Haar wavelet quasilinearization method for numerical solution of Emden-Fowler type equations

Journal Pre-proof Haar wavelet quasilinearization method for numerical solution of Emden-Fowler type equations Randhir Singh, Vandana Guleria, Mehakpreet Singh

PII: DOI: Reference:

S0378-4754(20)30043-4 https://doi.org/10.1016/j.matcom.2020.02.004 MATCOM 4948

To appear in:

Mathematics and Computers in Simulation

Received date : 22 August 2019 Revised date : 13 February 2020 Accepted date : 14 February 2020 Please cite this article as: R. Singh, V. Guleria and M. Singh, Haar wavelet quasilinearization method for numerical solution of Emden-Fowler type equations, Mathematics and Computers in Simulation (2020), doi: https://doi.org/10.1016/j.matcom.2020.02.004. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. c 2020 International Association for Mathematics and Computers in Simulation (IMACS). ⃝ Published by Elsevier B.V. All rights reserved.

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Haar Wavelet Quasilinearization Method for Numerical Solution of Emden-Fowler Type Equations Randhir Singha , Vandana Guleriaa , Mehakpreet Singh∗b b Bernal

a Department of Mathematics, Birla Institute of Technology Mesra, India Institute, Department of Chemical Science, University of Limerick, V94 T9PX Limerick, Ireland

Abstract

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In this paper, an efficient method for solving the nonlinear Emden-Fowler type boundary value problems with Dirichlet and Robin-Neumann boundary conditions is introduced. The present method is based on the Haar-wavelets and quasilinearization technique. The quasilinearization technique is adopted to linearize the nonlinear singular problem. Numerical solution of linear singular problem is obtained by the Haar wavelet method. The numerical study is further supported by examining thoroughly the convergence of the Haar wavelet method and the quasilinearization technique. In order to check the accuracy of the proposed method, the numerical results are compared with both existing methods and exact solutions. Keywords: Doubly Singular Boundary Value Problem; Haar Wavelets; Quasilinearization; Generalized Emden-Fowler; Convergence Analysis.

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Mathematics Subject Classification (AMC): 34B05, 34B15, 34B16, 34B18, 34B27. 1. Introduction

In this work, we consider a general class of Lane-Emden-Fowler types boundary value problems [7–9]  1   (p(x)y 0 (x))0 = f (x, y(x)), x ∈ (0, 1),   q(x)   (1) y(0) = α1 , y(1) = β1 ,       y 0 (0) = 0, α y(1) + β y 0 (1) = γ . 2 2 2 Email address: [email protected](Corresponding Author) (Mehakpreet Singh∗ )

Preprint submitted to Mathematics and Computers in Simulation

February 13, 2020

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Here p(0) = 0 and q(x) may be discontinuous at x = 0. We assume that (a) p(x) ∈ C[0, 1] ∩ C 1 (0, 1], p(x) > 0, and q(x) > 0,

(b) f (x, y), fy (x, y) ∈ C(Ω) and fy (x, y) ≥ 0 on Ω := {(0, 1] × R}, 1 ∈ L1 [0, 1] and (c) p(x)

1

(d) q(x) ∈ L [0, 1] and

Z1

1 p(t)

Z1

1 p(t)

 Z1 x

0

0

 Zx 0

 q(s) ds dt < ∞, (for the Dirichlet boundary conditions),

 q(s) ds dt < ∞, (for the Neumann-Robin boundary conditions),

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where α1 , β1 , α2 , β2 and γ2 are any real constants.

In the fields of mathematical biology and chemistery, there are some problems, which are modelled by equation (1). The Thomas-Fermi equations [10, 11] is modelled using the Lane-Emden-Fowler type (1), 1

3

where p(x) = 1, q(x) = x− 2 and f (x, y) = y 2 . Thomas [10] and Fermi [11] independently derived a boundary value problem for determining the electrical potential in an atom. The analysis leads to the boundary value problems y 00 = x−1/2 y 3/2 , y(0) = 1 and y(1) = 0. For p(x) = q(x) = x2 , equation (1) was used to solve problem arises in oxygen diffusion in a spherical cell [12] and modelling of heat conduction in human head [13, 14] with f (x, y) =

ny y+k

and f (x, y) = −δe−θy , respectively.

Moreover, in the areas of mathematical physics and astrophysics, there are many problems which are modelled using Lane-Emden-Fowler type equation (1) [6]. For example, the equation (1) with f = −y m and p(x) =

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q(x) = x2 models the equilibrium of isothermal gas spheres see [15]. Equation (1) with f (x, y) = ey and p(x) = q(x) = x2 was used to model the isothermal gas spheres [16]. In addition, equation (1) with

f (x, y) = ey , p(x) = q(x) = xα and α = 0, 1, 2 arises in physics and it is known as Poisson-Boltzmann differential equation.

In the recent years, the Haar wavelet method has been popular in the field of numerical approximations. The basic idea of the Haar wavelets and its applications to different problems can be found in [1–3, 30–36]. The collocation methods based on the Haar wavelets have been used by many researchers due to their properties like simple applicability, orthogonality and compact support. The different boundary conditions can easily be inserted in the numerical algorithms due to the compact support of the Haar basis. Since the Haar

2

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wavelet basis are linear and piecewise functions, therefore, the integration approach is applied instead of the differential approach to compute the unknown coefficients.

The Haar wavelet method for boundary value problems is more complicated than for initial value problems. The quasilinearization approach was introduced by Bellman and Kalaba [37] to solve the individual or systems of nonlinear ordinary and partial differential equations. The application of the Haar wavelet method quasilinearization technique for solving different models can be found in [38–40]. In [24], the Haar wavelet collocation method has been utilized to find the numerical solution the Lane-Emden equation, where authors converted the nonlinear Lane-Emden equations into the system nonlinear algebraic equations, then they were solved by the Newton’s method.

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In this work, an efficient numerical method based on the Haar-wavelets quasilinearization technique is presented for numerical solutions of the Lane-Emden-Fowler types boundary value problems on a uniform grid. The quasilinearization technique is used to linearize the nonlinear term in equation. In each iteration of the quasilinearization technique, the numerical solution is updated by the Haar-wavelet method. The accuracy of the proposed scheme is demonstrated by solving different singular problems, which contain various forms of nonlinearity. The results obtained using the proposed method are compared rather with exact solutions and results obtained using other numerical methods. It is shown that the proposed method shows better accuracy for approximating these problems than the existing methods. It is also shown that the use of Haar wavelet is flexible, convenient and the numerical results can be obtained by consuming less computational

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time.

2. Haar Wavelets

In this section, we define the Haar wavelet family on the interval [0, 1) as   1, 0 ≤ x < 1, h1 (x) =  0, elsewhere

3

(2)

and for i = 2, 3, . . .

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   1,   hi (x) = −1,     0,

where ξ1 =

ξ1 ≤ x < ξ2 ,

ξ2 ≤ x < ξ3 ,

(3)

elsewhere,

k + 0.5 k+1 k , ξ2 = , ξ3 = ; m = 2j , j = 0, 1, . . . , k = 0, 1, . . . , m − 1. m m m

Integer j indicates the level of the wavelet resolution and k is the translation parameter. The relation

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between i, m and k is given by i = m + k + 1.

Note that the Haar functions may also be constructed from the following relations j

hi (x) = 2 2 H(2j x − k), k = 0, 1, . . . , 2j − 1, j = 0, 1, . . .  k + 21 k   1, ≤ x < ,    2j 2j  k + 21 H(2j x − k) = k+1  −1, ≤x< ,  j  2 2j    0, elsewhere.

(4)

(5)

Any function y(x) ∈ L2 [0, 1] may be approximated by a finite sum of the Haar wavelets as follows

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y(x) ≈

2M X

ai hi (x),

(6)

i=1

where J is the maximum value of j and M = 2J . For simplicity, we introduce the following notation pi,1 (x) = Z pi,2 (x) =

Z

x

hi (t)dt,

(7)

pi,1 (t)dt,

(8)

pi,1 (t)dt.

(9)

0 x

0

Ci,1 =

Z

0

4

1

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Integrals (7)–(9) can be evaluated by using equation (3)    x − ξ1 ,   pi,1 (x) = ξ3 − x,     0, and

and given by

ξ1 ≤ x < ξ2 ,

ξ2 ≤ x < ξ3 , elsewhere

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 1   (x − ξ1 )2 , ξ1 ≤ x < ξ2 ,   2     1 1    − (ξ3 − x)2 , ξ2 ≤ x < ξ3 , 2 4m 2 pi,2 (x) =  1   , ξ3 ≤ x < 1,   4m2       0, elsewhere.

The Haar wavelet functions satisfy the following properties  Z 1  2−j , i = l = 2j + k, hi (x)hl (x)dx =  0, 0 i= 6 l and

Z

0

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3. Quasilinearization

1

(10)

(11)

(12)

  1, if i = 1, hi (x)dx =  0, if i = 2, 3, . . . .

(13)

In this section, we follow the quasilinearization technique [37] to transform (1) into the sequence of linear problem as

0 (p(x)yn+1 )0 = q(x) (f (x, yn ) + (yn+1 − yn )fy (x, yn )) ,

n = 0, 1, 2 . . . .

(14)

The sequence of linear problems (14) may be written as 00 0 p(x)yn+1 + p0 (x)yn+1 + r(x)yn+1 = g(x),

5

n = 0, 1, 2 . . . .

(15)

where r(x) and g(x) are given by r(x) = −q(x)fy (x, yn ),

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g(x) = q(x)(f (x, yn ) − yn fy (x, yn )),

n = 0, 1, 2 . . . .

(16)

The boundary conditions provided in equation (1) take the form yn+1 (0) = α1 , 0 yn+1 (0) = 0,

yn+1 (1) = β1 ,

0 α2 yn+1 (1) + β2 yn+1 (1) = γ2 .

(17) (18)

By following the procedure in [8, 9], the equivalent integral form of (15) with (17) and (18) is provided as G(x, s)q(s)[f (s, yn ) + (yn+1 − yn )fy (s, yn )]ds,

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yn+1 = v(x) +

Z1 0

n = 0, 1, 2 . . . ,

(19)

where v(x) and G(x, s) corresponding to boundary conditions (17) are given by b(x) , v(x) = α1 + (β1 − α1 ) b(1)    b(x)   b(1) − b(s) , x ≤ s,  b(1)   G(x, s) = b(s)   b(1) − b(x) , s ≤ x,  b(1)

and the corresponding boundary conditions (18) are given by

γ2 , α2  β b0 (1)   b(1) − b(s) + 2 , α2 G(x, s) = 0   b(1) − b(x) + β2 b (1) , α2

with

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v(x) =

b(x) =

Zx 0

ds , b(1) = p(s)

Z1

ds p(s)

x ≤ s, s≤x

and b0 (1) =

1 . p(1)

0

4. Solution Method

In this section, the Haar wavelet quasilinearization technique is discussed for equation (15) with boundary conditions (17) and (18). 6

4.1. Dirichlet Boundary Conditions

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We first approximate the second order derivative term by the Haar wavelet series as 00 yn+1 (x) ≈

2M X

ai hi (x),

i=1

Let us define the collocation points as xj =

j − 0.5 , 2M

n = 0, 1, 2 . . . .

j = 1, 2, . . . , 2M.

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Integrating equation (20) twice from 0 to x and utilizing the boundary conditions (17), we get   2M X yn+1 (x) ≈ α1 + (β1 − α1 )x + ai pi,2 (x) − Ci,1 x ,

(20)

(21)

(22)

i=1

0 yn+1 (x) ≈ β1 − α1 +

2M X i=1

  ai pi,1 (x) − Ci,1 .

(23)

0 00 Substituting the approximate expressions yn+1 (x), yn+1 (x) and yn+1 (x) from equations (20), (22) and (23)

into equation (15), the following is obtained:

00 0 p(x)yn+1 (x) + p0 (x)yn+1 (x) + r(x)yn+1 (x) = g(x),

n = 0, 1, 2, . . . .

(24)

Inserting collocation points (21), we obtain the following linear system of algebraic equations 00 0 p(xj )yn+1 (xj ) + p0 (xj )yn+1 (xj ) + r(xj )yn+1 (xj ) = g(xj ),

2M X

Jou

where

00 yn+1 (xj ) ≈

n = 0, 1, 2, . . . , j = 1, 2 . . . , 2M,

ai hi (xj ),

i=1

0 yn+1 (xj ) ≈ β1 − α1 +

2M X i=1

(25)

(26)   ai pi,1 (xj ) − Ci,1 ,

yn+1 (xj ) ≈ α1 + (β1 − α1 )xj +

2M X i=1



(27) 

ai pi,2 (xj ) − Ci,1 xj .

(28)

For each n, we solve the linear system of equations (25) to obtain the unknown coefficients a1 , a2 , . . . , a2M , which can be substituted in equation (22) to find the approximate solution of equation (15). In the above process, we start with the initial approximation y0 to get y1 , at next iteration we get y2 and so on. 7

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4.2. The Neumann-Robin Types Boundary Conditions

We integrate equation (20) twice from 0 to x and apply the boundary conditions (18) to get 2M X
γ2 β2 − a1 + ai pi,2 (x) − Ci,1 , α2 α2 i=1

yn+1 (x) ≈

2M X

0 yn+1 (x) ≈

i=1

ai pi,1 (x).

(29)

(30)

Substituting equations (20), (29) and (30) into equations (15) and inserting collocation points (21), a linear system of algebraic equations is obtained

where

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00 0 p(xj )yn+1 (xj ) + p0 (xj )yn+1 (xj ) + r(xj )yn+1 (xj ) = g(xj ),

00 yn+1 (xj ) ≈

0 yn+1 (xj ) ≈

2M X

n = 0, 1, 2, . . . , j = 1, 2 . . . , 2M,

(31)

ai hi (xj ),

(32)

ai pi,1 (xj ),

(33)

i=1

2M X i=1

  2M X γ2 β2 yn+1 (xj ) ≈ ai pi,2 (xj ) − Ci,1 . − a1 + α2 α2 i=1

(34)

For each n, we solve linear system of equations (31) to obtain the unknown coefficients a1 , a2 , . . . , a2M which can be substituted in equation (29) to find the approximate solution of equation (15). In the above process,

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we start with the initial approximation y0 to get y1 , at next iteration we get y2 and so on. 5. Convergence Analysis

Theorem 5.1. Convergence of the Quasilinearization Technique: The sequence {yn } of solutions defined in equation (19) shows second order convergence. Proof. From (19), we have ∆yn+1 =

Z1 0

  G(x, s)q(s) ∆fn + ∆yn+1 fy (s, yn ) − ∆yn fy (s, yn−1 ) ds, 8

(35)

where ∆yn = yn − yn−1 ,

∆fn = fn (x, yn ) − fn−1 (x, yn−1 ).

From the mean-value theorem, we know that ∆fn = ∆yn fy (x, yn−1 ) +

(∆yn )2 fyy (x, θ), 2

Using equation (36), the equation (35) reduces to ∆yn+1 =

Z1

G(x, s)q(s)

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yn−1 < θ < yn .

 (∆yn )2 fyy (s, θ) + ∆yn+1 fy (s, yn ) ds. 2

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Equation (37) implies

  Z1 |∆yn |2 fyy (s, θ) + ∆yn+1 fy (s, yn ) ds. |∆yn+1 | ≤ G(x, s)q(s) 2

(36)

(37)

(38)

0

Hence, we have

  Z1 2 G(x, s)q(s) |∆yn | fyy (s, θ) + ∆yn+1 fy (s, yn ) ds k∆yn+1 k ≤ max 2 x∈[0,1] 0

k1 g1 ≤ k∆yn k2 + m1 g1 k∆yn+1 k, 2

where

kyk = max |y(x)|,

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x∈[0,1]

k1 = max |fyy (y)|, y

m1 = max{|f (y)|, |fy (y)|} < ∞,

Thus, we obtain the estimate

y

Z1 g1 = max G(x, s)q(s)ds < ∞. x∈[0,1] 0

k∆yn+1 k ≤

k1 g1 k∆yn k2 . 2(1 − m1 g1 )

This shows that there is quadratic convergence, if there is convergence at all. 9

(39)

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Theorem 5.2. Convergence of the Haar Wavelet Method: Suppose that y(x) ∈ L2 [0, 1] and y(x) satisfies the Lipschitz’s condition: |y(x1 ) − y(x2 )| ≤ L|x1 − x2 |. Babolian and Shahsavaran [41] gave L2 -error norm for the Haar wavelet approximation, which is given by ky − yM k22 ≤ or

L2 1 3 M2

ky − yM k2 = O P2M

i=1

ai hi (x).

 1 , M

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where M = 2J and yM (x) =



(40)

(41)

6. Numerical Examples

In this section, we examine the accuracy and efficiency of the Haar wavelet quasilinearization technique by solving several singular problems. For the sake of comparison, the cubic spline interpolation is used to obtain the solution at any points in the interval [0, 1]. We define the absolute error as ea (x) := |y(x) − yh (x)|

and Ean (x) := |y(x) − ψn (x)|,

x ∈ [0, 1],

(42)

and the maximum absolute error L∞ as

L∞ := max |y(x) − yh (x)|,

(43)

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0≤x≤1

where y(x) is the exact solution, yh (x) is the Haar solution and ψn (x) is the Adomian decomposition method (ADM) solution [27]. All computational work has been done with the help of MATLAB software. Example 6.1. Consider the following Emden-Fowler equation with the Dirichlet type boundary conditions [42]

   y 00 (x) + 0.5 y 0 (x) = 1 ey(x) − e2y(x) , x 2   y(0) = ln 2, y(1) = 0. 10

x ∈ (0, 1),

(44)

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Table 1: Numerical results with J = 3 and n = 7 of Example 6.1

yh (x)

y(x)

ADM [26]

ea (x)

Ea5 (x) [26]

0.1

0.68320

0.68330

0.68367

0.00011

0.00105

0.2

0.65393

0.65408

0.65462

0.00015

0.00152

0.3

0.60697

0.60715

0.60785

0.00018

0.00193

0.4

0.54473

0.54492

0.54579

0.00019

0.00234

0.5

0.47020

0.47020

0.47125

0.00019

0.00276

0.6

0.38566

0.38584

0.38707

0.00018

0.00312

0.7

0.29437

0.29452

0.29585

0.00015

0.00330

0.8 0.9

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x

0.19845

0.19855

0.19979

0.00010

0.00304

0.09982

0.09988

0.10066

5.42×10−05

0.00197

Table 2: Maximum absolute L∞ for J = 2, 3, . . . , 6 and n = 7 of Example 6.1

J L∞

2

3

−04

7.80×10

The exact solution is y(x) = ln

2 x2 +1

4

−04

1.96×10


4.75×10

−05

5 1.25×10

6 −05

3.07×10−06

. The numerical results of the Haar solution yh (x), the exact solution

y(x) and the ADM solution ψ5 (x) [26] along with the absolute errors ea (x) and Ea5 (x) are reported in Table 1. The maximum absolute error L∞ is reported in Table 2.

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Example 6.2. Consider the Thomas-Fermi boundary value problem [10, 11] as   y 00 (x) = x−1/2 y 3/2 (x), x ∈ (0, 1),  y(0) = 1, y(1) = 0.

(45)

The numerical results of the Haar solution and the ADM solution [43] are shown in Table 3. Example 6.3. The equilibrium of isothermal gas spheres can be modelled by the following Emden-Fowler equation with Neumann-Robin type boundary conditions [15]    y 00 (x) + 2 y 0 (x) = −y 5 (x), x ∈ (0, 1), x   y 0 (0) = 0, y(1) = (3/4)1/2 . 11

(46)

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The exact solution is

yh (J = 3)

ADM [43]

0.84909

0.84950

0.72680

0.72729

0.61888

0.61937

0.52004

0.52051

0.42723

0.42765

0.33842

0.33879

0.25220

0.25249

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Table 3: Numerical results with J = 2, 3 and n = 7 of Example 6.2

x

yh (J = 2)

0.1

0.84976

0.2

0.72627

0.3

0.61829

0.4

0.51951

0.5

0.42672

0.6

0.33801

0.7

0.25187

0.8

0.16728

0.16751

0.16773

0.9

0.08351

0.08361

0.08374

 − 12 x2 y(x) = 1 + . 3

The numerical results of the Haar solution, the exact solution, the ADM solution [27], the absolute errors ea (x) and Ea5 (x) are reported in Table 4. The maximum absolute error L∞ is reported in Table 5. Table 4: Numerical results with J = 3 and n = 6 of Example 6.3

yh

y

ADM [27]

ea (x)

Ea5 (x) [27]

0.99834

0.99858

0.99796

0.00024

0.00199

0.2

0.9934

0.99362

0.99304

0.00023

0.00187

0.3

0.98533

0.98554

0.98501

0.00021

0.00167

0.4

0.97435

0.97455

0.97409

0.00019

0.00144

0.5

0.96077

0.96093

0.96055

0.00016

0.00116

0.6

0.94491

0.94505

0.94474

0.00014

0.00088

0.7

0.92715

0.92725

0.92703

0.00010

0.00062

0.8

0.90784

0.90791

0.90777

6.99×10−05

0.00038

0.88732

−05

0.00017

x

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0.1

0.9

0.88736

0.88739

12

3.27×10

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Table 5: Errors L∞ (for J = 2, 3, . . . , 6) and n = 6 of Example 6.3

J

2

3

L∞

1.97×10−03

4.91×10−04

4

5

6

1.23×10−04

3.12×10−05

8.92×10−06

Example 6.4. We consider the following Emden-Fowler equation with boundary conditions which arises in electro-hydrodynamics [16, 27]

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 1   y 00 (x) + y 0 (x) = −ey(x) , x   y 0 (0) = 0, y(1) = 0.

The exact solution is y = 2 ln

µ+1 µx2 +1




x ∈ (0, 1),

(47)

√ , µ = 3 − 2 2. Table 6 shows the comparison of the Haar solution,

the exact solution and the ADM solution [27], the absolute errors ea (x) and Ea5 (x). The maximum absolute error L∞ is given in Table 7.

Table 6: Numerical results with J = 3 and n = 6 of Example 6.4

x 0.1 0.2 0.3 0.4

y

ADM [27]

ea (x)

Ea5 (x) [27]

0.31327

0.31327

0.31326

8.34×10−06

0.00016

0.30302

0.30302

0.30301

8.29×10−06

0.00015

−06

0.00014

0.28605

0.28606

0.28604

8.08×10

0.26253

0.26254

0.26253

7.76×10−06

0.00012

0.23270

0.23270

0.23269

7.22×10−06

0.00010

−06

8.21×10−05

Jou

0.5

yh

0.6

0.19683

0.19683

0.19682

6.47×10

0.7

0.15525

0.15525

0.15524

5.41×10−06

6.01×10−05

0.8

0.10832

0.10833

0.10832

4.00×10−06

3.87×10−05

0.9

0.05644

0.05644

0.05644

2.21×10−06

1.86×10−05

Example 6.5. The distribution of heat sources in the human head [14] is modelled by the following Em-

13

repro of

Journal Pre-proof

Table 7: Maximum error L∞ (for J = 2, 3, . . . , 6) and n = 6 of Example 6.4

J L∞

2

3

3.85×10−05

8.35×10−06

4

5

6

2.07×10−06

5.25×10−07

1.31×10−07

den-Fowler equation with Neumann-Robin type boundary conditions    y 00 (x) + 2 y 0 (x) = −e−y(x) , x ∈ (0, 1), x   y 0 (0) = 0, 2y(1) + y 0 (1) = 0.

(48)

rna lP

The numerical results of the Haar solution, the ADM [27], finite difference method (FDM) [44] and the tangent chord method (TCM) [14] are presented in Table 8.

Table 8: Numerical results with J = 3 and n = 7 of Example 6.5

yh

ADM [27]

TCM [14]

FDM [44]

0.1

0.26866

0.26862

0.26907

0.26875

0.2

0.26484

0.26480

0.26525

0.26493

0.3

0.25845

0.25841

0.25886

0.25853

0.4

0.24945

0.24943

0.24986

0.24954

0.5

0.23782

0.23781

0.23822

0.23791

0.6

0.22349

0.22349

0.22388

0.22358

0.7

0.20640

0.20641

0.20677

0.20649

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x

0.8

0.18646

0.18648

0.18679

0.18655

0.9

0.16356

0.16359

0.16387

0.16365

Example 6.6. The following Emden-Fowler equation with Neumann-Robin type boundary conditions models oxygen diffusion in a spherical cell with oxygen uptake kinetics [12]    y 00 (x) + 2 y 0 (x) = 0.76129y(x) , x ∈ (0, 1), x y(x) + 0.03119   y 0 (0) = 0, 5y(1) + y 0 (1) = 5. 14

(49)

repro of

Journal Pre-proof

The numerical results of the Haar solution, the ADM [27], the variational iteration method (VIM) [28] and the cubic spline method (CSM) [22] in Table 9.

Table 9: Numerical results with J = 3 and n = 7 of Example 6.6

yh (x)

ADM [27]

VIM [28]

CSM [22]

0.1

0.82971

0.82970

0.82970

0.82970

0.2

0.83338

0.83337

0.83337

0.83337

0.3

0.83949

0.83949

0.83948

0.83948

0.4

0.84805

0.84805

0.84805

0.84805

0.5

0.85907

0.85906

0.85906

0.85906

rna lP

x

0.6

0.87253

0.87252

0.87252

0.87252

0.7

0.88845

0.88844

0.88844

0.88844

0.8

0.90682

0.90681

0.90681

0.90681

0.9

0.92765

0.92765

0.92765

0.92765

Example 6.7. The following Emden-Fowler equation with Neumann-Robin type boundary conditions mod-

(50)

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els the radial stress on a rotationally symmetric shallow membrane cap [27, 45]  1   y 00 (x) + 3 y 0 (x) = 1 − , x ∈ (0, 1), x 2 8y 2 (x)   y 0 (0) = 0, y(1) = 1.

The numerical results of the Haar solution, the ADM [27] and VIM [46] are reported in Table 10.

7. Conclusions

In this paper, the Haar-wavelets method has been coupled with quasilinearization technique for approximating the numerical solution of nonlinear doubly singular boundary value problems. The accuracy of the proposed method has been tested by implementing on different physical models and comparing with the existing methods [26–28]. It has been shown that the present method provides excellent results than the

15

repro of

Journal Pre-proof

Table 10: Numerical results with J = 3 and n = 7 of Example 6.7

yh

0.1

0.95459

0.2

0.95595

0.3

0.95822

0.4

0.96140

0.5

0.96550

0.6

0.97053

0.7

0.97648

0.8 0.9

ADM [27]

VIM [46]

0.95458

0.95263

0.95594

0.95408

0.95822

0.95649

0.96140

0.95986

0.96550

0.96420

0.97052

0.96948

0.97647

0.97571

rna lP

x

0.98337

0.98336

0.98288

0.99121

0.99120

0.99098

existing methods and show good agreement with the exact solutions. The proposed method provides a reliable technique, which requires less computational work compared to the FDM and CSM. The convergence analysis affirms that the proposed method shows second order convergence on uniform grids.

References

[1] A. Imran, S. ul-Islam, M. Asil, Haar wavelet collocation method for three-dimensional elliptic partial

Jou

differential equations, Computers & Mathematics with Applications 73 (9) (2017) 2023–2034. [2] A. Imran, S. ul-Islam, W. Khan, Quadrature rules for numerical integration based on Haar wavelets and hybrid functions, Computers & Mathematics with Applications 61 (9) (2011) 2770–2781. [3] S. ul-Islam, A. Imran, F. Haq, A comparative study of numerical integration based on Haar wavelets and hybrid functions, Computers & Mathematics with Applications 59 (6) (2017) 2026–2036. [4] A. Aslanov, An operational approach to the Emden–Fowler equation, Mathematical Methods in the Applied Sciences 39 (5) (2016) 1039–1042.

16

repro of

Journal Pre-proof

[5] G. Bengochea, L. Verde-Star, An operational approach to the Emden–Fowler equation, Mathematical Methods in the Applied Sciences 38 (18) (2015) 4630–4637.

[6] S ¸ . Y¨ uzba¸sı, A numerical approach for solving a class of the nonlinear Lane–Emden type equations arising in astrophysics, Mathematical Methods in the Applied Sciences 34 (18) (2011) 2218–2230.

[7] L. Bobisud, Existence of solutions for nonlinear singular boundary value problems, Applicable Analysis 35 (1-4) (1990) 43–57.

[8] R. Singh, J. Kumar, G. Nelakanti, Approximate series solution of singular boundary value problems

rna lP

with derivative dependence using Green’s function technique, Computational and Applied Mathematics 33 (2) (2014) 451–467.

[9] R. Singh, J. Kumar, The Adomian decomposition method with Green’s function for solving nonlinear singular boundary value problems, Journal of Applied Mathematics and Computing 44 (1-2) (2014) 397–416.

[10] L. Thomas, The calculation of atomic fields, in: Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 23, Cambridge Univ Press, 1927, pp. 542–548. [11] E. Fermi, Un metodo statistico per la determinazione di alcune priorieta dell’atome, Rend. Accad. Naz. Lincei 6 (602-607) (1927) 32.

Jou

[12] S. Lin, Oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics, Journal of Theoretical Biology 60 (2) (1976) 449–457.

[13] B. Gray, The distribution of heat sources in the human head-theoretical considerations, Journal of Theoretical Biology 82 (3) (1980) 473–476. [14] R. Duggan, A. Goodman, Pointwise bounds for a nonlinear heat conduction model of the human head, Bulletin of Mathematical Biology 48 (2) (1986) 229–236. [15] M. Chawla, R. Subramanian, H. Sathi, A fourth order method for a singular two-point boundary value problem, BIT Numerical Mathematics 28 (1) (1988) 88–97. 17

repro of

Journal Pre-proof

[16] J. Keller, Electrohydrodynamics. I. the equilibrium of a charged gas in a container, Tech. rep., New York Univ., New York. Inst. of Mathematical Sciences (1955).

[17] R. Russell, L. Shampine, Numerical methods for singular boundary value problems, SIAM Journal on Numerical Analysis 12 (1) (1975) 13–36.

[18] M. Chawla, C. Katti, Finite difference methods and their convergence for a class of singular two point boundary value problems, Numerische Mathematik 39 (3) (1982) 341–350.

[19] S. Iyengar, P. Jain, Spline finite difference methods for singular two point boundary value problems,

rna lP

Numerische Mathematik 50 (3) (1986) 363–376.

[20] M. K. Kadalbajoo, V. Kumar, B-spline method for a class of singular two-point boundary value problems using optimal grid, Applied mathematics and computation 188 (2) (2007) 1856–1869. [21] M. Kumar, Higher order method for singular boundary-value problems by using spline function, Applied Mathematics and Computation 192 (1) (2007) 175–179.

[22] A. Ravi Kanth, V. Bhattacharya, Cubic spline for a class of non-linear singular boundary value problems arising in physiology, Applied Mathematics and Computation 174 (1) (2006) 768–774. [23] A. R. Kanth, Cubic spline polynomial for non-linear singular two-point boundary value problems, Ap-

Jou

plied Mathematics and Computation 189 (2) (2007) 2017–2022. [24] R. Singh, H. Garg, V. Guleria, Haar wavelet collocation method for Lane–Emden equations with Dirichlet, Neumann and Neumann–Robin boundary conditions, Journal of Computational and Applied Mathematics 346 (2019) 150–161. [25] S. Khuri, A. Sayfy, A novel approach for the solution of a class of singular boundary value problems arising in physiology, Mathematical and Computer Modelling 52 (3) (2010) 626–636. [26] R. Singh, J. Kumar, G. Nelakanti, Numerical solution of singular boundary value problems using Green’s function and improved decomposition method, Journal of Applied Mathematics and Computing 43 (1-2) (2013) 409–425. 18

repro of

Journal Pre-proof

[27] R. Singh, J. Kumar, An efficient numerical technique for the solution of nonlinear singular boundary value problems, Computer Physics Communications 185 (4) (2014) 1282–1289.

[28] A. Wazwaz, The variational iteration method for solving nonlinear singular boundary value problems arising in various physical models, Communications in Nonlinear Science and Numerical Simulation 16 (10) (2011) 3881–3886.

[29] R. Singh, N. Das, J. Kumar, The optimal modified variational iteration method for the Lane–Emden equations with Neumann and Robin boundary conditions, The European Physical Journal Plus 132 (6)

rna lP

(2017) 251.

[30] C.-H. Hsiao, W.-J. Wang, Haar wavelet approach to nonlinear stiff systems, Mathematics and Computers in Simulation 57 (6) (2001) 347–353.

[31] C. Hsiao, Haar wavelet approach to linear stiff systems, Mathematics and Computers in simulation 64 (5) (2004) 561–567.

¨ Lepik, Numerical solution of differential equations using Haar wavelets, Mathematics and Computers [32] U. in Simulation 68 (2) (2005) 127–143.

¨ Lepik, Numerical solution of evolution equations by the Haar wavelet method, Applied Mathematics [33] U. and Computation 185 (1) (2007) 695–704.

Jou

ˇ [34] I. Aziz, B. Sarler, et al., The numerical solution of second-order boundary-value problems by collocation method with the Haar wavelets, Mathematical and Computer Modelling 52 (9) (2010) 1577–1590. [35] M. ur Rehman, R. A. Khan, A numerical method for solving boundary value problems for fractional differential equations, Applied Mathematical Modelling 36 (3) (2012) 894–907. [36] I. Aziz, F. Khan, et al., A new method based on haar wavelet for the numerical solution of twodimensional nonlinear integral equations, Journal of Computational and Applied Mathematics 272 (2014) 70–80. [37] R. E. Bellman, R. E. Kalaba, Quasilinearization and nonlinear boundary-value problems. 19

repro of

Journal Pre-proof

[38] R. Jiwari, A haar wavelet quasilinearization approach for numerical simulation of burgers’ equation, Computer Physics Communications 183 (11) (2012) 2413–2423.

[39] H. Kaur, R. Mittal, V. Mishra, Haar wavelet approximate solutions for the generalized Lane–Emden equations arising in astrophysics, Computer Physics Communications 184 (9) (2013) 2169–2177. [40] U. Saeed, M. ur Rehman, Haar wavelet–quasilinearization technique for fractional nonlinear differential equations, Applied Mathematics and Computation 220 (2013) 630–648.

[41] E. Babolian, A. Shahsavaran, Numerical solution of nonlinear fredholm integral equations of the second

rna lP

kind using Haar wavelets, Journal of Computational and Applied Mathematics 225 (1) (2009) 87–95. [42] M. Inc, D. Evans, The decomposition method for solving of a class of singular two-point boundary value problems, International Journal of Computer Mathematics 80 (7) (2003) 869–882. [43] R. Singh, J. Kumar, Solving a class of singular two-point boundary value problems using new modified decomposition method, ISRN Computational Mathematics 2013 (Article ID 262863,) (2013) 11–pages. [44] R. Pandey, A finite difference method for a class of singular two point boundary value problems arising in physiology, International Journal of Computer Mathematics 65 (1-2) (1997) 131–140. [45] R. Dickey, Rotationally symmetric solutions for shallow membrane caps, Quarterly of Applied Mathe-

Jou

matics 47 (3) (1989) 571–581.

[46] A. Ravi Kanth, K. Aruna, He’s variational iteration method for treating nonlinear singular boundary value problems, Computers & Mathematics with Applications 60 (3) (2010) 821–829. [47] R. Mohammadzadeh, M. Lakestani, M. Dehghan, Collocation method for the numerical solutions of Lane–Emden type equations using cubic Hermite spline function, Mathematical Methods in the Applied Sciences 37(9) (2014) 1303–1717. [48] A. Taghavi, S. Pearce, A solution to the Lane–Emden equation in the theory of stellar structure utilizing the Tau method, Mathematical Methods in the Applied Sciences 36(10) (2013) 1240–1247.

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