1 Introduction
The following singular boundary value problem,
with the following boundary conditions
is known as ThomasFermi equation ([Numerical1998]). This equation was studied by Thomas [Thomas1927] and Fermi [Fermi1927]. One can impose three types of boundary conditions, which reflects three different states of atoms:
a) ionized atom
b) neutral atom
c) isolated neutral atom
ThomasFermi equation may be written in a generalised form known as EmdenFowler equation ([Numerical1998, Numerical1996])
(1)  
(2)  
(3) 
where and are real numbers, , . Mehta et al. [MEHTA1971] studied it by using hypergeometric function and studied the EmdenFowler equations. Wong [JSWW1975] presented survey on EmdenFowler equations. Rosenau [ROSENAU1984] by simple change of variable generated a one parameter family of integrable EmdenFowler equations. P M Lima [Numerical1996, LIMA1999, LIMA2009] analyzed numerical methods based on finite differences/shooting methods for solving EmdenFowler equation. Mohammadi et al. [Mohammadi2019] used an iterative method to the singular and nonlinear fractional partial differential version of EmdenFowler equations. Haar wavelets operational matrix of fractional integration is be applied to compute the solution with the Picard technique.
Pikulin [Pikulin2019] considered the EmdenFowler equations on the halfline and on the interval and by assuming that the exponent in the coefficient of the nonlinear term is rational, he derived new parametric representations.
In this paper we consider (1) subject to the boundary conditions (2) and (3). We develop two approaches. In the first approach we use Newton’s quasilinearization and arrive at an iteration scheme which is linear at each iteration. At each iteration we apply Haar wavelet collocation approach. After certain iterations the solutions does not vary, so we stop and the solution that we get at this stage is the solution of the EmdenFowler equation. In the second approach we first use Haar wavelet collocation and arrive at the set of nonlinear equations. We then solve the system of nonlinear equation to get the solution of the EmdenFowler equation. To the best of our knowledge the results do not exists in literature.
2 Haar Wavelet
[Haar, pp.710] Let us consider the interval . Let us define ,where is maximum level of resolution. Let us divide into subintervals of equal length . The wavelet number is calculated by ,here and (here ). The Haar wavelet is defined as,
(4) 
where,
(5) 
Above equations are valid if . For , it is defined as,
(6) 
For ,
(7) 
The width of the wavelet is,
.
The integral is defined as,
(8) 
here is the order of integration.
Using (4), we will calculate these integrals analytically and by doing it we obtain,
(9) 
for . For we have and,
(10) 
Remark: In example we have used (for ) which is defined as,
and it is calculated with the help of (8) by keeping the limits as to .
For computation by using Haar wavelet, we make use of the method of collocation. Here, the grid points are defined by,
(11) 
and collocation points are defined by,
(12) 
For computational point of view, we introduce the Haar matrices . The order of these matrices are . These matrices are defined by,
For , the matrices , and are defined by,
, ,
For , the matrices , are defined by,
,
3 Haar Wavelet Collocation Method
In this section we develop two methods for solving generalized EmdenFowler equation.
3.1 Quasilinearization Approach
By quasilinearization ([An2013, Higher2019]) generalized EmdenFowler equation is linearized then at each iteration Haar wavelet collocation method is used to compute the solution. After some iteration the solution converges to the solution of the EmdenFowler equation. Consider the problem considerd by Lima [Numerical1996],
(13) 
(14) 
where and are real numbers, , . Now applying quasilinearization in above equation,
(15) 
(16) 
To apply Haar wavelet method [Haar_solving_HODE2008], we assume that
(17) 
where are wavelet coefficients. Now integrate above equation from to twice we get,
(18) 
(19) 
Now apply boundary conditions (16) in (19) and find then substitute (19),(17) in (15) and we get
(20) 
Now, we can solve above equation by collocation method.
3.2 Newton Raphson Method
[Haar, pp.33]
Consider a nonlinear ordinary differential equation,
(21) 
where is a nonlinear function. Now again applying Haar wavelet method [Haar_solving_HODE2008], let us assume
(22) 
and integrate above equation times. Obtained results are discretized by collocation method then substitute these results in (21). By doing this we will get a system of nonlinear equations,
(23) 
To solve above equation here we are using Newton’s method and using this method wavelet coefficients () can be calculated.
3.3 Convergence
The convergence of the proposed methods is an immediate consequence of the results stated by Majak et al. [BS2015].
Theorem 3.1.
([BS2015]) Let us assume that is a continuous function on and its first derivative is bounded for all , there exists such that , then both approaches discussed in this paper converge.
Proof. The proof is similar to the proof given in [Higher2019].
4 Numerical Illustration
4.1 Example 1
Consider (13), (14) with , and . Solving by quasilinearization method (subsection ) substitute , and and replace in (20) where are collocation points (section ). Now we get (20) in terms of ,, and where and
are row vectors from which unknowns
are calculated.Taking initial guess and by Haar wavelet collocation method, we will get required solution. Taking initial guess solution is given in Table 1 and graph for , , and is given in figure 1. Taking different initial guess and solution still remain same.
Now by Newton’s method (subsection ), assume
(24) 
where are wavelet coefficients. Now integrate above equation from to twice we get,
(25) 
(26) 
Now apply boundary conditions (16) in (26) and we get as,
(27) 
Substituting (27),(24) in (21), we get system of nonlinear equations. Thus we arrive at (23). To solve the nonlinear equation we use Newton’s method to calculate wavelet coefficients ().
For initial guess computed solution is given in Table 2 and graph for , , and is given in figure 2. Taking varying initial guesses to and we observe that solution does not change significantly.
Ealgo.[Numerical1996, pp.262]  
0.1  0.849094  0.849401  0.849464  0.84947  0.849474 
0.2  0.726803  0.727162  0.727222  0.727228  0.727231 
0.3  0.618867  0.619233  0.619286  0.619292  0.619294 
0.4  0.520041  0.520361  0.520408  0.520412  0.520414 
0.5  0.427227  0.427506  0.427544  0.427548  0.42755 
0.6  0.338421  0.338651  0.338682  0.338685  0.338686 
0.7  0.252197  0.252371  0.252395  0.252397  0.252398 
0.8  0.16751  0.167631  0.167647  0.167648  0.167649 
0.9  0.0836165  0.0836776  0.0836856  0.0836864  0.083686 

Ealgo.[Numerical1996, pp.262]  
0.1  0.849094  0.849401  0.849464  0.84947  0.849474 
0.2  0.726803  0.727162  0.727222  0.727228  0.727231 
0.3  0.618867  0.619233  0.619286  0.619292  0.619294 
0.4  0.520041  0.520361  0.520408  0.520412  0.520414 
0.5  0.427227  0.427506  0.427544  0.427548  0.42755 
0.6  0.338421  0.338651  0.338682  0.338685  0.338686 
0.7  0.252197  0.252371  0.252395  0.252397  0.252398 
0.8  0.16751  0.167631  0.167647  0.167648  0.167649 
0.9  0.0836165  0.0836776  0.0836856  0.0836864  0.083686 

4.2 Example 2
Consider (13), (14) with , and . Here we follow same procedure similar to example . For Quasilinearization method we take as initial guess. Computed solution is given in Table 3 and graph for , , and is given in figure 3. Taking different initial guesses and we observe that solution does not vary which proves that method is stable.
4.3 Example 3
Consider (13), (14) with , and . Here we follow same procedure similar to example . For Quasilinearization method we take as initial guess. Computed solution is displayed in Table 5 and graph for , , and is given in figure 5. Small perturbations in initial guesses and does not change the solution significantly.
For Newton’s method we take as initial guess. Computed solution is given in Table 6 and graph for , , and is given in figure 6. Taking different initial guess and we observe that solution remains same.
Ealgo.[Numerical1996, pp.264]  

0.1  0.781336  0.780201  0.78013  0.780126  0.780125 
0.2  0.657973  0.657498  0.657471  0.657469  0.657468 
0.3  0.558596  0.558365  0.55835  0.558349  0.558348 
0.4  0.470276  0.470119  0.470109  0.470109  0.470108 
0.5  0.387687  0.387587  0.387581  0.387581  0.38758 
0.6  0.308217  0.30815  0.308146  0.308146  0.308145 
0.7  0.23039  0.230345  0.230342  0.230342  0.230342 
0.8  0.153353  0.153328  0.153327  0.153326  0.153326 
0.9  0.0766365  0.0766246  0.0766239  0.0766238  0.076623 
Ealgo.[Numerical1996, pp.264]  
0.1  0.781336  0.780201  0.78013  0.780126  0.780125 
0.2  0.657973  0.657498  0.657471  0.657469  0.657468 
0.3  0.558596  0.558365  0.55835  0.558349  0.558348 
0.4  0.470276  0.470119  0.470109  0.470109  0.470108 
0.5  0.387687  0.387587  0.387581  0.387581  0.38758 
0.6  0.308217  0.30815  0.308146  0.308146  0.308145 
0.7  0.23039  0.230345  0.230342  0.230342  0.230342 
0.8  0.153353  0.153328  0.153327  0.153326  0.153326 
0.9  0.0766365  0.0766246  0.0766239  0.0766238  0.076623 

Ealgo.[Numerical1996, pp.265]  
0.1  0.716249  0.708156  0.7057  0.70517  0.704396 
0.2  0.598674  0.593057  0.591181  0.590768  0.590163 
0.3  0.508462  0.504055  0.502524  0.502185  0.501688 
0.4  0.428945  0.425335  0.424071  0.42379  0.423379 
0.5  0.354347  0.351425  0.350393  0.350164  0.349827 
0.6  0.282173  0.27987  0.279053  0.278872  0.278605 
0.7  0.211155  0.20944  0.208831  0.208696  0.208497 
0.8  0.140637  0.1395  0.139095  0.139005  0.138872 
0.9  0.0702995  0.0697316  0.0695291  0.0694841  0.069418 

Ealgo.[Numerical1996, pp.265]  
0.1  0.716249  0.708156  0.7057  0.70517  0.704396 
0.2  0.598674  0.593057  0.591181  0.590768  0.590163 
0.3  0.508462  0.504055  0.502524  0.502185  0.501688 
0.4  0.428945  0.425335  0.424071  0.42379  0.423379 
0.5  0.354347  0.351425  0.350393  0.350164  0.349827 
0.6  0.282173  0.27987  0.279053  0.278872  0.278605 
0.7  0.211155  0.20944  0.208831  0.208696  0.208497 
0.8  0.140637  0.1395  0.139095  0.139005  0.138872 
0.9  0.0702995  0.0697316  0.0695291  0.0694841  0.069418 

4.4 Example 4
Consider (13) with , and along with the boundary conditions,
(28) 
This problem is considered in [Implementation1980] to compute various atomic states. Now by Quasilinearization method, applying Haar wavelet method [Haar_solving_HODE2008], assume
(29) 
where are wavelet coefficients. Now integrate above equation from to twice we get,
(30) 
(31) 
Now apply same procedure similar to example . Taking initial guess we compute solution which is given in Table 7 and graph for , , and , is given in figure 7. Slight variations in initial guesses, say, and does not change the final computed solution significantly.
0.1  0.914582  0.910743  0.908939  0.908424 
0.2  0.822492  0.816856  0.814206  0.813445 
0.3  0.73796  0.732383  0.729788  0.729043 
0.4  0.666354  0.662663  0.660999  0.660526 
0.5  0.610785  0.61069  0.610805  0.610582 
0.6  0.573128  0.578278  0.580985  0.581791 
0.7  0.554554  0.566517  0.572592  0.574382 
0.8  0.555746  0.575998  0.586173  0.589163 
0.9  0.57701  0.606936  0.621895  0.626285 

Now by Newton’s method, applying Haar wavelet method [Haar_solving_HODE2008], assume
(33) 
where are wavelet coefficients. Now integrate above equation from to twice we get,
(34) 
(35) 
Further we follow same procedure similar to example . Taking as initial guess, solution is given in Table 8 and graph for , , and , is given in figure 8. To check the continuous dependence on data and stability of the proposed method we take different initial guesses and , and observe that final computed solution does not vary significantly.
0.1  0.914582  0.910743  0.908939  0.908424 
0.2  0.822492  0.816856  0.814206  0.813445 
0.3  0.73796  0.732383  0.729788  0.729043 
0.4  0.666354  0.662663  0.660999  0.660526 
0.5  0.610785  0.61069  0.610805  0.610582 
0.6  0.573128  0.578278  0.580985  0.581791 
0.7  0.554554  0.566517  0.572592  0.574382 
0.8  0.555746  0.575998  0.586173  0.589163 
0.9  0.57701  0.606936  0.621895  0.626285 

5 Conclusion
In this paper we have used Haar wavelet collocation method on a class of EmdenFowler equation which originates in Physics. These equations are also referred as ThomasFermi equations. We have proposed two methods: in one approach we first linearise it by Newton’s quasilinearization method and then solve it by Haar wavelet collocation method. In the second approach we use Haar wavelet collocation method an arrive at system of nonlinear equation which we then solve by NewtonRaphson method. We observed that as value of is increased beyond 3 computed solution does not vary significantly. We also observed that small perturbation in initial guess does not vary final computed solution at all. So therefore both methods are stable and robust. Higher the resolution we use, our computed solution is closer to the exact solution.