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Relation between Lane–Emden solutions and radial solutions to the elliptic Heavenly equation on a disk
New Astronomy 37 (2015) 42–47
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New Astronomy journal homepage: www.elsevier.com/locate/newast
Relation between Lane–Emden solutions and radial solutions to the elliptic Heavenly equation on a disk Robert A. Van Gorder Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom
h i g h l i g h t s A map between the elliptic Heavenly and Lane–Emden equations is provided. Simple analytical solutions are constructed through this transform. Quasi-stationary symmetric Heavenly solutions can be viewed as Lane–Emden solutions.
a r t i c l e
i n f o
Article history: Received 22 October 2014 Accepted 2 December 2014 Available online 9 December 2014 Communicated by W. Soon Keywords: Heavenly equation Gravitational instanton Lane–Emden equation of the second kind Analytical approximation
a b s t r a c t We provide a transformation between a type of solution to a Lane–Emden equation of second kind and a solution of the elliptic Heavenly equation on a disk. By doing so, we show that any solution of this Lane– Emden equation of second kind corresponds to an infinite family of solutions to the Heavenly equation. This Lane–Emden equation is naturally formulated as a boundary value problem, which makes it somewhat distinct from the initial value problem versions in the literature. We obtain simple analytical solutions of this Lane–Emden equation and associated boundary value problem, and then we use these analytical solutions to construct a family of solutions for the elliptic Heavenly equation. The obtained solutions are radial solutions to the Heavenly equation; that is, they exhibit radial symmetry. In effect, we obtain a relation between radially-symmetric self-dual gravitational instantons and the Lane–Emden approximation to the structure of a neutron star. In other words, the radially symmetric neutron star under the Lane–Emden model can be seen as a special type of gravitational instanton. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction The Heavenly equation (also known as the Boyer-Finley equation, dispersionless Toda equation, and SU(1)-Toda equation in some contexts) is a nonlinear complex partial differential equation often defined on complex variables (Manas and Alonso, 2002). Regarding one application area, the Heavenly equation can be used to describe self-dual vacuum Einstein spaces (Boyer and Finley, 1982; Gibbons and Hawking, 1978). More broadly, theses kinds of equations govern the dynamics of gravitational instantons (Gibbons and Hawking, 1978; Gibbons and Hawking, 1979). We shall consider the elliptic Heavenly equation in real variables, which reads
uxx þ uyy ¼ jðeu Þtt ;
ð1:1Þ
where j is a non-zero parameter, and u : D ½0; 1Þ, where D is a disk. Without loss of generality, we take D to be the unit disk. We E-mail address: [email protected] http://dx.doi.org/10.1016/j.newast.2014.12.001 1384-1076/Ó 2014 Elsevier B.V. All rights reserved.
take uðx; y; tÞ to be constant on the boundary of D (for any fixed time t), so we take say uðx; y; 0Þj@D ¼ C, as we assume initial data which is radially symmetric. The value of C can be scaled out of the boundary value problem (and can be taken to be 1 or 0 as warranted). Note that with an appropriate rescaling of the length coordinates, we can consider j 2 f1; 1g (that is, only the sign of j matters). As j ¼ 0 gives simply the well-known Laplace equation, we omit this case from consideration. The Eq. (1.1) is elliptic in the space variables. An analogous hyperbolic equation,
uxx þ uyy ¼ jðeu Þtt ;
ð1:2Þ
can be defined and this gives the Boyer-Finley equation (or dispersionless Toda equation, in some contexts). For references, see (Boyer and Finley, 1982; Calderbank and Tod, 2001; Ward, 1990). We derive a transformation between the elliptic Heavenly Eq. (1.1 and a Lane–Emden equation of the second kind (in two spatial dimensions). The Lane–Emden equation of the second kind typically models an isothermal, self-gravitating gas sphere (Van
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R.A. Van Gorder / New Astronomy 37 (2015) 42–47
Gorder, 2011; Van Gorder, 2011). This equation has previously been used to study neutron stars. In the present paper, we are able to show that any solution to the Lane–Emden equation of the second kind satisfying relevant boundary conditions gives a threeparameter family of radial solutions to the Heavenly equation. The resulting solutions are effectively pseudo-stationary states of the Heavenly equation. The outline of this paper is as follows. In Section 2, we derive the transformations between solutions of the Lane–Emden equation of the second kind and the elliptic form of the Heavenly Eq. (1.1). As mentioned in Manas and Alonso (2002), the complex Heavenly equation can be put into the form of a nonlinear Liouville equation. Here, we show that quasi-stationary radially symmetric solutions of the elliptic Heavenly equation can be put into correspondence with a particular class of Lane–Emden equations. In Section 3, we set out to analytically solve the resulting Lane– Emden equations for the j ¼ 1 and j ¼ 1 regimes. A series approach with a shooting method is first applied, and is shown the give sufficiently low residual errors. Then, the homotopy analysis method is used to obtain solutions with marginally lower residual errors. This allows us to analytically describe the solutions of quasi-stationary radially symmetric Heavenly equations for both the j ¼ 1 and j ¼ 1 cases. Concluding remarks are given in Section 4.
The solution to the Heavenly Eq. (1.1) is then reduced to the solution of (2.5). Appropriate boundary conditions on / (upon scaling the constant boundary data) are
/0 ð0Þ ¼ 0 and /ð1Þ ¼ 1 when a2 > 0 and
/0 ð0Þ ¼ 0 and /ð1Þ ¼ 0
3. Analytic approximation to the Lane–Emden equation of the second kind Various approximations to the Lane–Emden equation of the second kind have been considered in the literature, albeit these solutions are usually for the initial value problem formulation. As we are interested in the boundary value problem arising from mapping the elliptic Heavenly equation on the disk into a Lane– Emden problem, it is necessary to use the boundary value problem formulation. Here we shall determine a residual error minimizing solution. We must solve the boundary value problem 2
Let us first introduce the logarithmic change of dependent variable u ¼ ln v , so that (1.1) is transformed into
v xx þ v yy v þ v v v2 2 x
2 y
¼ jv tt :
ð2:1Þ
Consider a separable solution (a quasi-stationary solution) In order for (2.1) to become stationary, we need
v ðx; y; tÞ ¼ f ðtÞWðx; yÞ. f ðtÞ ¼
a2 2
t 2 þ a1 t þ a0 ;
ð2:2Þ
a three-parameter family of functions. In this case, we find that (2.1) reduces to the eigenvalue problem 2
2
Wxx þ Wyy Wx þ Wy ¼ ja2 W: W W2
ð2:3Þ
It is natural to make the transformation Wðx; yÞ ¼ expðja2 Uðx; yÞÞ, which puts (2.3) into the form
Uxx þ Uyy þ expðja2 UÞ ¼ 0: Finally, assuming a radial pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ y2 , we obtain
ð2:4Þ solution
Uðx; yÞ ¼ /ðrÞ
where
r¼ 2
d / dr
2
þ
1 d/ þ expðja2 /Þ ¼ 0: r dr
ð2:5Þ
This is a Lane–Emden equation of the second kind in two dimensions. It differs from the familiar three-dimensional case, in that the singular term is 2=r in the three-dimensional model. In general, this term takes the form ðd 1Þ=r for d-dimensional Cartesian space. Unwrapping the transformations, we find that any solution /ðrÞ to the Lane–Emden equation of the second kind gives us a threeparameter family of exact pseudo-stationary solutions u to the Heavenly equation, via the transformation
a
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 uðx; y; tÞ ¼ ja2 /ð x2 þ y2 Þ þ ln t þ a1 t þ a0 : 2
ð2:6Þ
ð2:8Þ
when a2 < 0. In the following section, we shall obtain solutions to (2.5) through analytical methods. We then use these solutions to construct solutions of the elliptic Heavenly equation on the disk.
d / 2. Transformation between the Heavenly equation and the Lane–Emden equation of second kind
ð2:7Þ
dr
2
þ
1 d/ þ expðja2 /Þ ¼ 0; r dr
/0 ð0Þ ¼ 0;
/ð1Þ ¼ 1:
ð3:1Þ
To simplify calculations, we pick the constant a2 so that a2 ¼ sgnðjÞ=j. For any other values of a2 , we can always rescale the time variable to reduce the problem to one of these two cases. This has the effect of ensuring that the temporal factor always exists (the coefficient of t 2 must always be positive), and this always means that one may take the condition /ð1Þ ¼ 1 with the scaling C ¼ 1 (as we commented before, the boundary value C can always be scaled in this manner). In the event that the boundary condition on the Heavenly equation satisfies C ¼ 0 it is most simple to take /ð1Þ ¼ 0 as the relevant boundary condition. Computationally, this problem will differ from the standard Lane–Emden problem of the second kind. The reason for this is because the standard Lane–Emden problem of the second kind is given as an initial value problem, with conditions /ð0Þ ¼ 0 and /0 ð0Þ ¼ 0. This initial value problem has been solved in a variety of ways in the literature. By considering a boundary value problem, we complicate the solution procedure a bit, as the solution must satisfy set conditions at two end points. This originates from the fact that the Heavenly equation we consider takes the form of an elliptic boundary value problem. To deal with this type of problem, we shall consider two analytical approaches, first a shooting method and second the homotopy analysis method. Both of these approaches enable us to deal with the two-point boundary conditions. Therefore, while there is plenty of literature on the Lane–Emden initial value problem of the second kind, the present results are needed to study the relevant boundary value problem, as it naturally arises upon transforming the elliptic Heavenly equation with homogeneous boundary data. 3.1. A shooting method approach If
j > 0, then we solve the boundary value problem
1 /00 þ /0 þ e/ ¼ 0; r
/0 ð0Þ ¼ 0;
/ð1Þ ¼ 1
ð3:2Þ
by use of a Taylor series approach, employing a shooting parameter /ð0Þ ¼ A. In order to use the Taylor series, we need to construct it at a single point (we take r ¼ 0), so a condition like /ð0Þ ¼ A is necessary. We then determine the value of A by enforcing the
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R.A. Van Gorder / New Astronomy 37 (2015) 42–47
condition /ð1Þ ¼ 1. In terms of A, the Taylor series solution to (3.2) is given by
/ðrÞ ¼ A
eA 2
2
r2
e2A 6
2
r4
e3A 32
r6
8
e4A 213
r8 ;
ð3:3Þ
where we have truncated the expansion at order eight. The condition /ð1Þ ¼ 1 implies
1¼A
eA 2
2
e2A 2
6
e3A 32
8
e4A
213
ð3:4Þ
;
and we find that this equation is satisfied for A ¼ 1:086205077. We find that the maximal residual error of this solution over the domain r 2 ½0; 1 is 5:6089 106 . We plot this solution in Fig. 1. If j < 0, then a2 ¼ 1=j and we solve the boundary value problem
1 /00 þ /0 þ e/ ¼ 0; r
/0 ð0Þ ¼ 0;
/ð1Þ ¼ 0:
ð3:5Þ
Note that we have modified the boundary data on the boundary of the disk. This is because of the fact that solutions may not exist in the j < 0 case when the boundary data is too large. We obtain the solution
/ðrÞ ¼ A
e
A
22
r2 þ
e
2A
26
r4
3A
e
3 28
r6 þ
4A
e
213
r8 ;
Fig. 2. Plot of the solution /ðrÞ given in Eq. (3.6) when A ¼ 0:3166207777.
ð3:6Þ
where we have truncated the expansion at order eight. The value of A for which /ð1Þ ¼ 0 is found to be A ¼ 0:3166207777. We plot this solution in Fig. 2. The residual errors increase monotonically over the domain, and the maximal residual error of this solution over the domain r 2 ½0; 1 is 4:9777513 103 . So, while the j > 0 Taylor series solution was reasonable with an error near 106 , the j < 0 Taylor series solution is not nearly as accurate.
differential equations in the past (Abbasbandy, 2006; Abbasbandy, 2007; Liao and Campo, 2002; Liao, 2003). This approach has even been applied to study various Lane–Emden type equations in the literature (Bataineh et al., 2007; Bataineh et al., 2009; Van Gorder and Vajravelu, 2008; Liao, 2003). First take the case j > 0. Consider the auxiliary linear operator 2
L½/ ¼
d / dr
2
þ
1 d/ ; r dr
ð3:7Þ
3.2. A homotopy approach
and let N denote the nonlinear operator
In the present section, we apply the homotopy analysis method (Liao, 2003; Liao, 2012; Van Gorder and Vajravelu, 2009; Vajravelu and Van Gorder, 2013) in order to obtain approximate analytical solutions of the relevant Lane–Emden boundary value problems. This approach has been used to study a number of nonlinear
N½/ ¼
2
d / dr
2
þ
1 d/ þ expð/Þ: r dr
ð3:8Þ
We then construct the homotopy
HðqÞ½/ ¼ ð1 qÞL½/ hqN½/:
ð3:9Þ
We assume a five term approximation to the solution of the form
^ /ðrÞ ¼ /0 ðrÞ þ /1 ðrÞq þ /2 ðrÞq2 þ /3 ðrÞq3 þ /4 ðrÞq4 ;
ð3:10Þ
and we shall make use of the convergence control parameter, h, in order to minimize the residual error in this approximation. The terms in the approximation are found by plugging (3.10) into (3.9) and then collecting terms with respect to q in order to obtain linear ODEs for each /k ðrÞ. The solutions may be found successively like
/00 ð0Þ ¼ 0;
L½/0 ¼ 0;
/0 ðrÞ
L½/1 ¼ he
;
/0 ð1Þ ¼ 1;
/01 ð0Þ ¼ 0;
ð3:11Þ
/1 ð1Þ ¼ 0;
ð3:12Þ
L½/2 ¼ ð1 þ hÞL½/1 h/1 ðrÞe/0 ðrÞ ¼ ðð1 þ hÞh h/1 ðrÞÞe/0 ðrÞ ;
/02 ð0Þ ¼ 0;
/2 ð1Þ ¼ 0; ð3:13Þ
and so on. Solving these equation successively, we find that
/0 ðrÞ ¼ 1; /1 ðrÞ ¼ Fig. 1. Plot of the solution /ðrÞ given in Eq. (3.3) when A ¼ 1:086205077.
ð3:14Þ h ð1 r2 Þ; 4e
ð3:15Þ
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R.A. Van Gorder / New Astronomy 37 (2015) 42–47
/2 ðrÞ ¼
h 2 ð1 r2 Þðhr 16ð1 þ hÞe 3hÞ; 64e2
ð3:16Þ
¼ ðð1 þ hÞh þ h/1 ðrÞÞe/0 ðrÞ ;
Note that h remains a free parameter. We shall now apply the so-called optimal homotopy analysis method (Liao, 2010; Van Gorder, 2012; Van Gorder, 2012; Ghoreishi et al., 2012; Van Gorder, 2012), in which one selects the convergence control parameter, h, in a way that minimizes the error inherent in the approximate analytical solution. In order to minimize the error of this approximation, we consider the absolution residual error at any point in the domain, given by
^ hÞj: Resðr; hÞ ¼ jN½/ðr;
ð3:17Þ
Let us define
ðhÞ ¼ maxResðr; hÞ:
ð3:18Þ
r2½0;1
In particular, we shall seek h so that the maximal residual error ðhÞ corresponding to each h is minimal. Since we are working over the compact set r 2 ½0; 1, it is possible to perform this type of optimization, and we find that h ¼ 0:9393 gives the minimal error, with ð0:9393Þ ¼ 4:9943 106 . This error is somewhat lower than that of the Taylor series expansion. The approximate solution is then
^ /ðrÞ ¼ 1:086204345 0:08437252357r 2 0:001780496632r 4 0:00004958392363r 6 0:000001740401376r 8 :
ð3:19Þ
In Fig. 3, we plot the absolute error between this solution and the analytical solution obtained via the shooting method. As we see, both methods agree nicely over the spatial domain. The agreement is to within 7:33 107 . We next consider the case j < 0. We keep L as it is, and modify the nonlinear operator accordingly: 2
N½/ ¼
d / dr
2
þ
1 d/ þ expð/Þ: r dr
/00 ð0Þ ¼ 0; /0 ðrÞ
L½/1 ¼ he
;
ð3:20Þ
/0 ð1Þ ¼ 0;
/01 ð0Þ ¼ 0;
/02 ð0Þ ¼ 0;
/2 ð1Þ ¼ 0;
ð3:23Þ
and so on. Solving these equation successively, we find that
/0 ðrÞ ¼ 0;
ð3:24Þ
/1 ðrÞ ¼
he ð1 r 2 Þ; 4
ð3:25Þ
/2 ðrÞ ¼
h 2 2 ð1 r 2 Þðhe r 2 þ 16ð1 þ hÞe 3he Þ; 64
ð3:26Þ
We find that the minimal residual errors occur at h ¼ 1:228. At this value of h, the maximal residual error is 2:744594 103 . This is nearly half of the maximal residual error of the Taylor series solution. In Fig. 4, we plot the absolute error between this solution and the analytical solution obtained via the shooting method. As we see, both methods agree nicely over the spatial domain. The agreement is to within 5 104 . 3.3. Radial quasi-stationary solutions of the Heavenly equation From the analytical solutions to the Lane–Emden equation of the second kind with boundary conditions given in this section, one may recover the radially symmetric solutions to the elliptic Heavenly equation through the relation (2.6). In terms of the Taylor series approximations found above, we may write
sgnðjÞe2A sgnðjÞ eA sgnðjÞ 2 2 uðx; y; tÞ ¼ sgnðjÞ A x þ y 22 26 o 2 sgnðjÞ 2 2 3 t þ a1 t þ a0 ; þ ln x þ y2 þ O x2 þ y2 2j ð3:27Þ
Again, in order to obtain a solution, we shall consider the boundary data /ð1Þ ¼ 0. The rest of the solution procedure follows as before. The solutions may be found successively like
L½/0 ¼ 0;
L½/2 ¼ ð1 þ hÞL½/1 þ h/1 ðrÞe/0 ðrÞ
/1 ð1Þ ¼ 0;
ð3:21Þ
where A will depend on the choice of j (as shown in the previous subsection). While the solutions exhibit growth in time, this rate of growth gradually decreases. Indeed, the growth rate of the quasistationary solutions of the Heavenly equation scales like
@u 1 @t t
as
t ! 1:
ð3:28Þ
ð3:22Þ
^ hÞj between the analytical solution /ðrÞ Fig. 3. Absolute error EðrÞ ¼ j/ðrÞ /ðr; ^ hÞ. given in Eq. (3.3) when A ¼ 1:086205077 and the homotopy solution /ðr;
^ hÞj between the analytical solution /ðrÞ Fig. 4. Absolute error EðrÞ ¼ j/ðrÞ /ðr; ^ hÞ. given in Eq. (3.6) when A ¼ 0:3166207777 and the homotopy solution /ðr;
46
R.A. Van Gorder / New Astronomy 37 (2015) 42–47
Fig. 5. Plot of the stationary part of the solution uðx; y; tÞ to the elliptic Heavenly equation when j > 0 and the boundary condition takes the form uðx; y; 0Þj@D ¼ 1.
Fig. 6. Plot of the stationary part of the solution uðx; y; tÞ to the elliptic Heavenly equation when j < 0 and the boundary condition takes the form uðx; y; 0Þj@D ¼ 0.
We see that the j ¼ 1 solutions have a maximal value on the boundary and a minimal value at the origin, at any fixed value of time. Conversely, the j ¼ 1 solutions have a minimal value on the boundary and a maximal value at the origin at any fixed value of time. In either case, the solutions exhibit logarithmic growth in time. In Fig. 5, we plot the representative j > 0 solution, while in Fig. 6 we plot the representative j < 0 solution. 4. Conclusions We have obtained a mapping between the elliptic Heavenly equation and a type of Lane–Emden equation (of the second kind) on a disk. Doing so, we obtain solutions for the Heavenly equation in terms of the solutions to this Lane–Emden equation. In effect, we
obtain a relation between self-dual gravitational instantons and the Lane–Emden approximation to the structure of a neutron star. This makes sense, as both equations are essentially variations of a more general type of nonlinear Liouville partial differential equations. The types of solutions to the Heavenly equation we obtain maintain radial symmetry for all time values, which makes sense as the Lane–Emden model implicitly assumes a radially symmetric self-gravitating gas ball. We therefore obtain a relation between radially-symmetric quasi-stationary self-dual gravitational instantons and the Lane–Emden approximation to the structure of a neutron star (which itself assumes radial symmetry). In other words, physically the radially symmetric neutron star under the Lane– Emden model can be seen as a special type of gravitational instanton, corresponding to a solution of the elliptic Heavenly equation. In order to analytically study solutions of the elliptic Heavenly equation, we have applied two distinct analytical methods. In the first method, we obtain a series solution centered around r ¼ 0 assuming some boundary value /ð0Þ ¼ A > 0. Assuming this series is valid over the unit interval r 2 ½0; 1, we pick A such that the conditions at r ¼ 1 are satisfied. This gives a fairly good approximate analytical solution with relatively few terms. We also apply the homotopy analysis method in order to obtain approximate analytical solutions. The technique here is rather different. One constructs a homotopy between a linear differential operator and the nonlinear Lane–Emden equation, and uses this to iteratively construct an approximate solution to the nonlinear equation of interest. The method depends on a convergence control parameter, h, and this parameter is selected so that the residual errors are minimized. In this way, what we do constitutes an application of the optimal homotopy analysis method; see (Liao, 2010). Comparing the analytical solutions obtained by each approach, we see that there is strong agreement between the two solutions (as shown in Figs. 2 and 4). Regarding the actual solutions, for each of the j ¼ 1 and j ¼ 1 cases the solutions monotonically decrease or increase over the domain, so that the solutions are maximal at either r ¼ 0 on along the boundary (as seen in Figs. 5 and 6). The time effects globally increase the value of the solutions in a uniform manner across the domain. Since the time effects are uniform, the solutions are indeed quasi-stationary. If we interpret the function uðx; y; tÞ as a density, then the density of the gas cloud is maximal either at the center (j < 0) or at the boundary (j > 0) of the circular cloud. In the former case, the gas is attractive, and therefore a density maximum occurs at the center of the cloud. This is the case more relevant to the physics of stars. For the case j > 0 case, one can view the obtained as a gaseous ring, with relatively minimal density in the interior of the ring and most of the gas density occurring near the boundary. As we see for the mathematics, both types of solutions to the Heavenly equation can be obtained from simpler Lane–Emden equations (in the case of radial symmetry), with the difference being in the value of the boundary condition away from the origin. If this boundary condition is large (in a relative sense), then the solutions will radially increase (corresponding to a ring-like density configuration). On the other hand, if the boundary condition is small, the solutions will radially decrease (corresponding to a density maximum at the origin). In this way, we have some qualitative understanding of the quasi-stationary radial solutions possible from the Heavenly equation. References Abbasbandy, S., 2006. Phy. Lett. A 360, 109. Abbasbandy, S., 2007. Int. Commun. Heat Mass Transfer 34, 380. Bataineh, A.S., Noorani, M.S.M., Hashim, I., 2007. Phy. Lett. A 371, 72. Bataineh, A.S., Noorani, M.S.M., Hashim, I., 2009. Commun. Nonlinear Sci. Numer. Simul. 14, 1121. Boyer, C.P., Finley, J.D., 1982. J. Math. Phys. 23, 1126.
R.A. Van Gorder / New Astronomy 37 (2015) 42–47 Calderbank, D.M.J., Tod, P., 2001. Differ. Geom. Appl. 14, 199. Ghoreishi, M., Ismail, A.I.B., Alomari, A.K., Sami Bataineh, A., 2012. Commun. Nonlinear Sci. Numer. Simul. 17, 1163. Gibbons, G.W., Hawking, S.W., 1978. Phys. Lett. B 78, 430. Gibbons, G.W., Hawking, S.W., 1979. Commun. Math. Phy. 66, 291. Liao, S.J., 2003. J. Fluid Mech. 488, 189. Liao, S., 2003. Appl. Math. Comput. 142, 1. Liao, S.J., 2003. Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman & Hall/CRC Press, Boca Raton. Liao, S., 2010. Commun. Nonlinear Sci. Numer. Simul. 15, 2315. Liao, S.J., 2012. Homotopy Analysis Method in Nonlinear Differential Equations. Springer & Higher Education Press, Heidelberg. Liao, S.J., Campo, A., 2002. J. Fluid Mech. 453, 411.
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Manas, M., Alonso, L.M., 2002. A hodograph transformation which applies to the Heavenly Eq. arXiv:nlin/0209050v1. Vajravelu, K., Van Gorder, R.A., 2013. Nonlinear Flow Phenomena and Homotopy Analysis: Fluid Flow and Heat Transfer. Springer, Heidelberg. Van Gorder, R.A., 2011. Celestial Mech. Dyn. Astron. 109, 137. Van Gorder, R.A., 2011. New Astron. 16, 65. Van Gorder, R.A., 2012. Int. J. Non-Linear Mech. 47, 1. Van Gorder, R.A., 2012. Commun. Nonlinear Sci. Numer. Simul. 17, 1233. Van Gorder, R.A., 2012. Numer. Algorithms 61, 613. Van Gorder, R.A., Vajravelu, K., 2008. Phy. Lett. A 372, 6060. Van Gorder, R.A., Vajravelu, K., 2009. Commun. Nonlinear Sci. Numer. Simul. 14, 4078. Ward, R.S., 1990. Classical Quantum Gravity 7, L95.
Contents lists available at ScienceDirect
New Astronomy journal homepage: www.elsevier.com/locate/newast
Relation between Lane–Emden solutions and radial solutions to the elliptic Heavenly equation on a disk Robert A. Van Gorder Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom
h i g h l i g h t s A map between the elliptic Heavenly and Lane–Emden equations is provided. Simple analytical solutions are constructed through this transform. Quasi-stationary symmetric Heavenly solutions can be viewed as Lane–Emden solutions.
a r t i c l e
i n f o
Article history: Received 22 October 2014 Accepted 2 December 2014 Available online 9 December 2014 Communicated by W. Soon Keywords: Heavenly equation Gravitational instanton Lane–Emden equation of the second kind Analytical approximation
a b s t r a c t We provide a transformation between a type of solution to a Lane–Emden equation of second kind and a solution of the elliptic Heavenly equation on a disk. By doing so, we show that any solution of this Lane– Emden equation of second kind corresponds to an infinite family of solutions to the Heavenly equation. This Lane–Emden equation is naturally formulated as a boundary value problem, which makes it somewhat distinct from the initial value problem versions in the literature. We obtain simple analytical solutions of this Lane–Emden equation and associated boundary value problem, and then we use these analytical solutions to construct a family of solutions for the elliptic Heavenly equation. The obtained solutions are radial solutions to the Heavenly equation; that is, they exhibit radial symmetry. In effect, we obtain a relation between radially-symmetric self-dual gravitational instantons and the Lane–Emden approximation to the structure of a neutron star. In other words, the radially symmetric neutron star under the Lane–Emden model can be seen as a special type of gravitational instanton. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction The Heavenly equation (also known as the Boyer-Finley equation, dispersionless Toda equation, and SU(1)-Toda equation in some contexts) is a nonlinear complex partial differential equation often defined on complex variables (Manas and Alonso, 2002). Regarding one application area, the Heavenly equation can be used to describe self-dual vacuum Einstein spaces (Boyer and Finley, 1982; Gibbons and Hawking, 1978). More broadly, theses kinds of equations govern the dynamics of gravitational instantons (Gibbons and Hawking, 1978; Gibbons and Hawking, 1979). We shall consider the elliptic Heavenly equation in real variables, which reads
uxx þ uyy ¼ jðeu Þtt ;
ð1:1Þ
where j is a non-zero parameter, and u : D ½0; 1Þ, where D is a disk. Without loss of generality, we take D to be the unit disk. We E-mail address: [email protected] http://dx.doi.org/10.1016/j.newast.2014.12.001 1384-1076/Ó 2014 Elsevier B.V. All rights reserved.
take uðx; y; tÞ to be constant on the boundary of D (for any fixed time t), so we take say uðx; y; 0Þj@D ¼ C, as we assume initial data which is radially symmetric. The value of C can be scaled out of the boundary value problem (and can be taken to be 1 or 0 as warranted). Note that with an appropriate rescaling of the length coordinates, we can consider j 2 f1; 1g (that is, only the sign of j matters). As j ¼ 0 gives simply the well-known Laplace equation, we omit this case from consideration. The Eq. (1.1) is elliptic in the space variables. An analogous hyperbolic equation,
uxx þ uyy ¼ jðeu Þtt ;
ð1:2Þ
can be defined and this gives the Boyer-Finley equation (or dispersionless Toda equation, in some contexts). For references, see (Boyer and Finley, 1982; Calderbank and Tod, 2001; Ward, 1990). We derive a transformation between the elliptic Heavenly Eq. (1.1 and a Lane–Emden equation of the second kind (in two spatial dimensions). The Lane–Emden equation of the second kind typically models an isothermal, self-gravitating gas sphere (Van
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R.A. Van Gorder / New Astronomy 37 (2015) 42–47
Gorder, 2011; Van Gorder, 2011). This equation has previously been used to study neutron stars. In the present paper, we are able to show that any solution to the Lane–Emden equation of the second kind satisfying relevant boundary conditions gives a threeparameter family of radial solutions to the Heavenly equation. The resulting solutions are effectively pseudo-stationary states of the Heavenly equation. The outline of this paper is as follows. In Section 2, we derive the transformations between solutions of the Lane–Emden equation of the second kind and the elliptic form of the Heavenly Eq. (1.1). As mentioned in Manas and Alonso (2002), the complex Heavenly equation can be put into the form of a nonlinear Liouville equation. Here, we show that quasi-stationary radially symmetric solutions of the elliptic Heavenly equation can be put into correspondence with a particular class of Lane–Emden equations. In Section 3, we set out to analytically solve the resulting Lane– Emden equations for the j ¼ 1 and j ¼ 1 regimes. A series approach with a shooting method is first applied, and is shown the give sufficiently low residual errors. Then, the homotopy analysis method is used to obtain solutions with marginally lower residual errors. This allows us to analytically describe the solutions of quasi-stationary radially symmetric Heavenly equations for both the j ¼ 1 and j ¼ 1 cases. Concluding remarks are given in Section 4.
The solution to the Heavenly Eq. (1.1) is then reduced to the solution of (2.5). Appropriate boundary conditions on / (upon scaling the constant boundary data) are
/0 ð0Þ ¼ 0 and /ð1Þ ¼ 1 when a2 > 0 and
/0 ð0Þ ¼ 0 and /ð1Þ ¼ 0
3. Analytic approximation to the Lane–Emden equation of the second kind Various approximations to the Lane–Emden equation of the second kind have been considered in the literature, albeit these solutions are usually for the initial value problem formulation. As we are interested in the boundary value problem arising from mapping the elliptic Heavenly equation on the disk into a Lane– Emden problem, it is necessary to use the boundary value problem formulation. Here we shall determine a residual error minimizing solution. We must solve the boundary value problem 2
Let us first introduce the logarithmic change of dependent variable u ¼ ln v , so that (1.1) is transformed into
v xx þ v yy v þ v v v2 2 x
2 y
¼ jv tt :
ð2:1Þ
Consider a separable solution (a quasi-stationary solution) In order for (2.1) to become stationary, we need
v ðx; y; tÞ ¼ f ðtÞWðx; yÞ. f ðtÞ ¼
a2 2
t 2 þ a1 t þ a0 ;
ð2:2Þ
a three-parameter family of functions. In this case, we find that (2.1) reduces to the eigenvalue problem 2
2
Wxx þ Wyy Wx þ Wy ¼ ja2 W: W W2
ð2:3Þ
It is natural to make the transformation Wðx; yÞ ¼ expðja2 Uðx; yÞÞ, which puts (2.3) into the form
Uxx þ Uyy þ expðja2 UÞ ¼ 0: Finally, assuming a radial pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ y2 , we obtain
ð2:4Þ solution
Uðx; yÞ ¼ /ðrÞ
where
r¼ 2
d / dr
2
þ
1 d/ þ expðja2 /Þ ¼ 0: r dr
ð2:5Þ
This is a Lane–Emden equation of the second kind in two dimensions. It differs from the familiar three-dimensional case, in that the singular term is 2=r in the three-dimensional model. In general, this term takes the form ðd 1Þ=r for d-dimensional Cartesian space. Unwrapping the transformations, we find that any solution /ðrÞ to the Lane–Emden equation of the second kind gives us a threeparameter family of exact pseudo-stationary solutions u to the Heavenly equation, via the transformation
a
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 uðx; y; tÞ ¼ ja2 /ð x2 þ y2 Þ þ ln t þ a1 t þ a0 : 2
ð2:6Þ
ð2:8Þ
when a2 < 0. In the following section, we shall obtain solutions to (2.5) through analytical methods. We then use these solutions to construct solutions of the elliptic Heavenly equation on the disk.
d / 2. Transformation between the Heavenly equation and the Lane–Emden equation of second kind
ð2:7Þ
dr
2
þ
1 d/ þ expðja2 /Þ ¼ 0; r dr
/0 ð0Þ ¼ 0;
/ð1Þ ¼ 1:
ð3:1Þ
To simplify calculations, we pick the constant a2 so that a2 ¼ sgnðjÞ=j. For any other values of a2 , we can always rescale the time variable to reduce the problem to one of these two cases. This has the effect of ensuring that the temporal factor always exists (the coefficient of t 2 must always be positive), and this always means that one may take the condition /ð1Þ ¼ 1 with the scaling C ¼ 1 (as we commented before, the boundary value C can always be scaled in this manner). In the event that the boundary condition on the Heavenly equation satisfies C ¼ 0 it is most simple to take /ð1Þ ¼ 0 as the relevant boundary condition. Computationally, this problem will differ from the standard Lane–Emden problem of the second kind. The reason for this is because the standard Lane–Emden problem of the second kind is given as an initial value problem, with conditions /ð0Þ ¼ 0 and /0 ð0Þ ¼ 0. This initial value problem has been solved in a variety of ways in the literature. By considering a boundary value problem, we complicate the solution procedure a bit, as the solution must satisfy set conditions at two end points. This originates from the fact that the Heavenly equation we consider takes the form of an elliptic boundary value problem. To deal with this type of problem, we shall consider two analytical approaches, first a shooting method and second the homotopy analysis method. Both of these approaches enable us to deal with the two-point boundary conditions. Therefore, while there is plenty of literature on the Lane–Emden initial value problem of the second kind, the present results are needed to study the relevant boundary value problem, as it naturally arises upon transforming the elliptic Heavenly equation with homogeneous boundary data. 3.1. A shooting method approach If
j > 0, then we solve the boundary value problem
1 /00 þ /0 þ e/ ¼ 0; r
/0 ð0Þ ¼ 0;
/ð1Þ ¼ 1
ð3:2Þ
by use of a Taylor series approach, employing a shooting parameter /ð0Þ ¼ A. In order to use the Taylor series, we need to construct it at a single point (we take r ¼ 0), so a condition like /ð0Þ ¼ A is necessary. We then determine the value of A by enforcing the
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R.A. Van Gorder / New Astronomy 37 (2015) 42–47
condition /ð1Þ ¼ 1. In terms of A, the Taylor series solution to (3.2) is given by
/ðrÞ ¼ A
eA 2
2
r2
e2A 6
2
r4
e3A 32
r6
8
e4A 213
r8 ;
ð3:3Þ
where we have truncated the expansion at order eight. The condition /ð1Þ ¼ 1 implies
1¼A
eA 2
2
e2A 2
6
e3A 32
8
e4A
213
ð3:4Þ
;
and we find that this equation is satisfied for A ¼ 1:086205077. We find that the maximal residual error of this solution over the domain r 2 ½0; 1 is 5:6089 106 . We plot this solution in Fig. 1. If j < 0, then a2 ¼ 1=j and we solve the boundary value problem
1 /00 þ /0 þ e/ ¼ 0; r
/0 ð0Þ ¼ 0;
/ð1Þ ¼ 0:
ð3:5Þ
Note that we have modified the boundary data on the boundary of the disk. This is because of the fact that solutions may not exist in the j < 0 case when the boundary data is too large. We obtain the solution
/ðrÞ ¼ A
e
A
22
r2 þ
e
2A
26
r4
3A
e
3 28
r6 þ
4A
e
213
r8 ;
Fig. 2. Plot of the solution /ðrÞ given in Eq. (3.6) when A ¼ 0:3166207777.
ð3:6Þ
where we have truncated the expansion at order eight. The value of A for which /ð1Þ ¼ 0 is found to be A ¼ 0:3166207777. We plot this solution in Fig. 2. The residual errors increase monotonically over the domain, and the maximal residual error of this solution over the domain r 2 ½0; 1 is 4:9777513 103 . So, while the j > 0 Taylor series solution was reasonable with an error near 106 , the j < 0 Taylor series solution is not nearly as accurate.
differential equations in the past (Abbasbandy, 2006; Abbasbandy, 2007; Liao and Campo, 2002; Liao, 2003). This approach has even been applied to study various Lane–Emden type equations in the literature (Bataineh et al., 2007; Bataineh et al., 2009; Van Gorder and Vajravelu, 2008; Liao, 2003). First take the case j > 0. Consider the auxiliary linear operator 2
L½/ ¼
d / dr
2
þ
1 d/ ; r dr
ð3:7Þ
3.2. A homotopy approach
and let N denote the nonlinear operator
In the present section, we apply the homotopy analysis method (Liao, 2003; Liao, 2012; Van Gorder and Vajravelu, 2009; Vajravelu and Van Gorder, 2013) in order to obtain approximate analytical solutions of the relevant Lane–Emden boundary value problems. This approach has been used to study a number of nonlinear
N½/ ¼
2
d / dr
2
þ
1 d/ þ expð/Þ: r dr
ð3:8Þ
We then construct the homotopy
HðqÞ½/ ¼ ð1 qÞL½/ hqN½/:
ð3:9Þ
We assume a five term approximation to the solution of the form
^ /ðrÞ ¼ /0 ðrÞ þ /1 ðrÞq þ /2 ðrÞq2 þ /3 ðrÞq3 þ /4 ðrÞq4 ;
ð3:10Þ
and we shall make use of the convergence control parameter, h, in order to minimize the residual error in this approximation. The terms in the approximation are found by plugging (3.10) into (3.9) and then collecting terms with respect to q in order to obtain linear ODEs for each /k ðrÞ. The solutions may be found successively like
/00 ð0Þ ¼ 0;
L½/0 ¼ 0;
/0 ðrÞ
L½/1 ¼ he
;
/0 ð1Þ ¼ 1;
/01 ð0Þ ¼ 0;
ð3:11Þ
/1 ð1Þ ¼ 0;
ð3:12Þ
L½/2 ¼ ð1 þ hÞL½/1 h/1 ðrÞe/0 ðrÞ ¼ ðð1 þ hÞh h/1 ðrÞÞe/0 ðrÞ ;
/02 ð0Þ ¼ 0;
/2 ð1Þ ¼ 0; ð3:13Þ
and so on. Solving these equation successively, we find that
/0 ðrÞ ¼ 1; /1 ðrÞ ¼ Fig. 1. Plot of the solution /ðrÞ given in Eq. (3.3) when A ¼ 1:086205077.
ð3:14Þ h ð1 r2 Þ; 4e
ð3:15Þ
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R.A. Van Gorder / New Astronomy 37 (2015) 42–47
/2 ðrÞ ¼
h 2 ð1 r2 Þðhr 16ð1 þ hÞe 3hÞ; 64e2
ð3:16Þ
¼ ðð1 þ hÞh þ h/1 ðrÞÞe/0 ðrÞ ;
Note that h remains a free parameter. We shall now apply the so-called optimal homotopy analysis method (Liao, 2010; Van Gorder, 2012; Van Gorder, 2012; Ghoreishi et al., 2012; Van Gorder, 2012), in which one selects the convergence control parameter, h, in a way that minimizes the error inherent in the approximate analytical solution. In order to minimize the error of this approximation, we consider the absolution residual error at any point in the domain, given by
^ hÞj: Resðr; hÞ ¼ jN½/ðr;
ð3:17Þ
Let us define
ðhÞ ¼ maxResðr; hÞ:
ð3:18Þ
r2½0;1
In particular, we shall seek h so that the maximal residual error ðhÞ corresponding to each h is minimal. Since we are working over the compact set r 2 ½0; 1, it is possible to perform this type of optimization, and we find that h ¼ 0:9393 gives the minimal error, with ð0:9393Þ ¼ 4:9943 106 . This error is somewhat lower than that of the Taylor series expansion. The approximate solution is then
^ /ðrÞ ¼ 1:086204345 0:08437252357r 2 0:001780496632r 4 0:00004958392363r 6 0:000001740401376r 8 :
ð3:19Þ
In Fig. 3, we plot the absolute error between this solution and the analytical solution obtained via the shooting method. As we see, both methods agree nicely over the spatial domain. The agreement is to within 7:33 107 . We next consider the case j < 0. We keep L as it is, and modify the nonlinear operator accordingly: 2
N½/ ¼
d / dr
2
þ
1 d/ þ expð/Þ: r dr
/00 ð0Þ ¼ 0; /0 ðrÞ
L½/1 ¼ he
;
ð3:20Þ
/0 ð1Þ ¼ 0;
/01 ð0Þ ¼ 0;
/02 ð0Þ ¼ 0;
/2 ð1Þ ¼ 0;
ð3:23Þ
and so on. Solving these equation successively, we find that
/0 ðrÞ ¼ 0;
ð3:24Þ
/1 ðrÞ ¼
he ð1 r 2 Þ; 4
ð3:25Þ
/2 ðrÞ ¼
h 2 2 ð1 r 2 Þðhe r 2 þ 16ð1 þ hÞe 3he Þ; 64
ð3:26Þ
We find that the minimal residual errors occur at h ¼ 1:228. At this value of h, the maximal residual error is 2:744594 103 . This is nearly half of the maximal residual error of the Taylor series solution. In Fig. 4, we plot the absolute error between this solution and the analytical solution obtained via the shooting method. As we see, both methods agree nicely over the spatial domain. The agreement is to within 5 104 . 3.3. Radial quasi-stationary solutions of the Heavenly equation From the analytical solutions to the Lane–Emden equation of the second kind with boundary conditions given in this section, one may recover the radially symmetric solutions to the elliptic Heavenly equation through the relation (2.6). In terms of the Taylor series approximations found above, we may write
sgnðjÞe2A sgnðjÞ eA sgnðjÞ 2 2 uðx; y; tÞ ¼ sgnðjÞ A x þ y 22 26 o 2 sgnðjÞ 2 2 3 t þ a1 t þ a0 ; þ ln x þ y2 þ O x2 þ y2 2j ð3:27Þ
Again, in order to obtain a solution, we shall consider the boundary data /ð1Þ ¼ 0. The rest of the solution procedure follows as before. The solutions may be found successively like
L½/0 ¼ 0;
L½/2 ¼ ð1 þ hÞL½/1 þ h/1 ðrÞe/0 ðrÞ
/1 ð1Þ ¼ 0;
ð3:21Þ
where A will depend on the choice of j (as shown in the previous subsection). While the solutions exhibit growth in time, this rate of growth gradually decreases. Indeed, the growth rate of the quasistationary solutions of the Heavenly equation scales like
@u 1 @t t
as
t ! 1:
ð3:28Þ
ð3:22Þ
^ hÞj between the analytical solution /ðrÞ Fig. 3. Absolute error EðrÞ ¼ j/ðrÞ /ðr; ^ hÞ. given in Eq. (3.3) when A ¼ 1:086205077 and the homotopy solution /ðr;
^ hÞj between the analytical solution /ðrÞ Fig. 4. Absolute error EðrÞ ¼ j/ðrÞ /ðr; ^ hÞ. given in Eq. (3.6) when A ¼ 0:3166207777 and the homotopy solution /ðr;
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R.A. Van Gorder / New Astronomy 37 (2015) 42–47
Fig. 5. Plot of the stationary part of the solution uðx; y; tÞ to the elliptic Heavenly equation when j > 0 and the boundary condition takes the form uðx; y; 0Þj@D ¼ 1.
Fig. 6. Plot of the stationary part of the solution uðx; y; tÞ to the elliptic Heavenly equation when j < 0 and the boundary condition takes the form uðx; y; 0Þj@D ¼ 0.
We see that the j ¼ 1 solutions have a maximal value on the boundary and a minimal value at the origin, at any fixed value of time. Conversely, the j ¼ 1 solutions have a minimal value on the boundary and a maximal value at the origin at any fixed value of time. In either case, the solutions exhibit logarithmic growth in time. In Fig. 5, we plot the representative j > 0 solution, while in Fig. 6 we plot the representative j < 0 solution. 4. Conclusions We have obtained a mapping between the elliptic Heavenly equation and a type of Lane–Emden equation (of the second kind) on a disk. Doing so, we obtain solutions for the Heavenly equation in terms of the solutions to this Lane–Emden equation. In effect, we
obtain a relation between self-dual gravitational instantons and the Lane–Emden approximation to the structure of a neutron star. This makes sense, as both equations are essentially variations of a more general type of nonlinear Liouville partial differential equations. The types of solutions to the Heavenly equation we obtain maintain radial symmetry for all time values, which makes sense as the Lane–Emden model implicitly assumes a radially symmetric self-gravitating gas ball. We therefore obtain a relation between radially-symmetric quasi-stationary self-dual gravitational instantons and the Lane–Emden approximation to the structure of a neutron star (which itself assumes radial symmetry). In other words, physically the radially symmetric neutron star under the Lane– Emden model can be seen as a special type of gravitational instanton, corresponding to a solution of the elliptic Heavenly equation. In order to analytically study solutions of the elliptic Heavenly equation, we have applied two distinct analytical methods. In the first method, we obtain a series solution centered around r ¼ 0 assuming some boundary value /ð0Þ ¼ A > 0. Assuming this series is valid over the unit interval r 2 ½0; 1, we pick A such that the conditions at r ¼ 1 are satisfied. This gives a fairly good approximate analytical solution with relatively few terms. We also apply the homotopy analysis method in order to obtain approximate analytical solutions. The technique here is rather different. One constructs a homotopy between a linear differential operator and the nonlinear Lane–Emden equation, and uses this to iteratively construct an approximate solution to the nonlinear equation of interest. The method depends on a convergence control parameter, h, and this parameter is selected so that the residual errors are minimized. In this way, what we do constitutes an application of the optimal homotopy analysis method; see (Liao, 2010). Comparing the analytical solutions obtained by each approach, we see that there is strong agreement between the two solutions (as shown in Figs. 2 and 4). Regarding the actual solutions, for each of the j ¼ 1 and j ¼ 1 cases the solutions monotonically decrease or increase over the domain, so that the solutions are maximal at either r ¼ 0 on along the boundary (as seen in Figs. 5 and 6). The time effects globally increase the value of the solutions in a uniform manner across the domain. Since the time effects are uniform, the solutions are indeed quasi-stationary. If we interpret the function uðx; y; tÞ as a density, then the density of the gas cloud is maximal either at the center (j < 0) or at the boundary (j > 0) of the circular cloud. In the former case, the gas is attractive, and therefore a density maximum occurs at the center of the cloud. This is the case more relevant to the physics of stars. For the case j > 0 case, one can view the obtained as a gaseous ring, with relatively minimal density in the interior of the ring and most of the gas density occurring near the boundary. As we see for the mathematics, both types of solutions to the Heavenly equation can be obtained from simpler Lane–Emden equations (in the case of radial symmetry), with the difference being in the value of the boundary condition away from the origin. If this boundary condition is large (in a relative sense), then the solutions will radially increase (corresponding to a ring-like density configuration). On the other hand, if the boundary condition is small, the solutions will radially decrease (corresponding to a density maximum at the origin). In this way, we have some qualitative understanding of the quasi-stationary radial solutions possible from the Heavenly equation. References Abbasbandy, S., 2006. Phy. Lett. A 360, 109. Abbasbandy, S., 2007. Int. Commun. Heat Mass Transfer 34, 380. Bataineh, A.S., Noorani, M.S.M., Hashim, I., 2007. Phy. Lett. A 371, 72. Bataineh, A.S., Noorani, M.S.M., Hashim, I., 2009. Commun. Nonlinear Sci. Numer. Simul. 14, 1121. Boyer, C.P., Finley, J.D., 1982. J. Math. Phys. 23, 1126.
R.A. Van Gorder / New Astronomy 37 (2015) 42–47 Calderbank, D.M.J., Tod, P., 2001. Differ. Geom. Appl. 14, 199. Ghoreishi, M., Ismail, A.I.B., Alomari, A.K., Sami Bataineh, A., 2012. Commun. Nonlinear Sci. Numer. Simul. 17, 1163. Gibbons, G.W., Hawking, S.W., 1978. Phys. Lett. B 78, 430. Gibbons, G.W., Hawking, S.W., 1979. Commun. Math. Phy. 66, 291. Liao, S.J., 2003. J. Fluid Mech. 488, 189. Liao, S., 2003. Appl. Math. Comput. 142, 1. Liao, S.J., 2003. Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman & Hall/CRC Press, Boca Raton. Liao, S., 2010. Commun. Nonlinear Sci. Numer. Simul. 15, 2315. Liao, S.J., 2012. Homotopy Analysis Method in Nonlinear Differential Equations. Springer & Higher Education Press, Heidelberg. Liao, S.J., Campo, A., 2002. J. Fluid Mech. 453, 411.
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Manas, M., Alonso, L.M., 2002. A hodograph transformation which applies to the Heavenly Eq. arXiv:nlin/0209050v1. Vajravelu, K., Van Gorder, R.A., 2013. Nonlinear Flow Phenomena and Homotopy Analysis: Fluid Flow and Heat Transfer. Springer, Heidelberg. Van Gorder, R.A., 2011. Celestial Mech. Dyn. Astron. 109, 137. Van Gorder, R.A., 2011. New Astron. 16, 65. Van Gorder, R.A., 2012. Int. J. Non-Linear Mech. 47, 1. Van Gorder, R.A., 2012. Commun. Nonlinear Sci. Numer. Simul. 17, 1233. Van Gorder, R.A., 2012. Numer. Algorithms 61, 613. Van Gorder, R.A., Vajravelu, K., 2008. Phy. Lett. A 372, 6060. Van Gorder, R.A., Vajravelu, K., 2009. Commun. Nonlinear Sci. Numer. Simul. 14, 4078. Ward, R.S., 1990. Classical Quantum Gravity 7, L95.