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Int. J. Appl. Comput. MathDOI 10.1007/s40819-016-0144-0
ORIGINAL PAPER
Solution of the Nonlinear Kompaneets Equation Throughthe Laplace-Adomian Decomposition Method
O. González-Gaxiola1 · J. Ruiz de Chávez2 ·R. Bernal-Jaquez1
© Springer India Pvt. Ltd. 2016
Abstract The Kompaneets equation is a nonlinear partial differential equation that playsan important role in astrophysics as it describes the spectra of photons in interaction with ararefied electron gas. In spite of its importance, exact solution to this nonlinear equation arerarely found in literature. In this work, we solve this equation and present a new approach toobtain the solution bymeans of the combined use of the Adomian decomposition method andthe Laplace transform (LADM). Besides, we illustrate our approach solving two examples inwhich, two initial photon distributions, well known in astrophysics, are given. We comparethe behaviour of the solutions obtained with the only exact solutions given in the literaturefor the non-relativistic case. Our results indicate that LADM is highly accurate and can beconsidered a very useful and valuable method.
Keywords Nonlinear Kompaneets equation ·Comptonization processes · Photon diffusionequation · Adomian decomposition method · Laplace transform
Mathematics Subject Classification 35Q85 · 35K55 · 85A05
Introduction
Most of the phenomena that arise in the real world can be described by means of nonlinearpartial and ordinary differential equations and, in some cases, by integral or differo-integralequations. However, most of the mathematical methods developed so far, are only capable tosolve linear differential equations. In the 1980’s, George Adomian (1923–1996) introduced a
B O. Gonzá[email protected]
1 Departamento de Matemáticas Aplicadas y Sistemas, Universidad AutónomaMetropolitana-Cuajimalpa, Vasco de Quiroga 4871, Santa Fe,Cuajimalpa, 05300 Mexico, D.F., Mexico
2 Departamento de Matemáticas, Universidad Autónoma Metropolitana-Iztapalapa,San Rafael Atlixco 186, A.P. 55534, Col. Vicentina, Iztapalapa, 09340 Mexico, D.F., Mexico
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Int. J. Appl. Comput. Math
powerfulmethod to solve nonlinear differential equations. Since then, thismethod is known astheAdomiandecompositionmethod (ADM) [3,4]. The technique is based on a decompositionof a solution of a nonlinear differential equation in a series of functions. Each term of theseries is obtained from a polynomial generated by a power series expansion of an analyticfunction. The Adomian method is very simple in an abstract formulation but the difficultyarises in calculating the polynomials that becomes a non-trivial task. This method has widelybeen used to solve equations that come from nonlinear models as well as to solve fractionaldifferential equations [11,12,27].
In the presente work we will utilize the ADM in combination with the Laplace transform(LADM) [30] to solve the Kompaneets equation [22]. This equation is a nonlinear partialdifferential equation that, in astrophysics, is used tomodel the diffusion of photons in a plasmamade up of electrons [25]. We will decomposed the nonlinear terms of this equation usingthe Adomian polynomials and then, in combination with the use of the Laplace transform,we will obtain an algorithm to solve the problem subject to initial conditions. Finally, wewill illustrate our procedure and the quality of the obtained algorithm by means of thesolution of two examples in which the Kompaneets equation is solved for two differentinitial distributions of photons.
Our work is divided in several sections. In “The Adomian Decomposition Method Com-bined with Laplace Transform” section, we present, in a brief and self-contained manner,the LADM. Several references are given to delve deeper into the subject and to study itsmathematical foundation that is beyond the scope of the present work. In “The NonlinearKompaneets Equation” section, we give a brief introduction to the model described by theKompaneets equation and we will establish that LADM can be use to solve this equation inits nonlinear version. Besides, we will show by means of two examples, the quality and pre-cision of our method, comparing the obtained results with the only exact solutions availablein the literature, given in terms of Heun special functions and obtained using separation ofvariables [15]. Finally, in the “Conclusion” section, we summarise our findings and presentour final conclusions.
The Adomian Decomposition Method Combined with Laplace Transform
The ADM is a method to solve ordinary and nonlinear differential equations. Using thismethod is possible to express analytic solutions in terms of a series [4]. In a nutshell, themethod identifies and separates the linear and nonlinear parts of a differential equation.Inverting and applying the highest order differential operator that is contained in the linearpart of the equation, it is possible to express the solution in terms of the rest of the equationaffected by the inverse operator. At this point, the solution is proposed by means of a serieswith terms that will be determined and that give rise to the Adomian polynomials [29]. Thenonlinear part can also be expressed in terms of these polynomials. The initial (or the borderconditions) and the terms that contain the independent variables will be considered as theinitial approximation. In this way and by means of a recurrence relations, it is possible tofind the terms of the series that give the approximate solution of the differential equation.
Given a partial (or ordinary) differential equation
Fu(x, t) = g(x, t) (1)
with initial conditionu(x, 0) = f (x) (2)
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Int. J. Appl. Comput. Math
where F is a differential operator that could, in general, be nonlinear and therefore includessome linear and nonlinear terms.
In general, Eq. (1) could be written as
Ltu(x, t) + Ru(x, t) + Nu(x, t) = g(x, t) (3)
where Lt = ∂∂t , R is a linear operator that includes partial derivatives with respect to x , N is
a nonlinear operator and g is a non-homogeneous term that is u-independent.Solving for Ltu(x, t), we have
Ltu(x, t) = g(x, t) − Ru(x, t) − Nu(x, t). (4)
The LADM consists of applying Laplace transform [30] first on both sides of Eq. (4),obtaining
L {Ltu(x, t)} = L {g(x, t) − Ru(x, t) − Nu(x, t).} (5)
An equivalent expression to (5) is
su(x, s) − u(x, 0) = L {g(x, t) − Ru(x, t) − Nu(x, t)} (6)
In the homogeneous case, g(x, t) = 0, we have
u(x, s) = f (x)
s− 1
sL {Ru(x, t) + Nu(x, t)} (7)
now, applying the inverse Laplace transform to Eq. (7)
u(x, t) = f (x) − L −1[1sL {Ru(x, t) + Nu(x, t)}]. (8)
The ADM method proposes a series solution u(x, t) given by,
u(x, t) =∞∑
n=0
un(x, t) (9)
The nonlinear term Nu(x, t) is given by
Nu(x, t) =∞∑
n=0
An(u0, u1, . . . , un) (10)
where {An}∞n=0 is the so-called Adomian polynomials sequence established in [5] and [29]and, in general, give us term to term:
A0 = N (u0)
A1 = u1N′(u0)
A2 = u2N′(u0) + 1
2u21N
′′(u0)
A3 = u3N′(u0) + u1u2N
′′(u0) + 1
3!u31N
(3)(u0)
A4 = u4N′(u0) +
(1
2u22 + u1u3
)N ′′(u0) + 1
2!u21u2N
(3)(u0) + 1
4!u41N
(4)(u0)
...
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Int. J. Appl. Comput. Math
Other polynomials can be generated in a similar way. For more details, see [5] and [29] andreferences therein. Some other approaches to obtain Adomian’s polynomials can be foundin [13,14].
Using (9) and (10) into Eq. (8), we obtain,
∞∑
n=0
un(x, t) =∞∑
n=0
fn(x) − L −1
[1
sL
{
R∞∑
n=0
un(x, t) +∞∑
n=0
An(u0, u1, . . . , un)
}]
,
(11)here we are considering
f (x) =∞∑
n=0
fn(x),
and thus modifying the method proposed in [31]. From the Eq. (11) we deduce the followingrecurrence formulas{u0(x, t)= f0(x),un+1(x, t) = fn+1(x) − L −1
[ 1sL {Run(x, t)+An(u0, u1, . . . , un)}
], n = 0, 1, 2, . . .
(12)Using (12) we can obtain an approximate solution of (1), (2) using
u(x, t) ≈k∑
n=0
un(x, t), where limk→∞
k∑
n=0
un(x, t) = u(x, t). (13)
It becomes clear that, the ADM, combined with the Laplace transform needs less work incomparison with the traditional ADM. This method decreases considerably the volume ofcalculations. The decomposition procedure of Adomian will be easily set, without linearisingthe problem. In this approach, the solution is found in the form of a convergent series witheasily computed components; in many cases, the convergence of this series is very fastand only a few terms are needed in order to have an idea of how the solutions behave.Convergence conditions of this series are examined by several authors, mainly in [1,2,8,9].Additional references related to the use of the ADM, combined with the Laplace transform,can be found in [21,30].
The Nonlinear Kompaneets Equation
Compton scattering process plays an important role in the description of the interaction ofradiationwithmatter. Since its establishment, it has been applied both in problems that appearin nuclear engineering aswell as tomodel non-relativistic astrophysics phenomena. Scatteringof the Compton type mentioned above is modelled through the Kompaneets equation (alsoknown in astrophysics as photon diffusion equation) which expresses the time change rate ofthe photon number, due to the scattering of isotropic radiation by a non-relativistic isotropicelectron plasma.
The time-dependent Kompaneets equation was first obtained in [22] and, independently,a few years later in [32]. This equation is given by
∂u
∂t= 1
x2∂
∂x
[x4
(∂u
∂x+ u + u2
)]. (14)
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Int. J. Appl. Comput. Math
In the Eq. (14) the dependent variable u(x, t) is the density of the photon gas, the terms uand u2 take into account spontaneous scattering (Compton effect) and induced scattering,respectively. Here t = akTe/mc2 is the dimensionless time, x = h̄ν/kTe is the dimensionlessfrequency, kTe and Ne are the temperature and density of the electrons respectively, a =cNeσ t andσ = 8π
3 (e2/mc2)2 is theThomson scattering cross section. For the non-relativisticcase the Eq. (14) is valid if the two inequalities h̄ν � mc2 and kTe � mc2 are satisfied.
As far aswe know, no exact time-dependent solution of the nonlinearKompaneets equationhas yet been published. Although, recently, some particular solutions and the steady statecase was reported in [6] and in [16,19]. The relativistic case has been recently studied in[10]. Also, solutions have been obtained for the description of the interaction of supernovaremnants with molecular clouds in [20] and the linear case was studied in [24] and in [28].Finally, using separation of variables, analytical solutions were found in [15]. The solutionsgiven in [15] are written in terms of the doubly degenerated Huen functions and the Besselfunctions of first and second kind and are not easy to handle. Besides, others studies basedon the inverse operator method have been carried out as it is shown in [26].
Explicitly calculating the derivatives that appear in Eq. (14), we obtain
∂u
∂t= x2
∂2u
∂x2+ (x2 + 2x2u + 4x)
∂u
∂x+ 4x(u2 + u). (15)
Tomake the description of the the problemcomplete,wewill consider some initial distributionof photons
u(x, 0) = f (x)
In the following section we will develop an algorithm using the method described in “TheAdomian Decomposition Method Combined with Laplace Transform” section in order tosolve the nonlinearKompaneets equation (15)without resort to any truncationor linearization.We have to mention that in [23] it was affirmed that Eq. (15) would never be solved.
Solution of the Nonlinear Kompaneets Equation Through LADM
Comparing (15) with Eq. (4) we have that g(x, t) = 0, Lt and R becomes:
Ltu = ∂
∂tu, Ru =
[x2
∂2
∂x2+ (x2 + 4x)
∂
∂x+ 4x
]u, (16)
while the nonlinear term is given by
Nu = 2x2u∂u
∂x+ 4xu2. (17)
By using now Eq. (12) through the LADM method we obtain recursively{u0(x, t) = f0(x),
un+1(x, t) = fn+1(x)−L −1[1sL {Run(x, t)+An(u0, u1, . . . , un)}
], n = 0, 1, 2, . . .
(18)Note that, the nonlinear term Nu = 2x2u ∂u
∂x + 4xu2 can be split into two terms to facilitatecalculations
N1u = 2x2u∂u
∂x, N2u = 4xu2,
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Int. J. Appl. Comput. Math
from this, we will consider the decomposition of the nonlinear terms into Adomianpolynomials as
N1u = 2x2uux =∞∑
n=0
Pn(u0, u1, . . . , un) (19)
N2u = 4xu2 =∞∑
n=0
Qn(u0, u1, . . . , un). (20)
Using ADM, Eq. (9) gives
u =∞∑
n=0
un,
thus, evaluating we obtain
N1(u) = 2x2uux = 2x2(u0 + u1 + u2 + u3 + u4 + · · · )(u0x + u1x + u2x + u3x + u4x + · · · )
= 2x2{u0u2x + u0u3x + u1u2x + u0u4x
+ u1u3x + u2u2x + u0u5x + u1u4x + u2u3x + u3u2x
+ u1u5x + u2u4x + u3u3x + u4u2x + u2u5x + u3u4x + u4u3x
+ u5u2x + u3u5x + u4u4x + u5u3x
+ u4u5x + u5u4x + u5u5x + u0u0x + u1u0x + u2u0x
+ u3u0x + u4u0x + u5u0x + u0u1x + u1u1x
+ u2u1x + u3u1x + u4u1x + u5u1x + · · · } (21)
and
N2(u) = 4xu2 = 4x(u0 + u1 + u2 + u3 + u4 + u5 + · · · )2= 4x{u20 + 2u0u1 + 2u0u2 + 2u0u3 + 2u0u4 + 2u0u5 + u21
+ 2u1u2 + 2u1u3 + 2u1u4+ 2u1u5 + u22 + 2u2u3 + 2u2u4+ 2u2u5 + u23 + 2u3u4 + 2u3u5 + u24 + 2u4u5 + u25 + · · · } (22)
the above expressions can be rearranged by grouping terms in which the sum of subscriptsof un be the same. This procedure gives the Adomian polynomials for N1(u) and N2(u):
P0 = 2x2u0u0x
P1 = 2x2u0xu1 + 2x2u0u1x
P2 = 2x2u0xu2 + 2x2u1xu1 + 2x2u0u2x
P3 = 2x2u0xu3 + 2x2u1xu2 + 2x2u2xu1 + 2x2u3xu0
P4 = 2x2u0xu4 + 2x2u0u4x + 2x2u1xu3 + 2x2u1u3x + 2x2u2u2x
P5 = 2x2u0xu5 + 2x2u0u5x + 2x2u1xu4 + 2x2u4xu1 + 2x2u3xu2 + 2x2u2xu3
...
Q0 = 4xu20
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Int. J. Appl. Comput. Math
Q1 = 8xu0u1
Q2 = 4xu21 + 8xu0u2
Q3 = 8xu1u2 + 8xu0u3
Q4 = 8xu0u4 + 8xu1u3 + 4xu22
Q5 = 8xu0u5 + 8xu1u4 + 8xu2u3
....
Now, considering (19) and (20), we have
N (u) =∞∑
n=0
An(u0, u1, . . . , un) =∞∑
n=0
((Pn + Qn)(u0, u1, . . . , un)
), (23)
then, the Adomian polynomials are
A0 = 2x2u0u0x + 4xu20,
A1 = 2x2u0xu1 + 2x2u0u1x + 8xu0u1,
A2 = 2x2u0xu2 + 2x2u1xu1 + 2x2u0u2x + 4xu21 + 8xu0u2
A3 = 2x2u0xu3 + 2x2u1xu2 + 2x2u2xu1 + 2x2u3xu0 + 8xu1u2 + 8xu0u3,
A4 = 2x2u0xu4 + 2x2u0u4x + 2x2u1xu3 + 2x2u1u3x + 2x2u2u2x + 8xu0u4 + 8xu1u3
+4xu22,
....
Using the expressions obtained above for Eq. (14), we will illustrate, with two examples, theefectiveness of LADM to solve the nonlinear Kompaneets equation.
Example 1 Using Laplace Adomian decomposition method (LADM), we solve this Kom-paneets equation subject to the initial condition u(0, x) = f (x) = 400
3 x3e−2x . This initialdistribution is known as exponential-power distribution of photons [17].
To use ADM, the Eq. (15) is decomposed in the operators (16) and (17).
Through the LADM we obtain recursively
u0(x, t) = f (x),
u1(x, t) = L −1[1sL {x2u0,xx + (x2 + 4x)u0,x + 4xu0 + A0}
],
u2(x, t) = L −1[1sL {x2u1,xx + (x2 + 4x)u1,x + 4xu1 + A1}
],
......
un+1(x, t) = L −1[1sL {x2un,xx + (x2 + 4x)un,x + 4xun + An}
].
Besides
A0 = 2x2u0u0x + 4xu20 = 1600000
9x7e−4x − 640000
9x8e−4x ,
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Int. J. Appl. Comput. Math
A1 = 2x2u0xu1 + 2x2u0u1x + 8xu0u1 = t
27e−6x
(179.2 × 108x11 + 153.6 × 108x13
+ 307.2 × 107x13 + 172.8 × 106x7e2x − 206.4 × 106x8e2x + 729.6 × 105x9e2x
− 76.8 × 105x10e2x),
A2 = 320 000
81t2x7e−8x
⎛
⎜⎜⎜⎜⎜⎜⎝
153600000x13 − 422400000x12 − 40960000x11
+153600000x10 − 357800000x9 + 288000x8e2x
+531450 000x8 − 3292800x7e2x
+13070400x6e2x − 144x5e4x − 21276000x5e2x
+2592x4e4x + 11928000x4e2x − 16956x3e4x
+49698x2e4x − 64431xe4x + 29160e4x
⎞
⎟⎟⎟⎟⎟⎟⎠
.
With the above, we have
u0(x, t) = 400
3x3e−2x ,
u1(x, t) = L −1[1sL {x2u0,xx + (x2 + 4x)u0,x + 4xu0 + A0}
]
= L −1[1sL {400
9x3e−2x (4000x4e−2x − 39x − 1600x5e−2x + 6x2 + 54)}
]
= L −1[ 1
s2
(4009
x3e−2x (4000x4e−2x − 39x − 1600x5e−2x + 6x2 + 54))]
= 400
9t x3e−2x (4000x4e−2x − 39x − 1600x5e−2x + 6x2 + 54),
and proceeding in a similar way we get
u2(x, t) = L −1[1sL {x2u1,xx + (x2 + 4x)u1,x + 4xu0 + A1}
]
= t2
54e−6x
(3072000000x13 − 15360000000x15 + 17920000000x11
−30720000x10e2x + 261120000x9e2x − 668160000x8e2x + 14400x7e4x
+ 508.8 × 106x7e2x − 230400x6e4x + 1166400x5e4x − 2152800x4e4x
+ 1166400x3e4x),
u3(x, t) = L −1[1sL {x2u2,xx + (x2 + 4x)u2,x + 4xu0 + A2}
]
= 320 000
243t3e−8x
⎛
⎜⎜⎜⎜⎜⎜⎝
153600000x20 − 422400000x19 − 40960000x18
+153600000x17 − 357800 000x16 + 288000x15e2x
+531 450000x15 − 3292 800x14e2x
+13070400x13e2x − 144x12e4x − 21276 000x12e2x
+2592x11e4x + 11928000x11e2x − 16956x10e4x
+49698x9e4x − 64431x8e4x + 29160x7e4x
⎞
⎟⎟⎟⎟⎟⎟⎠
−800
81t3e−6x
⎛
⎜⎜⎜⎜⎜⎜⎜⎜⎝
288000000x17 − 1776000000x16 + 2534400000x15
+312960000x14 − 735360000x13 + 230400x12e2x
+1579200000x12 − 3532800x11e2x − 1724800000x11
+19747200x10e2x − 18x9e4x − 49838400x9e2x
+513x8e4x + 56146800x8e2x − 5256x7e4x
−22260000x7e2x + 24318x6e4x − 52146x5e4x
+47151x4e4x − 13122x3e4x
⎞
⎟⎟⎟⎟⎟⎟⎟⎟⎠
.
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Int. J. Appl. Comput. Math
Fig. 1 Plot of the approximate solution uLADM given by Eq. (24) for (x, t) ∈ (0, 5] × [0, 0.004]
Thus, the solution approximate of the nonlinear Kompaneets equation (15) is given by:
uLADM = u0(x, t) + u1(x, t) + u2(x, t) + u3(x, t) (24)
In the other hand, in [15] and using the method of separation of variables, the authors haveobtained an exact solution, in terms of the doubly degenerated Huen functions and their firstderivatives [18], given by
u(x, t) = 150x
1 + e300(2t+3)
{[(2x5 − 2x4 − 4x3 + 4x4 + 2x − 2)HuenD2
+ 16x2HuenDHuenD′ + 16x2HuenDHuenD′]
×∫
dx
HuenD+ (x5 − x4 − 2x3 + 2x2 + x − 1)HuenD2
+ 8x2HuenDHuenD′ − 4x4 + 8x2 − 4
}× e300(2t+3)
×[3x(x2 − 1)2HuenD
(2
∫dx
xHuenD2 + 1)]−1
,
(25)
where
HuenD = HuenD(0,
1199
2,−2399
2, 600,
x2 + 1
x2 − 1
)
In Fig. 1 we plot the approximate solution given by Eq. (24) meanwhile, in Fig. 2 we plotthe exact solution recently obtained in [15] using separation of variables. Finally, in Fig. 3 weplot both, the approximate solution and the exact solution. The approximate solution appearsunder the exact solution but as it can be observed, the approximate solution, obtained usingLADM, converges to the exact solution in such a way that it becomes difficult to distinguishthem. All the numerical work and the graphics was accomplished with the Mathematicasoftware package.
As it is shown in Fig. 1, the approximate solution has the same profile as the profile of thenumerical solution obtained in [17] for a similar initial distribution.
Example 2 Using Laplace Adomian decomposition method (LADM), we solve this Kompa-neets equation subject to the initial condition u(0, x) = f (x) = 1
ex−1 ; this initial distributionis known as Bose-Einstein type distribution of photons.
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Int. J. Appl. Comput. Math
Fig. 2 Plot of the exact solution given by Eq. (25) for (x, t) ∈ (0, 5] × [0, 0.004]
Fig. 3 The plot of the exact solution given by (25) appears above the plot of the approximate solution givenin (24).
To use LADM, the Eq. (15) is decomposed in the operators (16) and (17). Here we willconsider modifying the method proposed in [31], i.e.
f (x) = 1
ex − 1= −1
2+ 1
x+ 1
12x − 1
720x3 + 1
30240x5 − 1
1209600x7 + O(x8)
Through the LADM we obtain recursively
u0(x, t) = f0(x),
u1(x, t) = f1(x) + L −1[1sL {x2u0,xx + (x2 + 4x)u0,x + 4xu0 + A0}
],
u2(x, t) = f2(x) + L −1[1sL {x2u1,xx + (x2 + 4x)u1,x + 4xu1 + A1}
],
......
un+1(x, t) = fn+1(x) + L −1[1sL {x2un,xx + (x2 + 4x)un,x + 4xun + An}
].
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Int. J. Appl. Comput. Math
Calculating
A0 = 2x2u0u0x + 4xu20 = x,
A1 = 2x2u0xu1 + 2x2u0u1x + 8xu0u1 = 5x2t − 3,
A2 = 2x2u0xu2 + 2x2u1xu1 + 2x2u0u2x + 4xu21 + 8xu0u2
= − 5
12x2 + 10t2x2 − 8t x + 2
x+ 6t2x3 + 6t,
A3 = 2x2u0xu3 + 2x2u1xu2 + 2x2u2xu1 + 2x2u3xu0 + 8xu1u2 + 8xu0u3 = 2
3x − 6t
− 8
xt − 5
3t x2 − 16t2x + t x3 + 20t2x2 + 40
3t3x2 + 8t3x3 − 14t3x4 − 6t2 + 7
720x4;
with the above, we have
u0(x, t) = −1
2,
u1(x, t) = 1
x+ L −1
[1sL {x2u0,xx + (x2 + 4x)u0,x + 4xu0 + A0}
]
= 1
x+ L −1
[1sL {−x}
]= 1
x− L −1
[ xs
]= 1
x− t x,
u2(x, t) = 1
12x + L −1
[1sL {x2u1,xx + (x2 + 4x)u1,x + 4xu1 + A1}
]
= 1
12x + L −1
[1sL
{− 2
x− 4t x
} ]= 1
12x − L −1
[ 1
s2
( 2x
)+ 1
s3(4x)
]
= 1
12x − 2
xt − 2t2x,
u3(x, t) = − 1
720x3 + L −1
[1sL {x2u2,xx + (x2 + 4x)u2,x + 4xu2 + A2}
]
= − 1
720x3 + L −1
[1sL
{−1
3x2 + 1
3x + 2
x+ t
(4
x− 8x + 8
)
+t2(8x2 − 8x + 6x3)} ]
= − 1
720x3+L −1
[L { 1
s2
(−1
3x2+ 1
3x+ 2
x
)+ 1
s3
(4
x− 8x+8
)
+ 1
s4(16x2−16x + 12x3)}
]
=− 1
720x3+t
(−1
3x2+ 1
3x+ 2
x
)+t2
(2
x−4x+4
)+ t3
(8
3x2 − 8
3x + 2x3
),
finally, proceeding in a similar way we get
u4(x, t) = 1
30240x5 + L −1
[1sL {x2u3,xx + (x2 + 4x)u3,x + 4xu2 + A3}
]
= 1
30240x5 + t
(2
3x − 1
40x3
)+ t2
(2
3x − 5
3x2 − 1
2x3 − 6
x
)
+ t3(
−32
3x − 8
3x
)+ t4(78x3 + 52x2 − 16x).
123
Int. J. Appl. Comput. Math
Thus the solution approximate of the nonlinear Kompaneets equation (15) is:
uLADM = u0(x, t) + u1(x, t) + u2(x, t) + u3(x, t) + u4(x, t) (26)
In the other hand, in [15], the authors have solved the nonlinear Kompaneets equation usingthe method of separation of variables. They have obtained the exact solution in terms of theBessel functions [7] of first Ivi and second Kvi kind given by
u(x, t) = 1√x3
{21e− 2t−7x
14
[(207x − 30
7
)x Iv1
(− x
2
)+
(5x2 − 120
7x + 60
7
)Iv0
(− x
2
)
+(207x2 − 60
7x)Iv−1
(− x
2
)+ 5
7x2 Iv−2
(− x
2
) ]+ 35e− 2t−7x
14
[(207x2− 30
7x)
× Kv1
(− x
2
)+
(5x2− 120
7x + 60
7
)Kv0
(− x
2
)+
(207x2 − 60
7x)Kv−1
(− x
2
)
+ 5
7x2Kv−2
(− x
2
) ]}. (27)
In the Tables1 and 2, we compare the solution of the nonlinear Kompaneets equation givenin (26) with the exact solution (27) recently obtained in [15] in which the solution of (15) isobtained directly under the assumption of separation of variables; as it can be observed, erroris smaller than 10−3. Moreover, in Fig. 4, we show the approximate solutions for successivevalues of time t = 0.01, t = 0.02, t = 0.03 and t = 0.05 and they are compared withthe exact solution (27) obtained in [15]. All the numerical work was accomplished with theMathematica software package.
From Tables1 and 2, we can conclude that the difference between the exact and theobtained LADM approximate solution is very small. This fact tells us about the effectivenessand accuracy of the LADM method.
Summary and Conclusions
Very few exact solutions of the nonlinear Kompaneets equation were known in the literature.In this work, we have obtained accurate approximate solutions for the Kompaneets nonlinearpartial differential equation using the ADM in combination with the Laplace transform,illustrating, in this way, the use of LADM in the solution of nonlinear partial differentialequations. We have chosen the Kompaneets equation due to its importance in astrophysicsas it describes the interaction between radiation and matter.
In order to show the accuracy and efficiency of our method, we have solved two examples,comparing our results with the exact solution of the equation that was obtained in [15].Our results show that LADM produces highly accurate solutions in complicated nonlinearproblems.
123
Int. J. Appl. Comput. Math
Tabl
e1
Tablefort=
0.01
andt=
0.02
xt=
0.01
t=
0.02
uLADM
uex
[15]
Error
uLADM
uex
[15]
Error
0.25
3.48
4414
8543
423.48
4421
2345
676.38
02×
10−6
3.50
7701
8876
763.50
8387
3109
516.85
42×
10−4
0.50
1.51
8013
3597
881.51
8023
2122
159.85
24×
10−6
1.53
2181
7864
551.53
2938
7765
327.56
99×
10−4
0.75
0.86
8060
4732
450.86
8076
7543
111.62
81×
10−5
0.88
3218
1510
230.88
3983
2588
917.65
11×
10−4
1.00
0.54
4606
6798
940.54
4624
5987
231.79
19×
10−5
0.56
4263
8532
280.56
5034
5656
817.70
71×
10−4
1.25
0.34
9447
4239
930.34
9478
9742
233.15
05×
10−5
0.37
5845
7906
600.37
6627
9265
397.82
14×
10−4
1.50
0.21
6349
2063
490.21
6384
7983
283.55
92×
10−5
0.25
1350
0952
380.25
2136
8425
797.86
75×
10−4
1.75
0.11
6799
4913
090.11
6846
7390
634.72
48×
10−5
0.16
2137
6275
990.16
2926
9211
027.89
29×
10−4
2.00
0.03
6447
0899
470.03
6898
4380
914.51
35×
10−4
0.09
3825
7566
140.10
1724
7943
107.89
01×
10−3
123
Int. J. Appl. Comput. Math
Tabl
e2
Tablefort=
0.03
andt=
0.05
xt=
0.03
t=
0.05
uLADM
uex
[15]
Error
uLADM
uex
[15]
Error
0.25
3.51
4898
7184
043.51
5212
5841
873.13
87×
10−4
3.51
9241
0960
093.52
0811
6641
871.57
06×
10−3
0.50
1.53
6820
2822
881.53
7493
0745
376.72
79×
10−4
1.53
9903
9864
551.54
1494
0825
361.59
01×
10−3
0.75
0.88
8619
9279
320.88
9456
0353
198.36
11×
10−4
0.89
2615
8871
340.89
5255
1344
022.63
92×
10−3
1.00
0.57
1738
4198
940.57
2558
5589
418.20
14×
10−4
0.57
7633
9865
610.58
1976
7068
694.34
27×
10−3
1.25
0.38
6331
0818
060.38
7150
2235
618.19
14×
10−4
0.39
4904
9656
600.40
1551
1184
936.64
62×
10−3
1.50
0.26
5685
8938
490.26
6524
5462
138.38
65×
10−4
0.27
7670
6507
940.28
7216
9167
889.54
63×
10−3
1.75
0.18
1143
6854
600.18
1980
2345
678.36
55×
10−4
0.19
7272
1429
760.21
0322
5165
471.30
50×
10−2
2.00
0.11
7892
3456
790.11
9174
3456
781.28
17×
10−3
0.13
9353
7566
140.15
6517
6427
491.71
64×
10−2
123
Int. J. Appl. Comput. Math
Approx. for t=0.05
Approx. for t=0.03
Approx. for t=0.02
Approx. for t=0.01
Exact
0.5 1.0 1.5 2.0x
0.5
1.0
1.5
2.0
2.5
3.0
3.5u(x,t)
Fig. 4 Graph of Example 2, the values of uLADM and uex for t = 0.01, 0.02, 0.03, 0.05
We therefore, conclude that the Laplace-ADM is a notable non-sophisticated powerful toolthat produces high quality approximate solutions for nonlinear partial differential equationsusing simple calculations and that attains converge with only few terms.
Acknowledgments We would like to thank anonymous referees for their constructive comments and sug-gestions that helped to improve the paper.
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