Haar wavelets collocation on a class of Emden-Fowler equation via Newton's quasilinearization and Newton-Raphson techniques

11/13/2019
by   Amit Kumar Verma, et al.
0

In this paper we have considered generalized Emden-Fowler equation, y”(t)+σ t^γ y^β(t)=0,         t ∈ ]0,1[ subject to the following boundary conditions y(0)=1, y(1)=0;  &  y(0)=1, y'(1)=y(1), where γ,β and σ are real numbers, γ<-2, β>1. We propsoed to solve the above BVPs with the aid of Haar wavelet coupled with quasilinearization approach as well as Newton-Raphson approach. We have also considered the special case of Emden-Fowler equation (σ=-1,γ=-1/2 and β=3/2) which is popularly, known as Thomas-Fermi equation. We have analysed different cases of generalised Emden-Fowler equation and for compared our results with existing results in literature. We observe that small perturbations in initial guesses does not affect the the final solution significantly.

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1 Introduction

The following singular boundary value problem,

with the following boundary conditions

is known as Thomas-Fermi 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

Thomas-Fermi equation may be written in a generalised form known as Emden-Fowler equation ([Numerical1998, Numerical1996])

(1)
(2)
(3)

where and are real numbers, , . Mehta et al. [MEHTA1971] studied it by using hypergeometric function and studied the Emden-Fowler equations. Wong [JSWW1975] presented survey on Emden-Fowler equations. Rosenau [ROSENAU1984] by simple change of variable generated a one parameter family of integrable Emden-Fowler equations. P M Lima [Numerical1996, LIMA1999, LIMA2009] analyzed numerical methods based on finite differences/shooting methods for solving Emden-Fowler equation. Mohammadi et al. [Mohammadi2019] used an iterative method to the singular and nonlinear fractional partial differential version of Emden-Fowler equations. Haar wavelets operational matrix of fractional integration is be applied to compute the solution with the Picard technique.

Pikulin [Pikulin2019] considered the Emden-Fowler equations on the half-line 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 Emden-Fowler 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 Emden-Fowler equation. To the best of our knowledge the results do not exists in literature.

The paper is divided into the following sections. In section 1 we discuss the literature survey. In section 2 we discuss basics of Haar Wavelets. In section 3 we discuss about proposed methodologies. Section 4 is devoted to computations. Finally in section 5 we conclude the paper.

2 Haar Wavelet

[Haar, pp.7-10] 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)

We substitute in (4),(9),(10).

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 Emden-Fowler equation.

3.1 Quasilinearization Approach

By quasilinearization ([An2013, Higher2019]) generalized Emden-Fowler 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 Emden-Fowler 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 non-linear 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 non-linear equations. Thus we arrive at (23). To solve the non-linear 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.

E-algo.[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


Table 1: Solutions for , and for , , and by Quasilinearization method
Figure 1: Graph of Quasilinearization method for , , , and E-algo ([Numerical1996]).
E-algo.[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


Table 2: Solutions for , and for , , and by Newton’s method
Figure 2: Graph of Newton’s method for , , , and E-algo ([Numerical1996]).

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.

For Newton’s method we take as initial guess. solution is given in Table 4 and graph for , , and is given in figure 4. Taking different initial guess and solution still remains same.

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.

E-algo.[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
Table 3: Solutions for , and for , , and by Quasilinearization method
Figure 3: Graph of Quasilinearization method for , , , and E-algo ([Numerical1996]).
E-algo.[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


Table 4: Solutions for , and for , , and by Newton’s method
Figure 4: Graph of Newton’s method for , , , and E-algo ([Numerical1996]).
E-algo.[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


Table 5: Solutions for , and for , , and by Quasilinearization method
E-algo.[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


Table 6: Solutions for , and for , , and by Newton’s method
Figure 5: Graph of Quasilinearization method for , , , and E-algo ([Numerical1996]).
Figure 6: Graph of Newton’s method for , , , and E-algo ([Numerical1996]).

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 boundary conditions (28) in (31) we get,

(32)

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

Table 7: Solutions for , and for , , and , by Quasilinearization method
Figure 7: Graph of Quasilinearization method for , , and

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)

Now by applying boundary conditions (28) in (36) we get,

(36)

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

Table 8: Solutions for , and for , , and , by Newton’s method
Figure 8: Graph of Newton’s method for , , and

5 Conclusion

In this paper we have used Haar wavelet collocation method on a class of Emden-Fowler equation which originates in Physics. These equations are also referred as Thomas-Fermi 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 non-linear equation which we then solve by Newton-Raphson 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.

References