A Brief Introduction to the Adomian Decomposition Method, with Applications in Astronomy and Astrophysics
aa r X i v : . [ a s t r o - ph . I M ] F e b A BRIEF INTRODUCTION TO THE ADOMIAN DECOMPOSITIONMETHOD, WITH APPLICATIONS IN ASTRONOMY ANDASTROPHYSICS
MAN KWONG MAK , CHUN SING LEUNG , TIBERIU HARKO Departamento de F´ısica, Facultad de Ciencias Naturales, Universidad de Atacama, Copayapu 485,Copiap´o, Chile, Email: [email protected] Department of Mathematics, Polytechnic University of Hong Kong, Hong Kong, Email:[email protected] School of Physics, Sun Yat-Sen University, Xingang Road, Guangzhou 510275, People’s Republic ofChina, Email: [email protected].
The Adomian Decomposition Method (ADM) is a very effective approachfor solving broad classes of nonlinear partial and ordinary differential equations, withimportant applications in different fields of applied mathematics, engineering, physicsand biology. It is the goal of the present paper to provide a clear and pedagogical in-troduction to the Adomian Decomposition Method and to some of its applications. Inparticular, we focus our attention to a number of standard first-order ordinary differ-ential equations (the linear, Bernoulli, Riccati, and Abel) with arbitrary coefficients,and present in detail the Adomian method for obtaining their solutions. In each casewe compare the Adomian solution with the exact solution of some particular differ-ential equations, and we show their complete equivalence. The second order and thefifth order ordinary differential equations are also considered. An important exten-sion of the standard ADM, the Laplace-Adomian Decomposition Method is also intro-duced through the investigation of the solutions of a specific second order nonlineardifferential equation. We also present the applications of the method to the Fisher-Kolmogorov second order partial nonlinear differential equation, which plays an im-portant role in the description of many physical processes, as well as three importantapplications in astronomy and astrophysics, related to the determination of the solu-tions of the Kepler equation, of the Lane-Emden equation, and of the general relativisticequation describing the motion of massive particles in the spherically symmetric andstatic Schwarzschild geometry.
Key words : Mathematical Methods in Physics – Ordinary Nonlinear Differential Equa-tions – Celestial Mechanics – Astronomy–General Relativity.
CONTENTS
Romanian Astron. J. , Vol. , No. 1, p. 1–41, Bucharest, 2019 The Adomian Decomposition Method with Applications 1 tial equations 7 dydx + P ( x ) y = Q ( x ) . . . . . . . . . . . . . . 83.1.1 Example: dydx + 2 xy = 4 x . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Bernoulli differential equation dydx + P ( x ) y = Q ( x ) y n . . . . . . . . . . 93.2.1 Example: dydx − xy = − x y . . . . . . . . . . . . . . . . . . . . . 113.3 Riccati differential equation dydx = P ( x ) + Q ( x ) y . . . . . . . . . . . . . 113.3.1 Example: dydx = 2 e x − e − x y . . . . . . . . . . . . . . . . . . . . . . 123.4 Abel differential equation dydx = M ( x ) + S ( x ) y + R ( x ) y + T ( x ) y dydx = x + 3 xy + 3 xy + xy . . . . . . . . . . . . . . . . 14 d ydx + 4 dydx + 3 y = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2 Example: d ydx + dydx + e x y + e x y = e x . . . . . . . . . . . . . . . . . . . . . 18 d ydx − d ydx − d ydx + 15 d ydx + 4 dydx − y = 0 . . . . . . . . . . 21 c (0 , x, y, z ) = x + y + z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 f ( y ) = P ml =0 a l +2 y l +2 . . . . . . . . . . . . . . . . . . . . . . . . . 27 i Man Kwong MAK, Chun Sing LEUNG, Tiberiu HARKO 3
1. INTRODUCTION
In recent years, a lot of consideration has been dedicated to the investiga-tions of the Adomian’s Decomposition Method (ADM) (Adomian and Rach, 1983;Adomian, 1988, 1994; Cherrualt et al. , 1995; Adomian and Rach, 1996; Duan et al. ,2012), which allows us to explore the solutions and properties of a large variety ofordinary and partial differential equations, as well as of integral equations, whichdescribe various mathematical problems, or can be used to mathematically modeldiverse physical processes. From a historical point of view, the ADM was first in-troduced, and extensively used in the 1980’s (Adomian and Rach, 1983; Adomian,1984a,b, 1985, 1986), and ever since many mathematicians and scientists have con-tinuously modified the ADM in an attempt to enhance its accuracy and/or to broadenthe applications of the initial method (Cherrualt et al. , 1995; Adomian and Rach,1996; Wazwaz, 1999a,b, 2005; Luo, 2005; Zhang et al. , 2006; Babolian and Javadi,2003; Babolian et al. , 2004; Jin and Liu, 2005; Jafari and Daftardar-Gejji, 2006a,b;Rach et al. , 1992; Wazwaz and EI-Sayed, 2001; Biazar et al. , 2004, 2003a,b; Sadat,2010; Bakodah, 2012).An important benefit of the Adomian Decomposition Method is that it can yieldanalytical approximations to quite extensive classes of nonlinear (and stochastic) dif-ferential equations without resorting to discretization, perturbation, linearization, orclosure approximations methods, which could result in the necessity of extensivenumerical computations. For most of the mathematical models used for the mathe-matical description of natural phenomena, in order to obtain the analytical solutionsof a nonlinear problem in a closed-form, and thus to make it solvable, it is usuallynecessary to make some simplifying assumptions, or t impose some restrictive con-ditions.It is worth to note that ADM can provide a solution of a differential/integralequation in the form of a series, whose terms are determined individually step by stepvia a recursive relation using the Adomian polynomials. The main advantage of theAdomian Decomposition Method is that the series solution of the differential/integralequation converges very quickly (Abbaoui and Cherrualt, 1994a,b; Cherrualt et al. ,1995), and therefore it saves significant amounts of computing time. On the otherhand, it is important to point out again that in the Adomian Decomposition Methodthere is no need to discretize or linearize the differential and integral equations. Onecan find reviews of ADM in applied mathematics, and its applications in science inAdomian (1988), Adomian (1994), and Haldar (2016), respectively.The basic nonlinear ordinary differential equations of mathematics (Riccati andAbel), as well as their physical and engineering applications have continuously at-tracted the interest of mathematicians and physicists (Mak and Harko, 2012, 2013a;Harko et al. , 2016; Mak et al. , 2001; Mak and Harko, 2002; Harko and Mak, 2003; ii The Adomian Decomposition Method with Applications 3
Harko et al. , 2013; Harko and Liang, 2016). These equations also proved to be a fer-tile investigation ground from the point of view of the ADM approach. Recently,using the ADM, the Riccati equation was solved in Gbadamosi et al. (2012). TheAbel differential equation, having constant coefficients, of the form dydt = M X k =0 f k y k , (1)was solved with the help of ADM in Al-Dosary et al. (2008). A modified versionof the Adomian Decomposition Mthod was introduced for solving second order or-dinary differential equation in Hassan and Zhu (2008) and Hosseini and Nasabzadeh(2007), respectively. A particular third order ordinary differential equation was in-vestigated by using a modified ADM for solving it in Mak et al. (2018a). The ADMwas applied to the third order ordinary differential equation y ′′′ = y − k , (2)representing a particular case of a generalized thin film equation describing the flowof a thin film downward of a vertical wall Momoniat et al. (2007). The ADM forsolving different classes of differential equations of importance in mathematical physicswas studied in Dit˘a and Grama (1997). The fourth order differential equation wassolved by ADM in Agom et al. (2016). The biharmonic nonlinear Schr¨odinger equa-tion, and its standing wave solutions were investigated, via the use of the Laplace-Adomian and Adomian Decomposition Methods, in Mak et al. (2018a).The Adomian Decomposition Method method was extensively applied in dif-ferent areas of science and technology, including the study of the dynamics of thepopulation growth models, which can be modelled by single partial or ordinary dif-ferential equations, or complex systems of such equations. A few example of spe-cific mathematical systems successfully explored by using the ADM are the shal-low water waves (Safari, 2011), the Brusselator model (Wazwaz, 2000), the Lotka-Volterra model (Ruan and Lu, 2007), and the Belousov-Zhabotinsky reduction model(Fatoorehchi et al. , 2015), respectively. The Adomian Decomposition Method wasapplied for the study of the Susceptible-Infected-Recovered (SIR) epidemic model,which is widely applied for the study of the spread of infectious diseases, in Harko and Mak(2020a) and Harko and Mak (2020b), respectively.The Adomian Decomposition Method did also find some important applica-tions in Physics. Nonlinear matrix differential equations of a new type, which emergein general relativity as well as other scientific fields, were investigated in Azreg-A¨ınou(2010). The solution of the nonlinear Klein-Gordon equation was obtained via theAdomian Decomposition Method in Ghasemi et al. (2014). The obtained semi-analyticalsolutions are in good accord with the full numerical solutions. The equations of mo-tion of the massive and massless particles in the spherically symmetric and static iii Man Kwong MAK, Chun Sing LEUNG, Tiberiu HARKO 5 Schwarzschild geometry of general relativity were studied extensively in Mak et al.(2018a) by using the Laplace-Adomian Decomposition Method. The physical prop-erties of vortices with arbitrary topological charges arising in weakly interactingBose-Einstein Condensates, described by differential equations of the form d R ( x ) dx + 1 x dR ( x ) dx − (cid:20) l x + ( v ( x ) − (cid:21) R ( x ) − R ( x ) = 0 , (3)where l is a constant, and v ( x ) = 0 and v ( x ) = x , were investigated using theAdomian Decomposition Method in Harko et al. (2020), where the nonlinear Gross-Pitaevskii equation was solved in polar coordinates. Series solutions using the Ado-mian Decomposition Method have been obtained for the Schr¨odinger-Newton- Λ sys-tem, described by the system of partial differential equations, i ~ ∂ ψ ( ~r, t ) ∂t = − ~ m ∇ ψ ( ~r, t ) + m Φ ( ~r, t ) ψ ( ~r, t ) , (4) ∇ Φ ( ~r, t ) = 4 π Gm | ψ ( ~r, t ) | − Λ c , (5)where by ψ ( ~r, t ) we have denoted the particle wave function, Φ ( ~r, t ) is the gravi-tational potential, ~ , G and Λ are the Planck, the gravitational and the cosmologicalconstants, respectively, while m is the mass of the particle, in Mak et al. (2020) andHarko et al. (2020), respectively.Despite the existence of a large literature on the ADM, to the best knowledgeof the authors no clearly written and pedagogical introduction to the method, whichwould be useful for a large audience of scientists from different fields, does existpresently. It is the purpose of the present paper to give such an introductory reviewof the Adomian Decomposition Method, and of the Laplace-Adomian Decomposi-tion Method, in which, by means of the detailed and explicit presentation of all thecalculations, and by providing a large number of examples, the power and efficiencyof the method is clearly outlined. Hopefully, such a presentation would be of interesteven for undergraduate students studying sciences and engineering, and will deter-mine them to proceed to the study and investigation of the advanced features of themethod.From the point of view of the applications of the Adomian DecompositionMethod in science we have chosen to present the analysis of the Fisher-Kolmogorovequation (Fisher, 1937; Kolmogorov et al. , 1937), which plays an essential role inmany physical and biological problems. But the main focus of the present paperare the potential astronomical and astrophysical applications of the Adomian De-composition Method, a field that has yet to be explored in detail. One importantastronomical problem that can be handled efficiently and effectively with the Ado-mian Decomposition Method is obtaining the solution of the Kepler equation, whichplays a fundamental role in the determination of the orbits of the celestial orbits. iv The Adomian Decomposition Method with Applications 5 The hyperbolic and the elliptic Kepler equations were investigated by using ADMin Ebaid et al. (2017) and Alshaery and Ebaid (2017), respectively. One of the ba-sic equations of Newtonian astrophysics is the Lane-Emden equation, which wasused, for example, for the study of the white dwarfs, which lead to the fundamentalChandrasekhar mass limit for this type of compact objects (Chandrasekhar, 1967).The Lane-Emden equation was intensively investigated by using the Adomian De-composition Method, which provides an efficient and computationally powerful pro-cedure to obtain its solutions, in (Adomian et al. ., 1995; Wazwaz and Rach, 2011;Wazwaz et al. , 2013; Hosseini and Abbasbandy, 2015; Rach et al. , 2015). Finally,we will consider the general relativistic motion of massive test particles in the staticand spherically symmetric Schwarzschild geometry, and present its Adomian seriessolution (Mak et al., 2018a). This approach can be used for the extremely preciseanalytical calculation of the orbit of the planet Mercury, for the study of its perihe-lion precession, as well as for the computation of the light deflection by the Sun.The solutions of the Kompaneets equation, a nonlinear partial differential equationthat plays an important role in astrophysics, describing the spectra of photons in in-teraction with a rarefied electron gas, were obtained, by using the Laplace-AdomianDecomposition Method, in Gonz´alez-Gaxiola et al. (2017).The present paper is organized as follows. We introduce the basics of the Ado-mian Decomposition Method in Section 2. In Section 3 we discuss the application ofthe ADM to the case of the first order differential equations. We begin our discussionwith the simplest case of the ordinary linear first order differential equation, whosesolution can be obtained exactly. The power series solution of the linear equation isobtained by using a power series expansion. We consider then a particular case, andwe compare the power series and the exact solutions. Next we proceed to the inves-tigation of the Bernoulli, Riccati and Abel type equations with constant coefficients,by using ADM, and the series solutions of these equations are obtained. In each casethe power series solution is compared with the exact solution of a particular differ-ential equation. The case of the second order differential equations is considered inSection 4. Two specific example are also presented, and discussed in detail. The fifthorder ordinary differential equation is analyzed in Section 5. As an example of the useof the ADM for solving nonlinear partial differential equations, in Section 6 we con-sider the case of the Fisher-Kolmogorov equation, a nonlinear differential equationwith many applications in biology. We present the Laplace-Adomian DecompositionMethod for second order nonlinear differential equations in Section 7. Astronomicaland astrophysical applications of the Adomian Decomposition Method (Kepler equa-tion, Lane-Emden equation, and the motion of massive particles in the Schwarzschildgeometry) are presented in Section 8. Finally, we discuss and conclude our results inSection 9. v Man Kwong MAK, Chun Sing LEUNG, Tiberiu HARKO 7
2. THE ADOMIAN DECOMPOSITION METHOD
We illustrate now the basic ideas of the Adomian Decomposition Method byconsidering the case of a nonlinear partial differential equation written in the generalform ˆ L t [ y ( x, t )] + ˆ R [ y ( x, t )] + ˆ N [ y ( x, t )] = f ( x, t ) , (6)where ˆ L t [ . ] = ∂/∂t [ . ] denotes the partial derivative operator with respect to the time t , while ˆ R [ . ] is the linear operator, generally containing partial derivatives with re-spect to x . Moreover, ˆ N [ . ] represents a nonlinear analytic operator, and f ( x, t ) is anon-homogeneous arbitrary function, assumed to be independent of y ( x, t ) . Eq. (6)has to be considered together with the initial condition y ( x,
0) = g ( x ) . In the follow-ing w assume that the operator ˆ L t is invertible, and therefore we can apply ˆ L − t toboth sides of Eq. (6), thus first obtaining y ( x, t ) = g ( x ) + ˆ L − t [ f ( x, t )] − ˆ L − t ˆ R [ y ( x, t )] − ˆ L − t ˆ N [ y ( x, t )] . (7)The ADM postulates the existence of a series solution of Eq. (6) in which y ( x, t ) can be represented by y ( x, t ) = ∞ X n =0 y n ( x, t ) . (8)Moreover, it is assumed that the nonlinear term ˆ N [ y ( x, t )] can be decomposed ac-cording to ˆ N [ y ( x, t )] = ∞ X n =0 A n ( y , y , ..., y n ) , (9)where { A n } ∞ n =0 are called the Adomian polynomials. They can be computed accord-ing to the simple rule (Adomian and Rach, 1983; Adomian, 1988, 1994; Cherrualt et al. ,1995; Adomian and Rach, 1996) A n ( y , y , ..., y n ) = 1 n ! d n d ǫ n ˆ N t, n X k =0 ǫ k y k ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 . (10)After the substitution of the series expansions (8) and (9) into Eq. (6), weobtain ∞ X n =0 y n ( x, t ) = g ( x ) + ˆ L − t [ f ( x, t )] ˆ L − t ˆ R " ∞ X n =0 y n ( x, t ) − ˆ L − t " ∞ X n =0 A n ( y , y , ..., y n ) . (11) vi The Adomian Decomposition Method with Applications 7 From the above equation we immediately obtain the following recurrence rela-tion, which gives the series solution of Eq. (6) as y ( x, t ) = g ( x ) + ˆ L − t [ f ( x, t )] , (12) y k +1 ( x, t ) = ˆ L − t ˆ R [ y k ( x, t )] − ˆ L − t [ A k ( y , y , ..., y n )] ,k = 0 , , , . . . (13)Therefore, an approximate solution of Eq. (6) is obtained as y ( x, t ) ≃ n X k =0 y k ( x, t ) , (14)and lim n →∞ n X k =0 y k ( x, t ) = y ( x, t ) . (15)For an arbitrary nonlinearity ˆ N [ y ( x, t )] , the Adomian polynomials can be ob-tained according to the rule A = ˆ N [ y ] , A = y ddy ˆ N [ y ] , (16) A = y ddy ˆ N [ y ] + y d dy ˆ N [ y ] , (17) A = y ddy ˆ N [ y ] + y y d dy ˆ N [ y ] + y d dy ˆ N [ y ] . (18)This procedure can be continued indefinitely. The greater the number of consideredterms in the Adomian Decomposition Method series expansion, the higher is thenumerical accuracy of the semi-analytical solution.In the following Sections we will present in detail the application of the Ado-mian Decomposition Method for a large class of nonlinear ordinary and partial dif-ferential equations.
3. THE ADOMIAN DECOMPOSITION METHOD FOR FIRST ORDER ORDINARYDIFFERENTIAL EQUATIONS
In the present Section we introduce the application of the ADM to the case offirst order differential equations. The linear, Bernoulli, Riccati and Abel differentialequations are considered in detail. vii Man Kwong MAK, Chun Sing LEUNG, Tiberiu HARKO 93.1. LINEAR DIFFERENTIAL EQUATION dydx + P ( x ) y = Q ( x ) The decomposition method can be used to solve the linear differential equa-tions. Consider that the differential equation takes the standard form of the first orderordinary differential equation, dydx + P ( x ) y = Q ( x ) , (19)where P ( x ) and Q ( x ) are arbitrary function of x . Eq. (19) must be solved togetherwith the initial condition y (0) = y . Assume that the solution of Eq. (19) can beobtained in power series form, y ( x ) = ∞ X n =0 y n ( x ) . (20)Now integrating Eq. (19) yields the integral equation y ( x ) = y (0) + Z x Q ( x ) dx − Z x P ( x ) ydx. (21)Substituting Eq. (20) into Eq. (21) gives the relation ∞ X n =0 y n ( x ) = y ( x ) + ∞ X n =1 y n ( x ) = y ( x ) + ∞ X n =0 y n +1 ( x )= y (0) + Z x Q ( x ) dx − Z x P ( x ) ∞ X n =0 y n ( x ) dx. (22)Next we rewrite Eq. (22) in the recursive forms y ( x ) = y (0) + Z x Q ( x ) dx, (23) y k +1 ( x ) = − Z x P ( x ) y k ( x ) dx. (24)From Eqs. (211) and (212) we can obtain the approximate semi-analyticalsolution of Eq. (19), as given by y ( x ) = ∞ X n =0 y n ( x ) . (25) dydx + 2 xy = 4 x Consider the differential equation dydx + 2 xy = 4 x , (26) viii0 The Adomian Decomposition Method with Applications 9 which we solve with the initial condition y (0) = 1 . Then its general solution is givenby y ( x ) = 3 e − x + 2 (cid:0) x − (cid:1) . (27)In the present case we have P ( x ) = 2 x and Q ( x ) = 4 x , respectively. Hence thepower series of the equation is obtained as y ( x ) = y (0) + Z x Q ( x ) dx = 1 + x , (28) y ( x ) = − Z x xy ( x ) dx = − x − x , (29) y ( x ) = − Z x xy ( x ) dx = x x , (30) y ( x ) = − Z x xy ( x ) dx = − x − x , (31) y ( x ) = − Z x xy ( x ) dx = x
24 + x . (32) y ( x ) ≈ y ( x ) + y ( x ) + y ( x ) + y ( x ) + y ( x ) = 1 − x + 3 x − x x .... (33)On the other hand by series expanding the exact solution (27) we obtain y ( x ) = 3 e − x + 2 (cid:0) x − (cid:1) = 1 − x + 3 x − x x ..., (34)Clearly, the solution (33) obtained by the Adomian Decomposition Method isidentical to the exact solution (34). dydx + P ( x ) y = Q ( x ) y n The Adomian Decomposition Method is very powerful for solving nonlinearordinary differential equations. Consider that the differential equation takes theBernoulli equation form dydx + P ( x ) y = Q ( x ) y n , (35)where P ( x ) and Q ( x ) are arbitrary function of x , and n is an arbitrary constant.Assume that the solution of Eq. (35) is given by the power series form y ( x ) = ∞ X n =0 y n ( x ) . (36) ix0 Man Kwong MAK, Chun Sing LEUNG, Tiberiu HARKO 11 The nonlinear term y n can be decomposed in terms of the Adomian polynomials A n ( x ) , given by y n ( x ) = ∞ X n =0 A n ( x ) , (37). Generally, for an arbitrary function f ( t, x ) , the Adomian polynomials are de-fined as (Adomian, 1994) A n = 1 n ! d n d ǫ n f t, ∞ X i =0 ǫ i y i !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 . (38)The first four Adomian polynomials can be obtained in the following form, A = f ( t, y ) , A = y f ′ ( t, y ) , A = y f ′ ( t, y ) + 12 y f ′′ ( t, y ) , (39) A = y f ′ ( t, y ) + y y f ′′ ( t, y ) + 16 y f ′′′ ( t, y ) . (40)For the function y n a few Adomian polynomials are (Wazwaz, 2005) A = y n , A = ny y n − , A = ny y n − + n ( n − y y n − , (41) A = ny y n − + n ( n − y y y n − + n ( n −
1) ( n − y y n − . (42)Now integrating Eq. (35) yields the integral equation y ( x ) = y (0) + Z x [ Q ( x ) y n − P ( x ) y ] dx, (43)where y (0) is the initial condition. Substituting Eqs. (36) and (37) into Eq. (43)gives the relation ∞ X n =0 y n ( x ) = y (0) + Z x Q ( x ) ∞ X n =0 A n ( x ) dx − Z x P ( x ) ∞ X n =0 y n ( x ) dx. (44)We rewrite Eq. (44) in the recursive forms y ( x ) = y (0) , (45) y k +1 ( x ) = Z x [ Q ( x ) A k ( x ) − P ( x ) y k ( x )] dx. (46)From Eqs. (45) and (46), we obtain the semi-analytical solution of Eq. (35), givenby y ( x ) = ∞ X n =0 y n ( x ) . (47) x2 The Adomian Decomposition Method with Applications 11 dydx − xy = − x y Consider now the differential equation dydx − xy = − x y , (48)with initial condition y (0) = 1 , having the general solution y ( x ) = 13 e − x + 2 ( x − . (49)In this case P ( x ) = − x and Q ( x ) = − x , respectively, and n = 2 . Next wecompute a few Adomian polynomials for y , A = y , A = 2 y y , A = 2 y y + y , A = 2 y y + 2 y y , (50)Hence we obtain y ( x ) = y (0) = 1 , (51) y k +1 ( x ) = Z x [ Q ( x ) A k ( x ) − P ( x ) y k ( x )] dx. (52)Eq. (46) can be written recursively for k = 0 , , , in the decomposed solutions y ( x ) = Z x (cid:2) − x A ( x ) + 2 xy ( x ) (cid:3) dx = x − x , (53) y ( x ) = Z x (cid:2) − x A ( x ) + 2 xy ( x ) (cid:3) dx = x − x x , (54) y ( x ) = Z x (cid:2) − x A ( x ) + 2 xy ( x ) (cid:3) dx = x − x
12 + 7 x − x , (55) y ( x ) = Z x (cid:2) − x A ( x ) + 2 xy ( x ) (cid:3) dx = x − x
60 + 25 x − x + x , (56) y ( x ) ≈ y ( x ) + y ( x )+ y ( x )+ y ( x )+ y ( x ) = 1+ x − x − x − x .... (57)On the other hand from the exact solution (49) it is easy to obtain y ( x ) = 13 e − x + 2 ( x −
1) = 1 + x − x − x − x .... (58)Clearly again, the solution (57) obtained by the Adomian Decomposition Method isidentical to the exact solution (58). dydx = P ( x ) + Q ( x ) y The reduced Riccati differential equation is given by (Kamke, 1959) dydx = P ( x ) + Q ( x ) y , (59) xi2 Man Kwong MAK, Chun Sing LEUNG, Tiberiu HARKO 13 where P ( x ) and Q ( x ) are two arbitrary functions of x , and which must be consid-ered together with the initial condition y = y (0) . Integrating Eq. (59) yields theequivalent integral equation y ( x ) = y (0) + Z x P ( x ) dx + Z x Q ( x ) y dx, (60)Substituting y ( x ) = P ∞ n =0 y n ( x ) and y = P ∞ n =0 A n ( x ) into Eq. (60) givesthe relation ∞ X n =0 y n ( x ) = y (0) + Z x P ( x ) dx + Z x Q ( x ) ∞ X n =0 A n dx. (61)Next we rewrite Eq. (61) in the recursive forms y ( x ) = y (0) + Z x P ( x ) dx, (62) y k +1 ( x ) = Z x Q ( x ) A k ( x ) dx. (63)From Eqs. (62) and (63), we obtain the semi-analytical solution of Eq. (59), givenby y ( x ) = P ∞ n =0 = y n ( x ) . dydx = 2 e x − e − x y We consider a particular Riccati equation that has the form dydx = 2 e x − e − x y , (64)and which must be solved together with the initial condition y (0) = 2 . The generalsolution of the equation is given by y ( x ) = e x (cid:18) − − e x (cid:19) . (65)The semi - analytic solution of this particular Riccati equation can be obtained as y ( x ) = y (0) + Z x P ( x ) dx = 2 + 2 Z x e x dx = 2 e x , (66) y k +1 ( x ) = Z x Q ( x ) A k ( x ) dx = − Z x e − x A k dx. (67)In view of Eqs. (66), (67), and (50), we have y ( x ) = Z x Q ( x ) A ( x ) dx = − Z x e − x A dx = − x − x − x − x − x − x − x ..., (68) xii4 The Adomian Decomposition Method with Applications 13 y ( x ) = Z x Q ( x ) A ( x ) dx = − Z x e − x A dx = 8 x + 8 x x x
15 + x
45 + x ..., (69) y ( x ) = Z x Q ( x ) A ( x ) dx = − Z x e − x A dx = − x − x − x − x − x ..., (70) y ( x ) = Z x Q ( x ) A ( x ) dx = − Z x e − x A dx = 32 x + 64 x − x ..., (71) y ( x ) ≈ y ( x ) + y ( x ) + y ( x ) + y ( x ) + y ( x ) = 2 − x + 7 x − x x .... (72)From the exact solution by series expansion it is easy to obtain y ( x ) = e x (cid:18) − − e x (cid:19) = 2 − x + 7 x − x x .... (73)Clearly, the solution (72) obtained by the Adomian decomposition method is identi-cal to the exact solution (73). dydx = M ( x ) + S ( x ) y + R ( x ) y + T ( x ) y The first kind Abel differential equation takes the form (Kamke, 1959) dydx = M ( x ) + S ( x ) y + R ( x ) y + T ( x ) y . (74)Integrating Eq. (74) yields the relation y ( x ) = y (0) + Z x M ( x ) dx + Z x (cid:2) S ( x ) y + R ( x ) y + T ( x ) y (cid:3) dx. (75)Inserting y ( x ) = P ∞ n =0 y n ( x ) , y = P ∞ n =0 A n ( x ) and y = P ∞ n =0 B n ( x ) intoEq. (75) gives the relation ∞ X n =0 y n ( x ) = y (0) + Z x M ( x ) dx + Z x " S ( x ) ∞ X n =0 y n ( x ) + R ( x ) ∞ X n =0 A n ( x ) + T ( x ) ∞ X n =0 B n ( x ) dx. (76)Then we have xiii4 Man Kwong MAK, Chun Sing LEUNG, Tiberiu HARKO 15 y ( x ) = y (0) + Z x M ( x ) dx, (77) y k +1 ( x ) = Z x [ S ( x ) y k ( x ) + R ( x ) A k ( x ) + T ( x ) B k ( x )] dx. (78)From Eqs. (77) and (78), we can obtain the semi-analytical solution of the Abel Eq.(74) as given by y ( x ) = P ∞ n =0 y n ( x ) . dydx = x + 3 xy + 3 xy + xy We consider now a first kind Abel equation that has the form dydx = x + 3 xy + 3 xy + xy = x (1 + y ) , (79)which should be solved with initial condition y (0) = 0 , or y (0) = − . Its generalsolution is given by y ( x ) = − ± √ − x . (80)Now M ( x ) = T ( x ) = x and S ( x ) = R ( x ) = 3 x , and a few Adomian polyno-mials of y are B = y , B = 3 y y , B = 3 y y + 3 y y , B = 3 y y + 6 y y y + y . (81)With the help of Eqs. (77), (78), (50), and (81), by taking y (0) = 0 , we obtain y ( x ) = y (0) + Z x M ( x ) dx = x , (82) y ( x ) = Z x [3 xy ( x ) + 3 xA ( x ) + xB ( x )] dx = 3 x x x , (83) y ( x ) = Z x [3 xy ( x ) + 3 xA ( x ) + xB ( x )] dx = 3 x
16 + 3 x
16 + 9 x
128 +3 x
256 + 3 x , (84) y ( x ) = Z x [3 xy ( x ) + 3 xA ( x ) + xB ( x )] dx = 9 x
128 + 99 x
640 + 15 x ..., (85) y ( x ) = Z x [3 xy ( x ) + 3 xA ( x ) + xB ( x )] dx = 27 x x x ..., (86) y ( x ) ≈ y ( x )+ y ( x )+ y ( x )+ y ( x )+ y ( x ) = x x x
16 + 35 x
128 + 63 x .... (87) xiv6 The Adomian Decomposition Method with Applications 15 From the exact solution it is easy to obtain y ( x ) = − √ − x = x x x
16 + 35 x
128 + 63 x .... (88)It immediately follows that the solution (87) obtained by the Adomian Decomposi-tion Method is identical to the exact solution (88).
4. SOLVING SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS VIA ADOMIANDECOMPOSITION METHOD
Consider a second order non-linear differential equation that takes the form d ydx + f ( x ) dydx + s ( x ) y + g ( x ) y n = k ( x ) , (89)and which must be solved together with the initial conditions y (0) and y ′ (0) , re-spectively, where f ( x ) , s ( x ) , g ( x ) and k ( x ) are arbitrary function of x , and n is aconstant. We define the integral operator L − as L − ( . ) = Z x e − R f ( x ) dx Z x e R f ( x ) dx ( . ) dxdx. (90)We consider first the action of the integral operator L − on the first two termsof the equation, which gives L − (cid:20) d ydx + f ( x ) dydx (cid:21) = Z x e − R f ( x ) dx Z x e R f ( x ) dx (cid:20) d ydx + f ( x ) dydx (cid:21) dxdx = Z x e − R f ( x ) dx (cid:18)Z x e R f ( x ) dx dy ′ + Z x e R f ( x ) dx f y ′ dx (cid:19) dx = Z x e − R f ( x ) dx nh e R f ( x ) dx y ′ i x o dx = Z x y ′ dx − h e R f ( x ) dx i x =0 y ′ (0) Z x e − R f ( x ) dx dx = y ( x ) − y (0) − y ′ (0) h e R f ( x ) dx i x =0 Z x e − R f ( x ) dx dx. (91)Then we have L − (cid:20) d ydx + f ( x ) dydx (cid:21) = L − [ k ( x ) − s ( x ) y − g ( x ) y n ] , (92) y ( x ) = φ ( x ) + Z x e − R f ( x ) dx (cid:26)Z x e R f ( x ) dx [ k ( x ) − s ( x ) y − g ( x ) y n ] dx (cid:27) dx, (93) xv6 Man Kwong MAK, Chun Sing LEUNG, Tiberiu HARKO 17 where we have denoted φ ( x ) as φ ( x ) = y (0) + y ′ (0) h e R f ( x ) dx i x =0 Z x e − R f ( x ) dx dx. (94)Hence we obtain ∞ X n =0 y n ( x ) = φ ( x ) + Z x e − R f ( x ) dx (cid:20)Z x e R f ( x ) dx k ( x ) dx (cid:21) dx − Z x e − R f ( x ) dx (Z x e R f ( x ) dx " s ( x ) ∞ X n =0 y n ( x ) + g ( x ) ∞ X n =0 A n ( x ) dx ) dx. (95)Then for the solution of the second order nonlinear differential equation we have y ( x ) = φ ( x ) + Z x e − R f ( x ) dx (cid:20)Z x e R f ( x ) dx k ( x ) dx (cid:21) dx, (96) y k +1 ( x ) = − Z x e − R f ( x ) dx (cid:20)Z x e R f ( x ) dx [ s ( x ) y k ( x ) + g ( x ) A k ( x )] dx (cid:21) dx. (97)From Eqs. (41)-(42), and (96), (97), we obtain the semi-analytical solution of Eq.(89), given by y = P ∞ n =0 y n ( x ) . d ydx + 4 dydx + 3 y = 3 As an example of the application of the ADM we consider a particular secondorder differential equation that takes the form d ydx + 4 dydx + 3 y = 3 , (98)which must be considered together with the initial conditions y (0) = 1 and y ′ (0) = 2 .The general solution of the equation is given by y ( x ) = − e − x + e − x + 1 , (99)From the equation we easily obtain e R f ( x ) dx = e R dx = e x , f ( x ) = 4 , g ( x ) = 0 , k ( x ) = 3 and s ( x ) = 3 , respectively. Then we have ∞ X n =0 y n ( x ) = φ ( x ) + Z x e − R f ( x ) dx (cid:20)Z x e R f ( x ) dx k ( x ) dx (cid:21) dx − Z x e − R f ( x ) dx "Z x e R f ( x ) dx s ( x ) ∞ X n =0 y n ( x ) dx dx. (100)We rewrite Eq. (100) in the recursive forms xvi8 The Adomian Decomposition Method with Applications 17 y ( x ) = y (0) + y ′ (0) h e R f ( x ) dx i x =0 Z x e − R f ( x ) dx dx + (101) Z x e − R f ( x ) dx (cid:20)Z x e R f ( x ) dx k ( x ) dx (cid:21) dx (102) = 2116 + 34 x − e − x , (103)and y k +1 ( x ) = − Z x e − x (cid:20)Z x e x y k ( x ) dx (cid:21) dx. (104)Hence we obtain y ( x ) = − Z x e − x (cid:20)Z x e x y ( x ) dx (cid:21) dx (105) = − x + x − x − x x − x
35 + 34 x ..., (106) y ( x ) = − Z x e − x (cid:20)Z x e x y ( x ) dx (cid:21) dx (107) = 3 x − x
20 + 27 x − x
28 + 9 x ..., (108) y ( x ) = − Z x e − x (cid:20)Z x e x y ( x ) dx (cid:21) dx (109) = − x
80 + 3 x − x x − x , (110) y ( x ) = − Z x e − x (cid:20)Z x e x y ( x ) dx (cid:21) dx (111) = 9 x − x
320 + 123 x .... (112)The semi-analytical solution of Eq. (98) is given by y ( x ) ≈ y ( x ) + y ( x ) + y ( x ) + y ( x ) + ... = 1 + 2 x − x + 13 x − x x − x
90 + 1093 x .... (113)On the other hand from the exact solution it is easy to obtain y ( x ) = − e − x + e − x + 1 = 1 + 2 x − x + 13 x − x x − x
90 + 1093 x .... (114) xvii8 Man Kwong MAK, Chun Sing LEUNG, Tiberiu HARKO 19 As one can easily see, the solution (113) of the second order differential Eq.(98) obtained by the Adomian decomposition method is identical to the exact solution(114). d ydx + dydx + e x y + e x y = e x As a second example of the application of the ADM for solving a nonlineardifferential equation we consider that the differential equation takes the form d ydx + dydx + e x y + e x y = e x , (115)and it must be solved together with the initial condition y (0) = 1 and y ′ (0) = 2 ,respectively. From the equation we obtain easily e R f ( x ) dx = e R dx = e x , f ( x ) = 1 ,g ( x ) = k ( x ) = s ( x ) = e x . Hence we immediately find y ( x ) = y (0) + y ′ (0) h e R f ( x ) dx i x =0 Z x e − R f ( x ) dx dx + Z x e − R f ( x ) dx (cid:20)Z x e R f ( x ) dx k ( x ) dx (cid:21) dx (116) = 1 + 2 Z x e − x dx + Z x e − x (cid:20)Z x e x dx (cid:21) dx = 2 − e − x + 12 e x . (117)and y k +1 ( x ) = − Z x e − R f ( x ) dx (cid:20)Z x e R f ( x ) dx [ s ( x ) y k ( x ) + g ( x ) A k ( x )] dx (cid:21) dx,y k +1 ( x ) = − Z x e − x (cid:20)Z x e x [ y k ( x ) + A k ( x )] dx (cid:21) dx, (118)From Eqs. (50), (117), and (118), we obtain y ( x ) = − Z x e − x (cid:20)Z x e x [ y ( x ) + A ( x )] dx (cid:21) dx (119) = − x − x − x − x − x − x − x − x − x ..., (120) y ( x ) = − Z x e − x (cid:20)Z x e x [ y ( x ) + A ( x )] dx (cid:21) dx (121) = x x
20 + 29 x
80 + 521 x x x x ..., (122) xviii0 The Adomian Decomposition Method with Applications 19 y ( x ) = − Z x e − x (cid:20)Z x e x [ y ( x ) + A ( x )] dx (cid:21) dx (123) = − x − x − x − x − x ..., (124) y ( x ) = − Z x e − x (cid:20)Z x e x [ y ( x ) + A ( x )] dx (cid:21) dx =27 x x x .... (125)Hence the semi-analytical solution of Eq. (115) is given by y ( x ) ≈ y ( x ) + y ( x ) + y ( x ) + y ( x ) + ... = 1 + 2 x − x − x − x x x
720 + 131 x − x − x − x .... (126)
5. THE FIFTH ORDER ORDINARY DIFFERENTIAL EQUATION VIA THE ADOMIANDECOMPOSITION METHOD
Consider the following fifth order ordinary differential equation, which takesthe form d ydx + a d ydx + a d ydx + a d ydx + a dydx + a y = 0 , (127)where a i , i = 0 , , , , are constants. Eq. (127) should be integrated with the initialconditions y (0) = y , y ′ (0) = y , y ′′ (0) = y , y ′′′ (0) = y , and y ( iv ) (0) = y ,respectively. Now applying the 5 fold integral operator L − , defined as L − ( . ) = Z x Z x Z x Z x Z x ( . ) dxdxdxdxdx, (128)to Eq. (127), yields the relation Z x Z x Z x Z x Z x (cid:18) d ydx + a d ydx + a d ydx + a d ydx + a dydx + a y (cid:19) d x = 0 . (129) xix0 Man Kwong MAK, Chun Sing LEUNG, Tiberiu HARKO 21 From Eq. (129), we obtain y ( x ) = y (0) + (cid:2) y ′ (0) + a y (0) (cid:3) x + (cid:2) y ′′ (0) + a y ′ (0) + a y (0) (cid:3) x (cid:2) y ′′′ (0) + a y ′′ (0) + a y ′ (0) + a y (0) (cid:3) x (cid:2) y ′′′′ (0) + a y ′′′ (0) + a y ′′ (0) + a y ′ (0) + a y (0) (cid:3) x − a Z x ydx − a Z x Z x ydxdx − a Z x Z x Z x ydxdxdx − a Z x Z x Z x Z x ydxdxdxdx − a Z x Z x Z x Z x Z x ydxdxdxdxdx. (130)Substituting y ( x ) = P ∞ n =0 y n ( x ) into Eq. (130) yields ∞ X n =0 y n ( x ) = y (0) + (cid:2) y ′ (0) + a y (0) (cid:3) x + (cid:2) y ′′ (0) + a y ′ (0) + a y (0) (cid:3) x (cid:2) y ′′′ (0) + a y ′′ (0) + a y ′ (0) + a y (0) (cid:3) x (cid:2) y ′′′′ (0) + a y ′′′ (0) + a y ′′ (0) + a y ′ (0) + a y (0) (cid:3) x − a Z x ∞ X n =0 y n ( x ) dx − a Z x Z x ∞ X n =0 y n ( x ) dxdx − a Z x Z x Z x ∞ X n =0 y n ( x ) dxdxdx − a Z x Z x Z x Z x ∞ X n =0 y n ( x ) dxdxdxdx − a Z x Z x Z x Z x Z x ∞ X n =0 y n ( x ) dxdxdxdxdx. (131)We rewrite Eq. (131) in the recursive forms y ( x ) = y (0) + (cid:2) y ′ (0) + a y (0) (cid:3) x + (cid:2) y ′′ (0) + a y ′ (0) + a y (0) (cid:3) x (cid:2) y ′′′ (0) + a y ′′ (0) + a y ′ (0) + a y (0) (cid:3) x (cid:2) y ′′′′ (0) + a y ′′′ (0) + a y ′′ (0) + a y ′ (0) + a y (0) (cid:3) x , (132) xx2 The Adomian Decomposition Method with Applications 21 and y k +1 ( x ) = − a Z x y k ( x ) dx − a Z x Z x y k ( x ) dxdx − a Z x Z x Z x y k ( x ) dxdxdx − a Z x Z x Z x Z x y k ( x ) dxdxdxdx − a Z x Z x Z x Z x Z x y k ( x ) dxdxdxdxdx. (133)From Eqs. (132) and (133), we can obtain the semi-analytical solution of Eq.(127), given by y ( x ) = P ∞ n =0 y n ( x ) . For the solution of a particular third-orderordinary differential equation see Pue-on and Viriyapong (2012). d ydx − d ydx − d ydx + 15 d ydx + 4 dydx − y = 0 In the following we consider a particular fifth order ordinary differential equa-tion that takes the form d ydx − d ydx − d ydx + 15 d ydx + 4 dydx − y = 0 , (134)and which should be solved together with the initial conditions y (0) = 1 , y ′ (0) = − , y ′′ (0) = 2 , y ′′′ (0) = − , y ′′′′ (0) = 3 . The coefficients a i of the equation aregiven by a = − , a = − , a = 15 , a = 4 and a = − , respectively. The generalsolution of the equation is given by y ( x ) = − e x + 1924 e − x + 13 e x + 15 e − x − e x . (135)By applying the ADM we have y ( x ) = 1 − x + 2 x − x , (136) y k +1 ( x ) = 3 Z x y k ( x ) dx + 5 Z x Z x y k ( x ) dxdx − Z x Z x Z x y k ( x ) dxdxdx − Z x Z x Z x Z x y k ( x ) dxdxdxdx +12 Z x Z x Z x Z x Z x y k ( x ) dxdxdxdxdx. (137) xxi2 Man Kwong MAK, Chun Sing LEUNG, Tiberiu HARKO 23 From Eqs. (136) and (137), we obtain y ( x ) = 3 x − x − x x x − x ..., (138) y ( x ) = 9 x − x − x
24 + 97 x
60 + 241 x ..., (139) y ( x ) = 9 x x − x − x ..., (140) y ( x ) = 27 x x − x .... (141)Thus we obtain the ADM solution of the equation as y ( x ) ≈ y ( x ) + y ( x ) + y ( x ) + y ( x ) + y ( x ) = 1 − x + x − x x .... (142)On the other hand from the exact solution it is easy to obtain y ( x ) = − e x + 1924 e − x + 13 e x + 15 e − x − e x = 1 − x + x − x x ..., (143)Hence, the solution (142) of the fifth order differential Eq. (134), obtained by theAdomian decomposition method, is identical to the exact solution (143).
6. SOLVING PARTIAL DIFFERENTIAL EQUATIONS VIA ADM: THEFISHER-KOLMOGOROV EQUATION
The three dimensional Fisher-Kolmogorov equation (Fisher, 1937; Kolmogorov et al. ,1937; Harko and Mak, 2015a), has many important applications in physics and biol-ogy. In particular, it can be used to describe the growth of glioblastoma (Harko and Mak,2015b). The Fisher-Kolmogorov equation is given by ∂c ( t, x, y, z ) ∂t = D ∆ c ( t, x, y, z ) + ac ( t, x, y, z ) (cid:20) − c ( t, x, y, z ) N (cid:21) , (144)where ∆ = ∂ ∂x + ∂ ∂y + ∂ ∂z , (145)and a , N and D are constants. Eq. (144) must be considered together with theinitial condition c (0 , x, y, z ) = c ( x, y, z ) . From the point of view of the ADMthe Fisher-Kolmogorov equation was studied in Wazwaz and Gorguis (2004) andBhalekar and Patade (2016), respectively. In order to apply the ADM method werewrite Eq. (144) as L t c = D ∆ c + F ( c ) , (146) xxii4 The Adomian Decomposition Method with Applications 23 where L t = ∂∂t and F ( c ) = ac (cid:2) − cN (cid:3) . Now applying the inverse operator L − t defined as L − t ( . ) = R t ( . ) dt to Eq. (146), the general solution of Eq. (146) can beobtained formally as c ( t, x, y, x ) = c ( x, y, z ) + DL − t ∆ c ( t, x, y, x ) + L − t F ( c ) . (147)According to the Adomian Decomposition Method we look for series solutionsof Eq. (147) of the form c ( t, x, y, z ) = ∞ X n =0 c n ( t, x, y, z ) , F ( c ) = ∞ X n =0 A n ( t, x, y, z ) , (148)where the Adomian polynomials A n ( t, x, y, z ) are defined as A n ( t, x, y, z ) = 1 n ! (cid:20) d n d λ n F ( c λ ) (cid:21) λ =0 , (149)where c λ = P ∞ i =0 λ i c i . The first few Adomian polynomials are given by A = F ( c ) = ac (cid:16) − c N (cid:17) , (150) A = c F ′ ( c ) = ac (cid:18) − c N (cid:19) , (151) A = c F ′ ( c ) + 12 c F ′′ ( c ) = ac (cid:18) − c N (cid:19) − aN c , (152) A = c F ′ ( c ) + c c F ′′ ( c ) + 16 c F ′′′ ( c ) (153) = ac (cid:18) − c N (cid:19) − aN c c . (154)Therefore, after substituting Eq. (148) into Eq. (147), the latter becomes ∞ X n =0 c n ( t, x, y, z ) = c ( x, y, z ) + DL − t ∆ " ∞ X n =0 c n ( t, x, y, z ) + L − t " ∞ X n =0 A n ( t, x, y, z ) . (155)We rewrite Eq. (155) as c ( t, x, y, z ) = c ( x, y, z ) , (156) c k +1 ( t, x, y, z ) = DL − t ∆ [ c k ( t, x, y, z )] + L − t [ A k ( t, x, y, z )] . (157) xxiii4 Man Kwong MAK, Chun Sing LEUNG, Tiberiu HARKO 25 From Eq. (157), we obtain c = Z t ( D ∆ c + A ) dt, c = Z t ( D ∆ c + A ) dt, c = Z t ( D ∆ c + A ) dt, (158) ... (159) c m +1 = Z t ( D ∆ c m + A m ) dt. (160)where k = 0 , , ...m. The approximate solution of the Fisher-Kolmogorov equationcan be written as c ( t, x, y, z ) = m +1 X i =0 c i ( t, x, y, z ) . (161) c (0 , x, y, z ) = x + y + z As an example of the application of the Adomian Decomposition Method, weconsider the case in which Eq. (144) should be solved together the initial condition c (0 , x, y, z ) = c ( x, y, z ) = x + y + z . Then we obtain c ( x, y, z ) = x + y + z , (162) c ( t, x, y, z ) = Z t ( D ∆ c + A ) dt = t " D + a (cid:0) x + y + z (cid:1) × (cid:18) − x + y + z N (cid:19) , (163) c ( t, x, y, z ) = Z t ( D ∆ c + A ) dt (164) = at N ( DN (cid:2) N − (cid:0) x + y + z (cid:1)(cid:3) + a (cid:0) x + y + z (cid:1) × h N − N (cid:0) x + y + z (cid:1) + 2 (cid:0) x + y + z (cid:1) i ) , (165) c ( t, x, y, z ) = Z t ( D ∆ c + A ) dt, c ( t, x, y, z ) = Z t ( D ∆ c + A ) dt, (166) c ( t, x, y, z ) ≈ c + c + c + c + c . (167) xxiv6 The Adomian Decomposition Method with Applications 25 Hence it follows that the Adomian method is also a very powerful approach forsolving partial differential equations. The solutions converge fast, thus saving a lotof computing time.
7. THE LAPLACE-ADOMIAN DECOMPOSITION METHOD
A very powerful version of the Adomian Decomposition Method is representedby the so-called Laplace-Adomian Decomposition Method (LADM) (Khuri, 2001,2004; Wazwaz, 2010; Manafianheris, 2012; Hamoud and Ghadle, 2017). We willintroduce this method by considering the particular example of a second order non-linear differential equation of the form d ydx + ω y + b + f ( y ) = 0 , (168)where ω and b are arbitrary constants, while f ( y ) is an nonlinear arbitrary functionof the dependent variable y . We will consider Eq. (168) together with the initialconditions y (0) = y = a , and y ′ (0) = 0 , respectively.We define the Laplace transform operator L x of an arbitrary function f ( x ) , as L x [ f ( x )]( s ) = R ∞ f ( x ) e − sx dx .The first, and essential step in the Laplace-Adomian Decomposition Method isto apply the Laplace transform operator L x to Eq. (168). Hence we obtain L x (cid:20) d ydx (cid:21) + ω L x [ y ] + L x [ b ] + L x [ f ( y )] = 0 . (169)By using the basic properties of the Laplace transform we straightforwardlyobtain (cid:0) s + ω (cid:1) L x [ y ] − sy (0) − y ′ (0) + b s + L x [ f ( y )] = 0 . (170)After explicitly taking into account the initial conditions for our problem weobtain the relation L x [ y ] = ass + ω − b s ( s + ω ) − s + ω L x [ f ( y )] . (171)We assume now that the solution of Eq. (168) can be represented in the formof an infinite series given by y ( x ) = ∞ X n =0 y n ( x ) , (172)where each term y n ( x ) can be calculated recursively. With respect to the nonlinear xxv6 Man Kwong MAK, Chun Sing LEUNG, Tiberiu HARKO 27 operator f ( y ) , we assume that it can be decomposed according to f ( y ) = ∞ X n =0 A n , (173)where the functions A n are the Adomian polynomials, which can be obtained fromthe general algorithm (Adomian and Rach, 1983; Adomian, 1994) A n = 1 n ! d n d ǫ n f ∞ X i =0 ǫ i y i !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 . (174)The first few Adomian polynomials are given by, A = f ( y ) , A = y f ′ ( y ) , A = y f ′ ( y ) + 12 y f ′′ ( y ) , (175) A = y f ′ ( y ) + y y f ′′ ( y ) + 16 y f ′′′ ( y ) , (176) A = y f ′ ( y ) + (cid:20) y + y y (cid:21) f ′′ ( y ) + 12! y y f ′′′ ( y ) + 14! y f (iv) ( y ) . (177)After substituting Eqs. (172) and (173) into Eq. (171) we find L x " ∞ X n =0 y n ( x ) = ass + ω − b s ( s + ω ) − s + ω L x [ ∞ X n =0 A n ] . (178)By matching both sides of Eq. (178) gives an iterative algorithm for obtainingthe power series solution of Eq. (168), which can be formulated as L x [ y ] = ass + ω − b s ( s + ω ) , (179) L x [ y ] = − s + ω L x [ A ] , (180) L x [ y ] = − s + ω L x [ A ] , (181) ... L x [ y k +1 ] = − s + ω L x [ A k ] . (182)To obtain the value of y we apply the inverse Laplace transform to Eq. (179).After substituting y into the first of Eqs. (175) we find the first Adomian polynomial A . The obtained expression of A is then substituted into Eq. (180), which allows tocompute the Laplace transforms of the quantities of its right-hand. Then the furtherapplication of the inverse Laplace transform gives the functional expressions of y .All the other terms y , y , . . ., y k +1 , ... of the series solution can be similarlycalculated recursively by using a step by step procedure. xxvi8 The Adomian Decomposition Method with Applications 277.1. EXAMPLE: f ( y ) = P ml =0 a l +2 y l +2 We will illustrate the applications of the Laplace-Adomian Decomposition Methodby considering the case of a second order nonlinear differential equation having theform (Mak et al., 2018a) d ydx + ω y + b + m X l =0 a l +2 y l +2 = 0 , (183)where ω , b , and a l +2 , l = 0 , ..., m are arbitrary constants. As usual, we considerEq. (183) together with the set of initial conditions y (0) = y = a , and y ′ (0) = 0 ,respectively. We investigate Eq. (183) by using the Laplace-Adomian DecompositionMethod. Hence, as a first step, we apply the Laplace transform to Eq. (183), thusfinding L x (cid:18) d ydx (cid:19) + ω L x ( y ) + b L x (1) + m X l =0 a l +2 L x h y l +2 i = 0 . (184)Next, by the use of the properties of the Laplace transform, we immediatelyobtain L x ( y ) (cid:0) s + ω (cid:1) = sy (0) + y ′ (0) − b s − m X l =0 a l +2 L x h y l +2 i = 0 , (185)and thus L x ( y ) = sy (0) + y ′ (0) s + ω − b s ( s + ω ) − s + ω m X l =0 a l +2 L x h y l +2 i . (186)Hence, from Eq. (186) y ( x ) in obtained in the form y ( x ) = L − x (cid:20) sy (0) + y ′ (0) s + ω − b s ( s + ω ) (cid:21) − L − x " s + ω m X l =0 a l +2 L x h y l +2 i . (187)We assume now that the solution y ( x ) of Eq. (183) can be represented as y ( x ) = P ∞ n =0 y n ( x ) . Moreover, we decompose the nonlinear terms according to y l +2 = ∞ X n =0 A n,l +2 ( x ) , (188) xxvii8 Man Kwong MAK, Chun Sing LEUNG, Tiberiu HARKO 29 where A n,l +2 are the Adomian polynomials determining y l +2 . Then we obtain ∞ X n =0 y n ( x ) = L − x (cid:20) sy (0) + y ′ (0) s + ω − b s ( s + ω ) (cid:21) −L − x ( s + ω m X l =0 a l +2 L " ∞ X n =0 A n,l +2 ( x ) . (189)We reformulate now Eq. (189) in the form y ( x ) + ∞ X n =0 y n +1 ( x ) = L − x (cid:20) sy (0) + y ′ (0) s + ω − b s ( s + ω ) (cid:21) − ∞ X n =0 L − x ( s + ω m X l =0 a l +2 L x [ A n,l +2 ( x )] ) . (190)Hence from Eq. (190) we obtain the set of recursive relations y ( x ) = L − x (cid:20) sy (0) + y ′ (0) s + ω − b s ( s + ω ) (cid:21) , (191) ...,y k +1 ( x ) = −L − x ( s + ω m X l =0 a l +2 L x [ A k,l +2 ( x )] ) . (192)A few Adomian polynomials for the function y l +2 are given by A ,l +2 = y l +20 , A ,l +2 = ( l + 2) y y l +10 , (193) A ,l +2 = ( l + 2) y y l +10 + ( l + 1) ( l + 2) y y l , (194) A ,l +2 = ( l + 2) y y l +10 + ( l + 1) ( l + 2) y y y l + l ( l + 1) ( l + 2) y y l − . (195)We find the first order approximation of the solution by taking k = 0 , thusobtaining y ( x ) = −L − x (cid:26) L x [ P ml =0 a l +2 A ,l +2 ] s + ω (cid:27) = −L − x ( L x (cid:0) a y + a y + a y + ... (cid:1) s + ω ) . (196) xxviii0 The Adomian Decomposition Method with Applications 29 For k = 1 we find y ( x ) as given by y ( x ) = −L − x (cid:26) L x [ P ml =0 a l +2 A ,l +2 ] s + ω (cid:27) = −L − x ( L x (cid:0) a y y + 3 a y y + 4 a y y + ... (cid:1) s + ω ) . (197)By fixing k as k = 2 yields y ( x ) = −L − x (cid:26) L x [ P ml =0 a l +2 A ,l +2 ] s + ω (cid:27) = −L − x ( L x (cid:2) a (cid:0) y y + y (cid:1) + 3 a (cid:0) y y + y y (cid:1) + a (cid:0) y y + 6 y y (cid:1) + ... (cid:3) s + ω ) . (198)As a last case we take k = 3 , and thus y ( x ) = −L − ( L x [ P ml =0 a l +2 A ,l +2 ] s + ω ) = −L − ( s + ω L " a ( y y + y y ) + a (cid:0) y y + 6 y y y + y (cid:1) + a (cid:0) y y + 12 y y y + 4 y y (cid:1) + ... . (199)Hence the truncated power series solution of Eq. (183) is given by y ( x ) = ∞ X n =0 y n ( x ) = y ( x ) + y ( x ) + y ( x ) + y ( x ) + y ( x ) + .... (200)
8. ASTRONOMICAL AND ASTROPHYSICAL APPLICATIONS
In the present Section we consider some astronomical and astrophysical ap-plications of the ADM. In particular, we will consider the solutions of the Keplerequation via ADM, the solutions of the Lane-Emden equation, and the study of themotion of massive particles in the Schwarzschild geometry.
In celestial mechanics, Kepler’s equation plays an essential role in the deter-mination of the orbit of an object evolving under the action of a central force. TheKepler equation for the hyperbolic case is (Ebaid et al. , 2017) e sin H ( t ) − H ( t ) = M ( t ) , (201) xxix0 Man Kwong MAK, Chun Sing LEUNG, Tiberiu HARKO 31 where e is the eccentricity of the orbit, H ( t ) is the eccentric anomaly, M ( t ) = p µ /a ( t − τ ) represents the mean anomaly, µ = GM , while a is the semi-majoraxis of the orbit. Moreover, the time interval for the passage through the closestpoint of approach to the focus of the orbit is denoted by τ . The Kepler equation(201) can be transformed into the forms ey ( t ) − arcsinh y ( t ) = M, (202)under the assumption y ( t ) = sinh H ( t ) , and y ( t ) = α + β arcsinh y ( t ) , (203)where ≤ α = M/e < ∞ , and ≤ β = 1 /e ≤ , respectively. In the AdomianDecomposition Method approach to the Kepler equation one assumes that y and arcsinh y can be decomposed as y ( t ) = P ∞ n =0 y n ( t ) , and arcsinhy = P ∞ n =0 A n ( t ) ,where A n ( t ) are the Adomian polynomials corresponding to arcsinh y . After sub-stituting the series expansions into Eq. (203) one arrives to the following recursionrelations, y = α , (204) y n +1 = β A n , n = 0 , , , .... (205)The Adomian polynomials for the function arcsinh y can be obtained as follows(Ebaid et al. , 2017), A = arcsinh y ( t ) , A = y (cid:0) y (cid:1) / , A = 2 (cid:0) y (cid:1) y − y y (cid:0) y (cid:1) / , (206) A = 6 (cid:0) y (cid:1) y − y (cid:0) y (cid:1) y y + (cid:0) y − (cid:1) y (cid:0) y (cid:1) / . (207)Then we obtain the following solution of the Kepler equation Φ ( t ) = P n − i =0 y i ( t ) (Ebaid et al. , 2017), Φ ( t ) = α + β arcsinh α , (208) Φ ( t ) = α + β arcsinh α + β arcsinh α (1 + α ) , (209) Φ ( t ) = α + β arcsinh α + β arcsinh α (1 + α ) +2 (cid:0) α (cid:1) / β arcsinh α − αβ (arcsinh α ) α ) / . (210)For the higher order terms in the Adomian expansion, the convergence of theseries and the comparison with the full numerical solution see Ebaid et al. (2017). xxx2 The Adomian Decomposition Method with Applications 31 For the study of the elliptical Kepler problem via the ADM see Alshaery and Ebaid(2017).
The astrophysical properties of the static Newtonian stars can be fully char-acterized by the two gravitational structure equations, which are represented by themass continuity equation, and the equation of the hydrostatic equilibrium, respec-tively, given by (Chandrasekhar, 1967; Horedt, 2004; Blaga, 2005; B ¨ohmer and Harko,2010) dm ( r ) dr = 4 πρ ( r ) r , (211) dp ( r ) dr = − Gm ( r ) r ρ ( r ) , (212)where ρ ( r ) ≥ is the matter density inside the star, p ( r ) ≥ is the thermodynamicpressure, while m ( r ) ≥ , ∀ r ≥ denotes the mass inside radius r , respectively. Toclose the system of structure equations one should assume an equation of state for theinterior stellar matter, p = p ( ρ ) , which is a functional relation between the thermo-dynamic pressure and the density of the matter inside the star. An important equationof state is the polytropic equation of state, for which the pressure can be expressedas a power law of the density, p = K ρ /n , (213)where K ≥ and n are constants, and n = 0 . After eliminating the mass function m ( r ) between the two structure equations (211) and (212), respectively, we obtain asingle second order non-linear differential equation given by r ddr (cid:18) r ρ dpdr (cid:19) = − π G ρ , (214)which describes the global properties of the Newtonian star. By introducing for thedensity a new dimensionless variable θ , so that ρ = ρ c θ n , (215)where ρ c is the central density, and n is the polytropic index, we obtain for the pres-sure the expression p = K ρ /nc θ n +1 . Next we introduce the dimensionless formof the radial coordinate ξ , defined as r = αξ , α = s ( n + 1) K ρ /n − c π G , n = − . (216) xxxi2 Man Kwong MAK, Chun Sing LEUNG, Tiberiu HARKO 33 In these dimensionless variables Eq. (214) takes the form of the Lane-Emden equa-tion of index n , θ ′′ + 2 ξ θ ′ + θ n = 0 . (217)To solve the Lane-Emden equation we adopt the initial conditions θ (0) = 1 and θ ′ (0) = 0 , respectively, where the prime represents the derivative with respect tothe dimensionless independent variable ξ .In the limit n → , the Lane-Emden equation has the solution θ ( ξ ) | n =0 =1 − ξ / . For n = 1 , the Lane-Emden equation (217) reduces to a linear ordi-nary differential equation, and it has the solution θ ( ξ ) | n =1 = sin( ξ ) / ξ . The non-linear Lane-Emden equation has only one known exact solution when n = 5 , givenby θ ( ξ ) | n =5 = 1 / p ξ / . For series solutions of the mass continuity and ofthe general relativistic hydrostatic equilibrium equation (the Tolman-Oppenheimer-Volkoff equation), describing the interior properties of high density compact objects,see Mak and Harko (2013b) and Harko and Mak (2016), respectively. The second order nonlinear ordinary differential equation of the form d ydx + kx dydx + y m = 0 , (218)where k > , is called the Lane–Emden equation of the first kind (Rach et al. , 2015).It has to be integrated together with the initial conditions y (0) = 1 and y ′ (0) = 0 ,respectively. The Lane-Emden equation of the second kind is given by d ydx + kx dydx + e y = 0 , (219)where k > , and the equation is considered together with the initial conditions y (0) = y ′ (0) = 0 . However, in the following we will consider the generalized Lane-Emden equation, given by (Rach et al. , 2015) d ydx + kx dydx + f ( y ) = 0 , (220)where k > , f ( y ) is an arbitrary analytic function of y , and which should be inte-grated with the initial conditions y (0) = α , and y ′ (0) = 0 , respectively. Eq. (220) can be reformulated as an integral equation as follows (Rach et al. ,2015). For k > , and k = 1 , Eq. (220) can be reformulated as (cid:16) x k y ′ (cid:17) ′ = − x k f ( y ) , (221) xxxii4 The Adomian Decomposition Method with Applications 33 where a prime denotes the differentiation with respect to x . Integrating once weobtain y ′ ( x ) = − x k Z x t k f ( y ( t )) dt. (222)Integrating again we find y ( x ) − α = − Z x x k Z x t k f ( y ( t )) dtdx = 1 k − Z x Z x t k f ( y ( t )) dtd (cid:18) x k − (cid:19) . (223)By using the Cauchy formula for repeated integration, Z xa Z x a ... Z x n − a f ( x n ) dx n ...dx dx = 1( n − Z xa ( x − t ) n − f ( t ) dt, (224)we immediately obtain the integral equation formulation of the Lane-Emden equationfor k = 1 as (Rach et al. , 2015) y ( x ) = α + 1 k − Z x t (cid:18) t k − x k − − (cid:19) f ( y ( t )) dt, k > , k = 1 . (225)For the case k = 1 we find (Rach et al. , 2015) y ( x ) = α + Z x t ln (cid:18) tx (cid:19) f ( y ( t )) dt, k = 1 . (226)These two cases can be unified in a single formulation once we introduce theintegral kernel K ( x, t ; k ) , defined as (Rach et al. , 2015), K ( x, t ; k ) = ( k − t (cid:16) t k − x k − − (cid:17) , k > , k = 1 ,t ln (cid:0) tx (cid:1) , k = 1 . (227)Then the Lane-Emden equation can be formulated generally in an integral formas (Rach et al. , 2015) y ( x ) = α + Z x K ( x, t ; k ) f ( y ( t )) dt. (228) As usual in the Adomian Decomposition Method, we assume that the solution y ( x ) of the Lane-Emden equation can be represented in the form of an infinite se-ries, y ( x ) = P ∞ n =0 y n ( x ) , while the nonlinear term f ( y ) is decomposed by usingthe Adomian polynomials, f ( y ( x )) = P ∞ n =0 A n ( y ( x ) , y ( x ) , ..., y n ( x )) . Then bysubstituting these expressions into Eq. (228) we find ∞ X n =0 y n ( x ) = α + Z x K ( x, t ; k ) ∞ X n =0 A n ( y ( t ) , y ( t ) , ..., y n ( t )) dt, k > . (229) xxxiii4 Man Kwong MAK, Chun Sing LEUNG, Tiberiu HARKO 35 By choosing y = α , we find the following set of recursive relations for theterms in the series solution of the Lane-Emden equation, y = α , (230) y m +1 = Z x K ( x, t ; k ) A m ( y ( t ) , y ( t ) , ..., y n ( t )) dt, m ≥ . (231)The above set of relations will lead to the complete determination of each ofthe components y n ( x ) of the solution y ( x ) . As a simple application of the AdomianDecomposition Method to the nonlinear Lane-Emden type equations we consider,following Rach et al. (2015), the case of the equation d ydx + 1 x dydx − x e − y = 0 , y (0) = 0 , y ′ (0) = 0 , (232)which has the exact solution y ( x ) = ln (cid:0) x (cid:1) . (233)The recursive Adomian relation is obtained as y ( x ) = 0 , (234) y m +1 ( x ) = Z x t ln (cid:18) tx (cid:19) (cid:0) − t A m ( t ) (cid:1) dt, m ≥ . (235)After computing the Adomian polynomials for the nonlinear term e − y , we obtain y ( x ) = 0 , y ( x ) = x , y ( x ) = − x , y ( x ) = 13 x , y ( x ) = − x , .... (236)It is easy to see by series expanding the exact solution (233) that the Adomianseries solution y ( x ) = x − x + 13 x − x + ..., (237)coincides with the exact solution. The equation of motion describing the general relativistic motion of a mas-sive celestial body (for example, a planet) in the spherically symmetric and staticSchwarzschild geometry, written in spherical coordinates ( r, ϕ , θ ) , is given by d ud ϕ + u = ML + 3 M u , (238)where u = 1 /r . For the details of Schwarzschild geometry and of the derivation ofEq. (238) see Mak et al. (2018a) and Harko and Lobo (2018). In the following weuse the natural system of units with G = c = 1 . xxxiv6 The Adomian Decomposition Method with Applications 35 To obtain a simpler mathematical formalism we rescale the function u = 1 /r according to u = 13 M U. (239)Thus Eq. (238) takes the form d Ud ϕ + U = b + U , (240)where we have denoted b = 3 M /L . We will consider Eq. (240) together with theinitial conditions U (0) = 3 M u (0) = a , and U ′ (0) = 0 , respectively. In the followingwe will obtain semi-analytical solutions of Eq. (240) by using the Laplace-AdomianMethod. We assume that the solution of Eq. (240) can be obtained in the form of apower series, so that U ( ϕ ) = ∞ X n =0 U n ( ϕ ) . (241)We apply now the Laplace transform operator L ϕ to Eq. (240), thus obtaining L ϕ (cid:20) d Ud ϕ (cid:21) + L ϕ [ U ] = b L ϕ [1] + L ϕ (cid:2) U (cid:3) . (242)By using the properties of the Laplace transform we find s L ϕ ( U ) − sU (0) − U ′ (0) + L ϕ ( U ) = b s + L ϕ (cid:2) U (cid:3) , (243)and L ϕ ( U ) = sU (0) + U ′ (0) s + 1 + b s ( s + 1) + 1 s + 1 L ϕ (cid:2) U (cid:3) , (244)respectively. The first four Adomian polynomials for U are given by A = U , A = 2 U U , A = 2 U U + U , A = 2 U U + 2 U U . (245)Now we substitute Eq. (241) and U = P ∞ n =0 A n ( ϕ ) into Eq. (244), and thus weobtain the relation L ϕ " ∞ X n =0 U n ( ϕ ) = sU (0) + U ′ (0) s + 1 + b s ( s + 1) + 1 s + 1 L ϕ " ∞ X n =0 A n ( ϕ ) , (246) xxxv6 Man Kwong MAK, Chun Sing LEUNG, Tiberiu HARKO 37 which can be written explicitly as U ( ϕ ) + ∞ X n =1 U n ( ϕ ) = U ( ϕ ) + ∞ X n =0 U n +1 ( ϕ ) = L − ϕ (cid:20) sU (0) + U ′ (0) s + 1 + b s ( s + 1) (cid:21) + ∞ X n =0 L − ϕ (cid:20) L ϕ [ A n ( ϕ )] s + 1 (cid:21) . (247)Now we can rewrite Eq. (247) in the following recursive forms U ( ϕ ) = L − ϕ (cid:20) sU (0) + U ′ (0) s + 1 + b s ( s + 1) (cid:21) , (248) ...,U k +1 ( ϕ ) = L − ϕ (cid:20) L ϕ [ A k ( ϕ )] s + 1 (cid:21) . (249) By using the explicit expressions of the Adomian polynomials, we can findthe analytical forms of the successive terms in the Adomian series expansion of thesolution of the general relativistic equation of motion of a planet in the sphericallysymmetric and static Schwarzschild geometry as follows. First of all, by neglectingthe nonlinear term in Eq. (240), we obtain the zeroth order approximation of thesolution as given by U ( ϕ ) = (cid:0) a − b (cid:1) cos ϕ + b . (250)Then for the first Adomian polynomial we obtain A = U = (cid:2)(cid:0) a − b (cid:1) cos ϕ + b (cid:3) , (251)Once A is known, for the first order approximation of the solution we find U ( ϕ ) = L − ϕ (cid:20) L ϕ [ A ( ϕ )] s + 1 (cid:21) = L − ϕ " L ϕ (cid:2) U (cid:3) s + 1 , (252)or, explicitly, U ( ϕ ) = 16 ( − (cid:0) a − ab + 4 b (cid:1) cos ϕ + 3 (cid:0) a − ab + 3 b (cid:1) + (cid:0) a − b (cid:1) (cid:2)(cid:0) b − a (cid:1) cos(2 ϕ ) + 6 b ϕ sin ϕ (cid:3) ) , (253) xxxvi8 The Adomian Decomposition Method with Applications 37 The Adomian polynomial A can be then obtained as A = 2 U U = 13 " (cid:0) a − b (cid:1) cos ϕ + b − (cid:0) a − ab + 4 b (cid:1) cos ϕ +3 (cid:0) a − ab + 3 b (cid:1) + (cid:0) a − b (cid:1) (cid:2)(cid:0) b − a (cid:1) cos(2 ϕ ) + 6 b ϕ sin ϕ (cid:3) ) , (254)giving for the second order approximation the expression U ( ϕ ) = L − ϕ (cid:20) L ϕ [ A ( ϕ )] s + 1 (cid:21) = 2 L − ϕ (cid:20) L ϕ [ U U ] s + 1 (cid:21) , (255)or, explicitly, U ( ϕ ) = 1144 ( (cid:0) a − ab + 7 b (cid:1) (cid:0) a − b (cid:1) cos(2 ϕ ) + cos ϕ " a − a b +3 ab (cid:0) − ϕ (cid:1) + b (cid:0) ϕ − (cid:1) + 12 ϕ (cid:0) a − a b + 41 ab − b (cid:1) × sin ϕ − (cid:0) a − a b + 12 ab − b (cid:1) − ϕ (cid:0) b − ab (cid:1) sin(2 ϕ ) +3 (cid:0) a − b (cid:1) cos(3 ϕ ) ) , (256)For the higher terms expansions of the solutions of the general relativistic equationof motion of a massive celestial object in Schwarzschild geometry see Mak et al.(2018a), where astrophysical applications of the method (motion of the planet Mer-cury, perihelion precession, and light deflection) are also presented, and discussed indetail. The study of the deflection of light can be done in a similar manner. Gener-ally, by using LADM we can obtain the power series representation of the solutionof the general relativistic equation of motion of planets in Schwarzschild geometryup to an arbitrary precision level as U ( ϕ ) = U ( ϕ ) + U ( ϕ ) + U ( ϕ ) + U ( ϕ ) + U ( ϕ ) + .... .
9. DISCUSSIONS AND CONCLUDING REMARKS
In the present paper we have presented, at an introductory level, some aspectsof the powerful method introduced by G. Adomian to solve nonlinear differential,stochastic and functional equations. Usually this method is known as the AdomianDecomposition Method, or ADM for short. The mathematical technique is essen-tially based on the decomposition of the solution of the nonlinear operator equationinto a series of analytic functions. Each term of the Adomian decomposition series is xxxvii8 Man Kwong MAK, Chun Sing LEUNG, Tiberiu HARKO 39 computed from a polynomial obtained from the power series expansion of an analyticfunction. The Adomian technique is very simple, efficient, and effective, but, on theother hand, it may raise the necessity of the in depth investigations of the convergenceof the series of functions representing the solution of the given nonlinear equationAbbaoui and Cherrualt (1994a,b). The Adomian Decomposition Method has beenused very successfully to obtain semianalytical solutions for many important classesof functional, differential, and integral equations, respectively, with important ap-plications in many fields of fundamental and applied sciences, and engineering, re-spectively. The key to the success of the method relies in the decomposition of thenonlinear term in the differential or integral equations into a series of polynomialsof the form P ∞ n =1 A n , where A n are polynomials known as the Adomian polyno-mials. A large number of algorithms and formulas that can calculate the Adomianpolynomials for all expressions of nonlinearity were introduced in Adomian (1988,1994).Even that the Adomian method is discussed in many articles, a systematic,simple and pedagogical introduction to the subject is still missing. It is the maingoal of the present paper to provide such an introduction, which may be useful forscientists who would like to learn about this method by investigating its simplest ap-plications, before proceeding to more advanced topics. After introducing the basicsof the method, we have discussed in detail the ADM for the standard differentialequations of mathematics, including the linear ordinary differential equation, and theBernoulli, Riccati and Abel equations, respectively. In each case we have describedin detail the general formalism and the particular method, and we have written downexplicitly the Adomian form of the solution. For each type of considered equationswe have also analyzed a concrete example, and we have shown that the Adomiansolution exactly coincides with the analytic solution that can be obtained by usingstandard mathematical methods. This full agreement explicitly indicates the powerof the Adomian Decomposition Method, which could lead to obtaining even the ex-act solution of a given complicated nonlinear ordinary differential equation, or of anintegral equation. We have performed a similar analysis for the second order andthe fifth order ordinary differential equations, by explicitly formulating the full pro-cess of obtaining the series solution. Specific example have been analyzed for eachcase. A very powerful extension of the ADM, the Laplace-Adomian DecompositionMethod was also introduced through the study of a particular example of a secondorder nonlinear differential equation.Finally, we have briefly considered the applications of the Adomian Decompo-sition Method to some important cases of differential equations that play an essentialrole in physics and astronomy. Thus, we have presented in detail the important caseof the Fisher-Kolmogorov equation, a fundamental equation in several fields of biol-ogy, medicine and population dynamics. In this case, after presenting the general al- xxxviii0 The Adomian Decomposition Method with Applications 39 gorithm for the solution, a particular example has been investigated in detail. We havealso described the applications of the ADM in three important fields of astronomy andastrophysics, namely, the determination of the orbits of celestial objects from the Ke-pler equation, obtaining the solutions of the nonlinear Lane-Emden equation, whichplays a fundamental role in the study of the stellar structure, and for the investigationof the general relativistic motion of celestial objects in the Schwarzschild geometry.In all these fields the Adomian Decomposition Method has proven to be a compu-tationally efficient and a highly precise theoretical tool for solving the complicatednonlinear equations describing astronomical and astrophysical phenomena.Certainly the Adomian decomposition method represents a valuable tool forphysicists and engineers working with real physical problems. Hopefully the presentintroduction to this subject will determine scientists working in various fields to be-come more involved in this interesting and fertile field of investigation, which is veryefficient and productive in dealing with large classes of differential/integral equationsand complicated mathematical models describing natural phenomena. REFERENCESAbbaoui, K. and Cherrualt, Y.: 1994a,
Comp. Math. Appl. , 103.Abbaoui, K. and Cherrualt, Y.: 1994b, Mathematical and Computer Modelling , 69.Adomian, G. and Rach, R.: 1983, J. Math. Anal. Applic. , 39.Adomian, G.: 1984a, J. Comput. Appl. Math. , 225.Adomian, G.: 1984b, J. Comput. Appl. Math. , 379.Adomian, G.: 1985, J. Math. Anal. Appl. , 105.Adomian, G.: 1986,
Internat. J. Math. Sci. , 731.Adomian, G.: 1988, J. Math. Anal. Appl. , 501.Adomian, G.: 1994,
Solving Frontier Problems of Physics: the Decomposition Method , Kluwer, Dor-drecht.Adomian, G., Rach, R., and Shawagfeh, N. T.: 1995,
Foundations of Physics Letters , 161.Adomian, G. and Rach, R.: 1996, Mathematical and Computer Modelling , 39.Agom, E. U., Ogunfiditimi, F. U., and Tahir, A.: 2016, British Journal of Mathematics and ComputerScience , 1.Al-Dosary, K. I., Al-Jubouri, N. K., and Abdullah, H. K.: 2008, Applied Mathematical Sciences ,2105.Alshaery, A. and Ebaid, A.: 2017, Acta Astronautica , 27Azreg-A¨ınou, M.: 2010,
Classical and Quantum Gravity , 015012.Babolian, E. and Javadi, S.: 2003, Appl. Math. Comput. , 533.Babolian, E., Javadi, S., and Sadehi, H.: 2004,
Appl. Math. Comput. , 353.Bakodah, H. O.: 2012,
International Journal of Contemporary Mathematical Sciences , 929.Bhalekar, S. and Patade, J.: 2016, American Journal of Computational and Applied Mathematics ,123.Biazar, J., Babolian, E., Nouri, A., and Islam, R.: 2003a, Appl. Math. Comput. , 1582.Biazar, J., Tango, M., Babolian, E., and Islam, R.: 2003b,
Appl. Math. Comput. , 433.Biazar, J., Babolian, E., and Islam, R.: 2004,
Appl. Math. Comput. , 713.Blaga, C.: 2005,
Politropi: o abordare dinamica , Editura Risoprint, Cluj-Napoca, Romania xxxix0 Man Kwong MAK, Chun Sing LEUNG, Tiberiu HARKO 41B¨ohmer, C. G. and Harko, T.: 2010,
Journal of Nonlinear Mathematical Physics International Journal of Bio-Medical Computing , 89.Chandrasekhar, S. 1967, An Introduction to the Study of Stellar Structure , Dover Publications, NewYorkDit˘a, P. and Grama, N.: 1997,
On Adomian’s Decomposition Method for solving differential equation ,arXiv: 9705008.Duan, J.-S., Rach, R., Baleanu, D., and Wazwaz, A.-M.: 2012,
Commun. Frac. Calc. Acta Astronautica , 1.Fatoorehchi, H., Abolghasemi, H., Zarghami, R., and Rach, R.: 2015,
The Canadian Journal of Chem-ical Engineering , 1212.Fisher, R. A.: 1937, Ann. Eugenics , 353.Ghasemi, H., Ghovatmand, M., Zarrinkamar, S., and Hassanabadi, H.: 2014, The European PhysicalJournal Plus , 32.Gbadamosi, B., Adebimpe, O., Akinola, E. I., and Olopade, L. A.: 2012,
Int. J. of Pure and AppliedMathematics , 409.Gonz´alez-Gaxiola, O., Ruiz de Ch´avez, J., and Bernal-Jaquez, R.: 2017, Int. J. Appl. Comput. Math. , 489.Haldar, K.: 2016, Decomposition Analysis Method in Linear and Nonlinear Differential Equations ,CRC Press, Taylor & Francis Group, Boca Raton, London, New YorkHamoud,, A. and Ghadle, K.: 2017,
Korean J. Math. , 323.Harko, T. and Mak, M. K.: 2003, Computers & Mathematics with Applications , 849.Harko, T., Lobo, F. S. N., and Mak, M. K.: 2013, Universal Journal of Applied Mathematics , 101).Harko. T. and Liang, S.-D.: 2016, J. Eng. Math. , 93.Harko, T., Lobo, F. S. N., and Mak, M. K.: 2016, J. Eng. Math. , 193.Harko, T. and Mak, M. K.: 2015a, Journal of Mathematical Physics , 111501.Harko, T. and Mak, M. K.: 2015b, Mathematical Biosciences and Engineering , 41.Harko, T. and Mak, M. K.:2016, Astrophysics and Space Science , 283.Harko, T. and Lobo, F. S. N.: 2018,
Extensions of f(R) Gravity: Curvature-Matter Couplings and Hy-brid Metric-Palatini Theory , Cambridge Monographs on Mathematical Physics, Cambridge, UnitedKingdomHarko, T., Mak, M. K., and Leung, C. S.: 2020,
Romanian Reports in Physics , 116.Harko, T. and Mak, M. K.: 2020a, accepted for publication in Romanian Reports in Physics ,arXiv:2006.07170 [q-bio.PE].Harko, T. and Mak, M. K.: 2020b, arXiv:2009.00434 [q-bio.PE].Harko, T., Mak, M. K., and Lake, M. J.: 2020, arXiv:2011.11072 [gr-qc].Hassan, Y. Q. and Zhu, L. M.: 2008,
Surveys in Mathematics and its applications , 183.Horedt, G. P.: 2004, Polytropes Applications in Astrophysics and Related Fields , Astrophysics andSpace Science Library, Kluwer Academic Publishers, Dordrecht/BostonLondonHosseini, M. M. and Nasabzadeh, H.: 2007,
Appl. Math. Comput. , 117.Hosseini, S. G. and Abbasbandy, S.: 2015,
Mathematical Problems in Engineering , 534754.Jafari, H. and Daftardar-Gejji, V.: 2006a,
Appl. Math. Comput. , 1.Jafari, H. and Daftardar-Gejji, V.: 2006b,
Appl. Math. Comput. , 598.Jin, C. and Liu, M.: 2005,
Appl. Math. Comput. , 953.Kamke, E.: 1959,
Differentialgleichungen: L¨osungsmethoden und L¨osungen , Chelsea, New York.Khuri, S. A.: 2001,
J. Math. Appl. , 141.Khuri, S. A.: 2004, Appl. Math. Comput. , 131.Kolmogorov, A., Petrovskii, I. and Piscounov, N.: 1937,
Bulletin de l’Universit´e d’ ´Etat a Moscou,S´erie Internationale, Section A Math´ematiques et M´ecanique , 1. xl2 The Adomian Decomposition Method with Applications 41Luo, X.-G.: 2005, Appl. Math. Comput. , 570.Mak, M. K., Chan, H. W., and Harko, T.: 2001,
Computers & Mathematics with Applications ,1395.Mak, M. K. and Harko, T.: 2002, Computers & Mathematics with Applications , 91.Mak, M. K. and Harko, T: 2012, Appl. Math. Comput. , 10974.Mak, M. K. and Harko, T.: 2013a,
Appl. Math. Comput. , 7465.Mak, M. K. and Harko, T.: 2013b,
The European Physical Journal
C 73 , 2585.Mak, M. K., Leung, C. S., and Harko, T.: 2018a,
Surveys in Mathematics and its Applications , 183(2018).Mak, M. K., Leung, C. S., and Harko, T.: 2018b, Advances in High Energy Physics , 7093592.Mak, M. K., Leung, C. S., and Harko, T.: 2020, to appear in
Mod. Phys. Lett. A , arXiv:2012.08239[gr-qc].Manafianheris, J.: 2012,
Journal of Mathematical Extension , 1.Momoniat, E., Selway, T. A., and Jina, K.: 2007, Nonlinear analysis: Theory, methods and applica-tions , 2315.Pue-on, P. and Viriyapong, N.: 2012, Applied Mathematical Sciences Comput. Math. Appl. , 17.Rach, R., Wazwaz, A.-M., and Duan, J.-S.: 2015, J. Appl. Math. Comput. , 365.Ruan, J. and Lu, Z.: 2007, Mathematical and Computer Modelling , 1214.Sadat, H.: 2010, Physica Scripta , 045004.Safari, M.: 2011, Advances in Pure Mathematics , 238.Wazwaz, A.-M.: 1999a, Appl. Math. Comput. , 11.Wazwaz, A.-M.: 1999b,
Appl. Math. Comput. , 77.Wazwaz, A.-M.: 2000,
Appl. Math. Comput. , 251.Warwaz, A.-M. and EI-Sayed, S. M.: 2001,
Appl. Math. Comput. , 393.Wazwaz, A.-M. and Gorguis, A.: 2004,
Appl. Math. Comput. , 609.Wazwaz, A.-M.: 2005,
Appl. Math. Comput. , 652.Wazwaz, A.-M.: 2010,
Appl. Math. Comput. , 1304.Wazwaz, A.-M. and Rach, R.: 2011,
Kybernetes , 1305.Wazwaz, A.-M., Rach, R., and Duan, J.-S.: 2013, Appl. Math. Comput , 5004.Zhang, B.-Q., Wu Q.-B., and Luo X.-G.: 2006,
Appl. Math. Comput. , 1495.