 . \)
Its true solution is y(t) = 1 + tanh(t)

Using ADM, solve the initial value problem:
\( y' = x+ e^{y} , \quad y(0) =0 . \)
Its true solution is \( y(x) = \frac{x^2}{2} + \ln \left( 1 +\sqrt{\frac{\pi}{2}}\,\mbox{erf} \left( \frac{x}{\sqrt{x}} \right) \right) \) for x > 1.2755, where erf(x) = \( \frac{2}{\sqrt{\pi}} \int_0^x e^{t^2}\,{\text d}t . \)
Complete List of Publications using Adomian Decomposition Method
 Duan, J.S., Convenient analytic recurrence algorithms for the Adomian polynomials, Applied Mathematics and Computation, 2011, Vol. 217, No. 13, pp. 63376348.
 Duan, J.S., New recurrence algorithms for the nonclassic Adomian polynomials, Computers and Mathematics with Applications, 2011, Vol. 62 No. 8, pp. 29612977.
 Duan, J.S. and Guo, A.P., Reduced polynomials and their generation in Adomian decomposition methods, CMES: Computer Modeling in Engineering and Sciences, 2010, Vol. 60, No. 2, pp. 139150.
 Duan, J.S. and Guo, A.P., Symbolic implementation of a new, fast algorithm for the multivariable Adomian polynomials, Proceedings of the 2011 World Congress on Engineering and Technology (CET 2011), Shanghai, China, October 28  November 2, 2011, Vol. 1, IEEE Press, Beijing, pp. 7274.
 Duan, J.S. and Rach, R., New higherorder numerical onestep methods based on the Adomian and the modified decomposition methods, Applied Mathematics and Computation, 2011, Vol. 218 No. 6, pp. 28102828.
 Duan, J.S. and Rach, R., The degenerate form of the Adomian polynomials in the power series method for nonlinear ordinary differential equations, Journal of Mathematics and System Science, Volume 5, Pages 411428, 2015.

Duan, J.S., Rach, R., Baleanu, D., and Wazwaz, A.M.,
A review of the Adomian decomposition method and its applications to
fractional differential equations
Communications in Fractional Calculus, Volume 3, Issue 2, 2012, Pages 7399.  Duan, J.S. and Rach, R., Higherorder numeric WazwazElSayed modified Adomian decomposition algorithms, Computers and Mathematics with Applications, 2012, Vol. 63 No. 11, pp. 15571568.
 Duan, J.S., Rach, R. and Wazwaz, A.M., Solution of the model of beamtype micro and nanoscale electrostatic actuators by a new modified Adomian decomposition method for nonlinear boundary value problems, International Journal of NonLinear Mechanics, 2013, Vol. 49, 159169.
 J.S. Duan, R. Rach, and A.M. Wazwaz, A new modified Adomian decomposition method for higherorder nonlinear dynamical systems, CMES: Computer Modeling in Engineering & Sciences, 2013, Vol. 94, No. 1, pp. 77118.
 J.S. Duan, R. Rach, A.M. Wazwaz, Temuer Chaolu, and Z. Wang, A new modified Adomian decomposition method and its multistage form for solving nonlinear boundary value problems with Robin boundary conditions, Applied Mathematical Modelling, 2013, Vol. 37, Nos. 20/21, Pages 86878708.
 A.M. Wazwaz, R. Rach, and J.S. Duan, Adomian decomposition method for solving the Volterra integral form of the LaneEmden equations with initial values and boundary conditions, Applied Mathematics and Computation, 2013, Volume 219, Issue 10, Pages 50045019; doi: 10.1016/j.amc.2012.11.012
 R. Rach, A.M. Wazwaz, and J.S. Duan, A reliable modification of the Adomian decomposition method for higherorder nonlinear differential equations, Kybernetes, 2013, Volume 42, Issues 1/2, Pages 282308
 JunSheng Duan, Zhong Wang, ShouZhong Fu, and Temuer Chaolu, Parameterized temperature distribution and efficiency of convective straight fins with temperaturedependent thermal conductivity by a new modified decomposition method, International Journal of Heat and Mass Transfer, Vol. 59 (2013) 137143.
 A.M. Wazwaz, R. Rach, L. Bougoffa, and J.S. Duan, Solving the LaneEmdenFowler type equations of higher orders by the Adomian decomposition method, CMES: Computer Modeling in Engineering & Sciences, Vol. 100, No. 6, pp. 507529, 2014, doi:10.3970/cmes.2014.100.507.
 J.S. Duan, R. Rach, and A.M. Wazwaz, A reliable algorithm for positive solutions of nonlinear boundary value problems by the multistage Adomian decomposition method, Open Engineering, Volume 5, Issue 1, 5974, 2014. doi: 10.1515/eng20150007
 R. Rach, A.M. Wazwaz, and J.S. Duan, A reliable analysis of oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics, International Journal of Biomathematics, Vol. 07, No. 02, March 2014, 1450020 (2014) [12 pages] doi: 10.1142/S179352451450020X.
 Rach, R. Duan,J.S., and Wazwaz, A.M., Solving coupled LaneEmden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method Journal of Mathematical Chemistry, Volume 52, Issue 1, 2014, Pages 255267.
 Wazwaz, A.M., Rach, R., and Duan, J.S., A study on the systems of the Volterra integral forms of the LaneEmden equations by the Adomian decomposition method, Mathematical Methods in the Applied Sciences, Vol. 37, Issue 1, 2014, Pages 1019.
 R. Rach, J.S. Duan, and A.M. Wazwaz, On the solution of nonisothermal reactiondiffusion model equations in a spherical catalyst by the modified Adomian method, Chemical Engineering Communications, Volume 202, Issue 8, 2015, Pages 10811088.
 J.S. Duan, R. Rach, and A.M. Wazwaz, Oxygen and carbon substrate concentrations in microbial floc particles by the Adomian decomposition method, MATCH Communications in Mathematical and in Computer Chemistry, Vol. 73, No. 3, 2015, pp. 785796.
 J.S. Duan, R. Rach, and A.M. Wazwaz, Steadystate concentrations of carbon dioxide absorbed into phenyl glycidyl ether solutions by the Adomian decomposition method, Journal of Mathematical Chemistry, Vol. 53, 2015, Pages 10541067.
 R. Rach, A.M. Wazwaz, and J.S. Duan, The Volterra integral form of the LaneEmden equation: New derivations and solution by the Adomian decomposition method, Journal of Applied Mathematics and Computing, Volume 47, Issue 1, 2015, Pages 365379.
 J.S. Duan, R. Rach, and A.M. Wazwaz, Higherorder numeric solutions of the LaneEmden type equations derived from the multistage modified decomposition method, International Journal of Computer Mathematics, International Journal of Computer Mathematics, Volume 94, No. 1, 2017, 197215.
 Abassy, T.A., Improved Adomian decomposition method, Computers and Mathematics with Applications, 2010, Vol.59, pp. 4254.
 DaftardarGejji, V. and Jafari, H., An iterative method for solving nonlinear functional equations, Journal of Mathematical Analysis and Applications, 2006, Vol. 316, No. 2, pp. 753763. https://doi.org/10.1016/j.jmaa.2005.05.009
 Rach, R., On the Adomian (decomposition) method and comparisons with Picard's method, Journal of Mathematical Analysis and Applications, 1987, Vol. 128, Issue 2, pp. 480483.
 Rach, R., A convenient computational form for the Adomian polynomials, Journal of Mathematical Analysis and Applications, 1984, Volume 102, Issue 2, pages 415419. https://doi.org/10.1016/0022247X(84)901811
 Rach, R., “A new definition of the Adomian polynomials”, Kybernetes, Volume 37, Issue 7, 2008, Pages 910955.
 Sari, M.R., Kezzar, M., Adjabi, R., A comparison of Adomian and generalized Adomian methods in solving the nonlinear problem of flow in convergentdivergent channels, Applied Mathematical Sciences, 2014, Vol. 8, no. 7, 321  336; http://dx.doi.org/10.12988/ams.2014.39495
 Abdelrazec, Ahmed H.M., Adomian decomposition method: convergence analysis and numerical approximations, Thesis, McMaster University, Ontario, Canada, 2008.
 Abdelrazec, A., and Pelinovsky, D., Convergence of the Adomian decomposition method for initialvalue problems, Numerical Methods for Partial Differential Equations, 2011, Vol. 27, No. 4, 749766, 2011. https://doi.org/10.1002/num.20549
 Abushammala, Mariam B.H., Iterative methods for the numerical solutions of boundary value problems, Thesis of the American University of Sharjah College of Arts and Sciences, Sharjah United Arab Emirates, June 2014.
 Golberg, M.A., "A note on the decomposition method for operator equation," Applied Mathematics and Computation, 1999, 106, 215220.
 Himoun, N., Abbaoui, K. and Cherruault, Y. (1999), “New results of convergence of Adomian’s method”, Kybernetes, Vol. 28 No. 4, pp. 4239.
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