A novel operational matrix method based on shifted Legendre polynomials for solving second-order boundary value problems involving singular, singularly perturbed and Bratu-type equations

Abstract

In this article, a new operational matrix method based on shifted Legendre polynomials is presented and analyzed for obtaining numerical spectral solutions of linear and nonlinear second-order boundary value problems. The method is novel and essentially based on reducing the differential equations with their boundary conditions to systems of linear or nonlinear algebraic equations in the expansion coefficients of the sought-for spectral solutions. Linear differential equations are treated by applying the Petrov-Galerkin method, while the nonlinear equations are treated by applying the collocation method. Convergence analysis and some specific illustrative examples include singular, singularly perturbed and Bratu-type equations are considered to ascertain the validity, wide applicability and efficiency of the proposed method. The obtained numerical results are compared favorably with the analytical solutions and are more accurate than those discussed by some other existing techniques in the literature.

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Web End = A novel operational matrix method based on shifted Legendre polynomials for solving second-order boundary value problems involving singular, singularly perturbed and Bratu-type equations

W. M. Abd-Elhameed1,2 Y. H. Youssri2 E. H. Doha2

Received: 31 July 2014 / Accepted: 23 May 2015 / Published online: 11 June 2015 The Author(s) 2015. This article is published with open access at Springerlink.com

Abstract In this article, a new operational matrix method based on shifted Legendre polynomials is presented and analyzed for obtaining numerical spectral solutions of linear and nonlinear second-order boundary value problems. The method is novel and essentially based on reducing the differential equations with their boundary conditions to systems of linear or nonlinear algebraic equations in the expansion coefcients of the sought-for spectral solutions. Linear differential equations are treated by applying the PetrovGalerkin method, while the nonlinear equations are treated by applying the collocation method. Convergence analysis and some specic illustrative examples include singular, singularly perturbed and Bratu-type equations are considered to ascertain the validity, wide applicability and efciency of the proposed method. The obtained numerical results are compared favorably with the analytical solutions and are more accurate than those discussed by some other existing techniques in the literature.

Keywords Shifted Legendre polynomials Second-order

equations Singular and singularly perturbed Bratu

equation PetrovGalerkin method Collocation method

Mathematics Subject Classication 65M70 65N35

35C10 42C10

Introduction

Spectral methods (see, for instance [16]) are one of the principal methods of discretization for the numerical solution of differential equations. The main advantage of these methods lies in their accuracy for a given number of unknowns. For smooth problems in simple geometries, they offer exponential rates of convergence/spectral accuracy. In contrast, nite difference and nite-element methods yield only algebraic convergence rates. The three spectral methods, namely, the Galerkin, collocation, and tau methods are used extensively in the literature. Collocation methods [7, 8] have become increasingly popular for solving differential equations, since they are very useful in providing highly accurate solutions to nonlinear differential equations. PetrovGalerkin method is widely used for solving ordinary and partial differential equations; see for example [913]. The PetrovGalerkin methods [14] have generally come to be known as stablized formulations, because they prevent the spatial oscillations and sometimes yield nodally exact solutions where the classical Galerkin method would fail badly. The difference between Galerkin and PetrovGalerkin methods is that the test and trial functions in Galerkin method are the same, while in PetrovGalerkin method, they are not.

The subject of nonlinear differential equations is a well-established part of mathematics and its systematic development goes back to the early days of the development of calculus. Many recent advances in mathematics, paralleled by a renewed and ourishing interaction between mathematics, the sciences, and engineering, have again shown that many phenomena in applied sciences, modeled by differential equations, will yield some mathematical explanation of these phenomena (at least in some approximate sense).

& Y. H. Youssri [email protected]

1 Department of Mathematics, Faculty of Science, University of Jeddah, Jeddah, Saudi Arabia

2 Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt

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94 Math Sci (2015) 9:93102

2x 1

cosh h4

; 2

where h is the solution of the nonlinear equation h

cosh

h 4

Even order differential equations have been extensively discussed by a large number of authors due to their great importance in various applications in many elds. For example, in the sequence of papers [12, 1517], the authors dealt with such equations by the Galerkin method. They constructed suitable basis functions which satisfy the boundary conditions of the given differential equation. For this purpose, they used compact combinations of various orthogonal polynomials. The suggested algorithms in these articles are suitable for handling one- and two-dimensional linear high even-order boundary value problems. In this paper, we aim to give some algorithms for handling both of linear and nonlinear second-order boundary value problems based on introducing a new operational matrix of derivatives, and then applying PetrovGalerkin method on linear equations and collocation method on nonlinear equations.

Of the important high-order differential equations are the singular and singular perturbed problems (SPPs) which arise in several branches of applied mathematics, such as quantum mechanics, uid dynamics, elasticity, chemical reactor theory, and gas porous electrodes theory. The presence of a small parameter in these problems prevents one from obtaining satisfactory numerical solutions. It is a well-known fact that the solutions of SPPs have a multi-scale character, that is, there are thin layer(s) where the solution varies very rapidly, while away from the layer(s) the solution behaves regularly and varies slowly.

Also, among the second-order boundary value problems is the one-dimensional Bratu problem which has a long history. Bratus own article appeared in 1914 [19]; generalizations are sometimes called the LiouvilleGelfand or LiouvilleGelfandBratu problem in honor of Gelfand [20] and the nineteenth century work of the great French mathematician Liouville. In recent years, it has been a popular testbed for numerical and perturbation methods [2127].

Simplication of the solid fuel ignition model in thermal combustion theory yields an elliptic nonlinear partial differential equation, namely the Bratu problem. Also due to its use in a large variety of applications, many authors have contributed to the study of such problem. Some applications of Bratu problem are the model of thermal reaction process, the Chandrasekhar model of the expansion of the Universe, chemical reaction theory, nanotechnology and radiative heat transfer (see, [2832]).

The Bratu problem is nonlinear (BVP) and extensively used as a benchmark problem to test the accuracy of many numerical methods. It is given by:

y00x k eyx 0; y0 y1 0; 0 6 x 6 1;

1

where k [ 0. The Bratu problem has the following analytical solution:

yx 2 ln

p cosh h.

Our main objectives in the present paper are:

Introducing a new operational matrix of derivatives based on using shifted Legendre polynomials and harmonic numbers.

Using PetrovGalerkin matrix method (PGMM) to solve linear second-order BVPs.

Using collocation matrix method (CMM) to solve a class of nonlinear second-order BVPs, including singular, singularly perturbed and Bratu-type equations.

The outlines of the paper is as follows. In Some properties and relations of Shifted Legendre polynomials and harmonic numbers, some relevant properties of shifted Legendre polynomials are given. Some properties and relations of harmonic numbers are also given in this section. In A shifted Legendre matrix of derivatives, and with the aid of shifted Legendre polynomials polynomials, a new operational matrix of derivatives is given in terms of harmonic numbers. In Solution of second-order linear two point BVPs, we use the introduced operational matrix for reducing a linear or a nonlinear second-order boundary value problems to a system of algebraic equations based on the application of PetrovGalerkin and collocation methods, and also we state and prove a theorem for convergence. Some numerical examples are presented in Numerical results and discussions to show the efciency and the applicability of the suggested algorithms. Some concluding remarks are given in Concluding remarks.

Some properties and relations of shifted Legendre polynomials and harmonic numbers

Shifted Legendre polynomials

The shifted Legendre polynomials L kx are dened on

[a, b] as:

L kx Lk

2k

2x a b

b a

; k 0; 1; . . .;

where Lkx are the classical Legendre polynomials. They

may be generated by using the recurrence relation

k 1 L k1x 2k 1

2x b a

b a

L kx k L k 1x; k 1; 2; . . .; 3

with L 0x 1; L 1x

2x b a

b a

:These polynomials are

orthogonal on [a, b], i.e.,

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Math Sci (2015) 9:93102 95

Zb

a

L mx L nx dx

8 <

:

b a

2n 1

; m n; 0; m 6 n:

Any function yx 2 L20a; b can be expanded as yx X

1 ci /ix; 8

where

ci

2i 1

b a Z

4

The polynomials L kx are eigenfunctions of the following

singular SturmLiouville equation:

D x ax b D /kx kk 1 /kx 0;

where D ddx.

Harmonic numbers

The nth harmonic number is the sum of the reciprocals of the rst n natural numbers, i.e.,

Hn X

n

i1

b

yx /ix

x a2b x2

dx yx; /ix wx

:

a

If the series in Eq. (8) is approximated by the rst N 1

terms, then

yx yNx X

N

i0

ci /ix CT Ux; 9

where

CT c0; c1; . . .; cN ; Ux /0x; /1x; . . .; /Nx T:

10

Now, we state and prove the basic theorem, from which a new operational matrix can be intoduced.

Theorem 1 Let /ix be as chosen in (7). Then for all

i 1, one has

D /ix

2b a X

i 1

1i : 5

The numbers Hn satisfy the recurrence relation

Hn Hn 1

1n ; n 1; 2; . . .;

and have the integral representation

Hn Z

1 1 xn

0 1 x

dx:

The following Lemma is of fundamental importance in the sequel.

Lemma 1 The harmonic numbers satisfy the following three-term recurrence relation:

2i 1 Hi 1 i 1 Hi 2 i Hi; i 2: 6 Proof The recurrence relation (6) can be easily proved with the aid of the relation (5). h:

A shifted Legendre matrix of derivatives

Consider the space (see, [33])

L20a; b f/x 2 L2a; b : /a /b 0g; and choose the following set of basis functions:

/ix x ab x L ix; i 0; 1; 2; . . .: 7 It is not difcult to show that the set of polynomials

f/kx : k 0; 1; 2; . . .g are linearly independent and

orthogonal in the complete Hilbert space L20a; b ; with

respect to the weight function wx

1

x a2 b x2

2 j 1 1 2 Hi 2 Hj

/jx dix; 11 where dix is given by

di

x

j 0 i j odd

12

Proof We proceed by induction on i. For i 1; it is clear

that the left hand side of (11) is equal to its right-hand side,

which is equal to: a b

6 x ab x

b a

a b 2 x; i even;

a b; i odd:

. Assuming that

relation (11) is valid for i 2 and i 1, we want to

prove its validity for i. If we multiply both sides of (3) by

x ax b and make use of relation (7), we get

/ix

2 i 1

i

/i 1x

2 x b a

b a

i 1

i

/i 2x; i 2; 3; . . .; 13 which immediately gives

D/ix

2 i 1

ib a

2x b aD/i 1x 2 /i 1x

, i.e.,

i 1

i

Z

b

/ix /jx dx

x a2 b x2

8 <

:

0; i 6 j;

a

b a

2 i 1

D/i 2x: 14

Now, application of the induction step on D/i 1x and

D/i 2x in (14), yields

; i j:

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96 Math Sci (2015) 9:93102

D/ix

22 i 12x b a

ib a2 X

i 2

mij

22j 1 b a

1

2i 2i 1Hi 1 i 1Hi 2

f g

j 0

i j even

2 j 1

22i 1

i2j 1

j Hj 1 j 1Hj1

2i 1

i Hj ;

1 2 Hi 1 2 Hj

/jx

2i 1

ib a X

i 3

/jx

j 0

i j odd

2 j 1 1 2 Hi 2 2 Hj

19

which can be simplied with the aid of Lemma 1, to take the form

mij

22j 1 b a

1 2Hi 2Hj :

Repeated use of Lemma 1 in (17), and after performing some manipulation, leads to

D/ix

2b a X

i 1

22i 1

ib a

/i 1x nix; 15

where

nix

2i 12x b a

ib a

8 <

:

2

666664

di 1x

i 1

i di 2x

x

a b 2x; i even;

2 i 12x b a2

ia b

i 1

i a b; i odd:

j 0 i j odd

2j 1 1 2Hi 2Hj

/

j

22i 1

i cix ab x nix;

16

Substitution of the recurrence relation (13) in the form

2x a b

b a

/jx

;

into relation (15), and after performing some rather lenghthy algebraic manipulations, give

D/ix X

i 3

20

j 1

2j 1

/j1x

jj 1

/j 1x

and by noting that

nix

42i 1

ib a

cix ab x dix;

then

D/ix

22i 1 b a

j 1

i j odd

mij /jx

2b a X

i 1

j 0

i j odd

1

2i 1

2 ci

b a

2i 1

i Hi 2

22i 1

i Hi 1

2j 1 1 2Hi 2Hj

/jx dix; 21 and this completes the proof of Theorem 1. h

Corollary 1 Let x 2 1; 1 a; b ; wix 1 x2

Lix: Then for all i > 1; one has

Dwix X

i 1

i Hi 1 Hi 2

/i 1x

3i 2

i

/0x nix;

17

where

mij

22j 1 b a

j

x cix;

22

j 0

i jodd

2j 1 1 2Hi 2Hj

w

1

22i 1j

i2j 1

Hj 1

2i 1

i Hj

22i 1j 1

i2j 1

Hj1:

18

where

cix

2i 1

i Hi 2

22i 1

i Hi 1 ;

2x; i even;

2; i odd:

ci

1; i odd, 0; i even.

Now, the elements mij in (18) can be written in the alternative form

Now, and based on Theorem 1, it can be easily shown that the rst derivative of the vector Ux dened in (10)

can be expressed in the matrix form:

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Math Sci (2015) 9:93102 97

dUx

dx M Ux d; 23

where

d d0x; d1x; . . .; dNx

T;

di

If we approximate y(x) as in (9), making use of relations (23) and (24), we have the following approximations for y(x), y0x and y00x:yx CT Ux; 27

y0x CT M Ux CT d; 28 y00x CT M2 Ux CT M d CT d0: 29

If we substitute the relations (27), (28) and (29) into Eq. (25), then the residual R(x), of this equation is given by:

Rx CT M2Ux CT M d CT d0 f1x

CT M Ux CT d

f2x CT Ux

a b 2x; i even;

a b; i odd;

and M m

ij

06i;j6N , is an N 1 N 1 matrix

whose nonzero elements can be given explicitly from relation (11) as:

mi;j

8 <

30

The application of Petrov Galerkin method (see, [1]) yields the following N 1 linear equations in the unknown

expansion coefcients, ci; namely,

Z

b

a Rx L ix dx 0; i 0; 1; . . .; N:

2b a

2 j 1 1 2 Hi 2 Hj ;

i [ j; i j odd; 0; otherwise:

:For example, for N 5, we have

M

gx:

0

B

B

B

B

B

B

B

B

B

B

B

B

B

@

31

Thus, Eq. (31) generates a set of N 1 linear equations

which can be solved for the unknown components of the vector C, and hence the approximate spectral solution yNx given in (9) can be obtained.

Linear second-order BVPs subjectto nonhomogeneous boundary conditions

Consider the following one-dimensional second-order equation:

u00x f1x u0x f2x ux g1x; x 2 a; b;

32

subject to the nonhomogeneous boundary conditions:

ua a; ub b: 33 It is clear that the transformation

ux yx

a b x b x a

b a

2b a

0 0 0 0 0 0

3 0 0 0 0 0 0 6 0 0 0 0 143 0

25

3 0 0 0

0 19

2 0

21

2 0 0

16730 0

1

C

C

C

C

C

C

C

C

C

C

C

C

C

A

:

77

6 0

63

5 0

Remark 1 The second derivative of the vector Ux is

given by

d2Uxdx2 M2 Ux M d d0;

24

where

d0 d00; d01; . . .; d0N

T

; d0i

2; i even; 0; i odd:

Solution of second-order linear two-point BVPs

In this section, both of linear and nonlinear second-order two-point BVPs are handled. For linear equations, a PetrovGalerkin method is applied, while for nonlinear equations, the typical collocation method is applied.

Linear second-order BVPs subject to homogenous boundary conditions

Consider the linear second-order boundary value problem

y00x f1x y0x f2x yx gx; x 2 a; b;

25

subject to the homogenous boundary conditions

ya yb 0: 26

;

turns the nonhomogeneous boundary conditions (33) into the homogeneous boundary conditions:

ya yb 0: 34 Hence it sufces to solve the following modied one-dimensional second-order equation:

y00x f1x y0x f2x yx gx; x 2 a; b;

35

subject to the homogeneous boundary conditions (34), where

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98 Math Sci (2015) 9:93102

gx g1x

b a

b a

f1x

a b x b x a

b a

f2x:

yx; yNx wx yx; X N

i0

ci /ix wx

X

N

i0

yx; /ix wx

X

Solution of second-order nonlinear two-point BVPs

Consider the nonlinear differential equation

y00x F x; yx; y0x

; 36 subject to the homogenous conditions

ya yb 0:If we follow the same procedure of Linear second-order BVPs subject to homogenous boundary conditions, and approximate y(x) as in (27), then after making use of the two relations (23) and (24), then we get the following nonlinear equation in the unknown vector C

CTM2Ux CT M d CT d0 F x; CT Ux;

CT M Ux CT d:

N

i0

ci ci X

N

i0 j

cij2:

We show that yNx is a Cauchy sequence in the complete

Hilbert space L20a; b and hence converges.

Now,

y

N

x yMx 2wx X

N

iM1 j

cij2:

From Bessels inequality, we have

X1i0 jcij2 is convergent,

which yields

x yMx 2wx ! 0 as M; N ! 1 and

hence yNx converges to say b(x). We prove that

bx yx;

bx yx; /ix wx bx; /ix wx yx; /ix wx

lim

N!1

yN; /ix wx

ci

y

N

37

To nd the numerical solution yNx, we enforce (37) to be

satised exactly at the rst N 1 roots of the polynomial

L N1x. Thus a set of N 1 nonlinear equations is

generated in the expansion coefcients, ci. With the aid of the well-known Newtons iterative method, this nonlinear system can be solved, and hence the approximate solution yNx can be obtained.

Remark 2 Following a similar procedure to that given in Linear second-order BVPs subject to nonhomogeneous boundary conditions, the nonlinear second-order Eq. (36) subject to the nonhomogeneous boundary conditions given as in (33) can be tackled.

Convergence analysis

In this section, we state and prove a theorem for convergence of the proposed method.

Theorem 2 The series solutions of Eqs. (25) and (36) converge to the exact ones.

Proof Let

yx X

lim

N!1

yN; /ix wx ci

0:

X1i0ci/ix converges to y(x). h

As the convergence has been proved, then consistency and stability can be easily deduced.

Numerical results and discussions

In this section, the presented algorithms in Solution of second-order linear two point BVPs are applied to solve regular, singular as well as singularly perturbed problems. As expected, the accuracy increases as the number of terms of the basis expansion increases.

Example 1 Consider the second-order nonlinear equation (see, [34]).

2 y00 y x 1

3; 0\x\1; y0 y1 0:

38

The exact solution of (38) is

yx

22 x

This proves

1 ci/ix;

yMx X

M

i0

ci/ix;

yNx X

N

i0

ci/ix;

be the exact and approximate solutions (partial sums) to Eqs. (25) and (36) with N > M. Then we have

x 1:

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Math Sci (2015) 9:93102 99

Table 1 Maximum absoluteerror E for Example 1 N 6 10 14 18 22

E 1.371 9 10 6 1.037 9 10 9 5.715 9 10 13 7.140 9 10 16 5.312 9 10 17

In Table 1, the maximum absolute error E is listed for various values of N, while in Table 2 a comparison between the numerical solution of problem (38) obtained by the application of CMM with the two numerical solutions obtained by using a sinc-collocation and a sinc-Galerkin methods in [34] is given.

Example 2 Consider the second-order nonlinear singular equation (see, [34]).

4 xs xr y0

0 sxrs 2 sxs ey r s 1

; 0\x\1;

y0 ln

1

1 x

2

2 1

ex

yx e

2 1

2 1

e

:

2 1

In Table 5, the maximum absolute error E is listed for various values of and N, while in Table 6, we give a comparison between the solution of Example 3 obtained by our method (PGMM) with the solution obtained by the shooting method given in [35].

Example 4 Consider the following nonlinear second-order boundary value problem:

y00x

y0x 2

; y1 ln

1 5

1 4

; s 3 r; r 2 0; 1;

with the exact solution

yx ln 4 xs

:In Table 3, the maximum absolute error E is listed for various values of r and N, while in Table 4 a comparison between the solution of Example 2 obtained by our method (CMM) with the two numerical solutions obtained in [34] is given for the case r 14. In addition, Fig. 1 illustrates the

absolute error resulting from the application of CMM for the two cases corresponding to N 10; r 14 and

N 15; r 14.

Example 3 Consider the following singularly perturbed linear second-order BVP (see, [35])

y00x y0x yx 0; 0\x\1;

y0

16 yx 2 16 x6; 1\x\1; y 1 y1 0;

39

with the exact solution yx x2 x4: Making use of (9)

with N 2 yieldsyNx CT Ux 1 x2 c

0 L0x c1 L1x c2 L2x : Moreover, in this case the two matrices M and M2 take the forms

M

0 0 0

3 0 0 0 6 0

0

B

@

1

C

A

; M2

0

B

@

0 0 0

0 0 0

18 0 0

1

C

A

:

2

e1 2

Now, with the aid of Eq. (37), we have

c02x2 1 c16x3 5x c26x4 5x2 1

1

2

2 1e

2 1

; y1 1;

2c0 x c1 3 c1 x2 6c2 x3 4c2 x 2 8x6 1:

40

We enforce (40) to be satised exactly at the roots of L3x, namely,

3 5

where

p 4 1; with the exact solution

Table 2 Comparison between different solutions for Example 1

Method CMM Sinc-collocation [34] Sinc-Galerkin [34]

N 22 130 130

E 5.312 9 10 17 9.159 9 10 16 9.992 9 10 16

Table 3 Maximum absoluteerror E for Example 2 N r E r E r E r E

10 5.844 9 10 9 3.234 9 10 7 1.291 9 10 6 4.698 9 10 7 15 1

100 7.338 9 10 10

14 4.927 9 10 8

1

2 2.441 9 10 7

: This immediately yields three nonlin

ear algebraic equations in the three unknowns, c0; c1 and c2. Solving these equations, we get

q

; 0;

q

3 5

99100 1.416 9 10 7 20 1.164 9 10 10 4.435 9 10 9 2.247 9 10 8 1.229 9 10 8

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100 Math Sci (2015) 9:93102

Table 4 Comparison between different solutions for Example 2, r 14

Method CMM Sinc-collocation [34] Sinc-Galerkin [34]

N 20 100 100E 4.435 9 10 9 1.552 9 10 6 1.552 9 10 6

Fig. 1 The absolute error of Example 2 for r 14

Table 5 Maximum absolute error E for Example 3

N E E

4 4.527 9 10 5 2.942 9 10 5

5 1.091 9 10 5 7.974 9 10 7

6 10 3 9.202 9 10 6 10 4 9.444 9 10 7

7 8.184 9 10 6 8.323 9 10 7

8 1.013 9 10 6 7.815 9 10 8

Table 6 Comparison between the best errors for Example 3

Method PGMM Shooting method [35]

Best error 7.815 9 10 8 3.677 9 10 5

Table 7 Maximum absolute error E for Example 5

N k E k E k E

2 7.065 9 10 5 9.420 9 10 4 4.904 9 10 2 4 2.274 9 10 7 7.828 9 10 6 1.833 9 10 3 6 2.102 9 10 9 1.437 9 10 7 7.008 9 10 5 8 1.715 9 10 11 2.758 9 10 9 3.761 9 10 6 10 1 2.217 9 10 13 2 8.082 9 10 11 3.51 3.740 9 10 7 12 1.984 9 10 15 1.461 9 10 12 2.355 9 10 8 14 2.983 9 10 16 5.342 9 10 14 2.410 9 10 9 16 3.053 9 10 16 1.644 9 10 15 2.975 9 10 10 18 2.983 9 10 16 4.024 9 10 16 2.747 9 10 11

With the analytical solution

yx 2 ln

cosh

h4 2x 1

cosh h4

; 42

where h is the solution of the nonlinear equation h

p cosh h. The presented algorithm in Section 4.3 is applied to numerically solve Eq. (41), for the three cases corresponding to k 1; 2 and 3.51 which yield h

1:51716; 2:35755 and 4.66781, respectively. In Table 7, the maximum absolute error E is listed for various values of N, and in Table 8, we give a comparison between the best errors obtained by various methods used to solve Example 5. This table shows that our method is more accurate compared with the methods developed in [2831]. In addition, Fig. 2 illustrates a comparison between different solutions obtained by our algorithm (CMM) in case of k 1 and N 1; 2; 3.

Concluding remarks

In this paper, a novel matrix algorithm for obtaining numerical spectral solutions for second-order boundary value problems is presented and analyzed. The derivation of this algorithm is essentially based on choosing a set of basis functions satisfying the boundary conditions of the given boundary value problem in terms of shifted Legendre polynomials. The two spectral methods, namely, PetrovGalerkin and collocation methods, are used for handling linear second-order and nonlinear second-order boundary value problems, respectively. One of the main advantages of the presented algorithms is their availability for application on both linear and nonlinear second-order boundary value problems including some important singular perturbed equations and also a Bratu-type equation. Another advantage of the developed algorithms is that high accurate approximate solutions are achieved by using

2k

c0

13 ; c1 0; c2

2

3 ;

and hence

yx

13 ; 0;

2

3

0

B

B

@

1 x2

x x3

1

C

C

A

x2 x4;

1

2 2x2

32 x4

which is the exact solution.

Example 5 Consider the following Bratu Equation (see, [2831]).

y00x k eyx 0; y0 y1 0; 0 6 x 6 1:

41

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Math Sci (2015) 9:93102 101

Table 8 Comparison between the best errors for Example 5 for k 1

Present error Error in [28] Error in [29] Error in [30] Error in [31]

2.98 9 10 16 1.02 9 10 6 8.89 9 10 6 1.35 9 10 5 3.01 9 10 3

Fig. 2 Different solutions of Example 5

a few number of terms of the suggested expansion. The obtained numerical results are comparing favorably with the analytical ones. We do believe that the proposed algorithms in this article can be extended to treat other types of problems including some two-dimensional problems.

Acknowledgments The author would like to thank the referee for carefully reading the paper and for his useful comments which have improved the paper.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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