A numerical scheme for solutions of a class of

Accepted Manuscript Title: A numerical scheme for solutions of a class of nonlinear differential equations Author: S¸uayip Yuzbas ¨ ¸i PII: DOI: Reference:

S1658-3655(17)30031-6 http://dx.doi.org/doi:10.1016/j.jtusci.2017.03.001 JTUSCI 368

To appear in: Received date: Revised date: Accepted date:

26-3-2016 13-7-2016 20-3-2017

Please cite this article as: {http://dx.doi.org/ This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. 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.

A numerical scheme for solutions of a class of nonlinear differential equations Şuayip YÜZBAŞI* Department of Mathematics, Faculty of Science, Akdeniz University, TR-07058 Antalya, Turkey [email protected]

*

Corresponding author. Tel.: + 90 242 310 23 83 fax: + 90 242 227 89 11. E-mail address: [email protected] (Ş. YÜZBAŞI).

Abstract In this paper, a collocation method based on the Bessel functions of the first kind is presented to compute the approximate solutions of a class of high-order nonlinear differential equations under the initial and boundary conditions. Firstly, the matrix forms of the Bessel functions of the first kind and their derivatives are constructed. Secondly, by using these matrix forms, collocation points and the matrix operations, nonlinear differential equation problem is converted to a system of nonlinear algebraic equations. The solutions of this system give the coefficients of assumed approximate solution. To demonstrate the validity and applicability of the technique, numerical examples are included and the comparisons are made with the existing results. The results show the efficiently and accuracy of the present work. Mathematics Subject Classification: 65L80; 65L60; 34B15; 65L10; 65L05. Keywords: Nonlinear differential equations; the Bessel functions of the first kind; collocation method; matrix method; numerical solutions.

1. Introduction Nonlinear differential equations play a significant role in many physical and technical applications [1-8]. For example, the simulation of electrical networks and mechanical systems, and the solution of the equations of fluid Dynamics. Most of these equations have not analytical solution and numerical techniques may be used to obtain approximate ones. For example, some nonlinear differential equations have been solved by using the Taylor matrix method [9], the closed-form [10], Chebyshev polynomial approximations [11], the variational iteration method [12], the subdomain finite element method [13], the differential quadrature method [14,15], the variational iteration method [16-17], He’s variational iteration method [18], the cubic B-spline scaling functions and Chebyshev cardinal functions [19], the homotopy perturbation method [20-21], the variation of parameters method [22], the Adomian decomposition method [23-24], the quintic B-spline differential quadrature method [25], the variational iteration method, the power series method [26], the Adomian decomposition method [27], the pade series method [28], the Legendre polynomial fucntion approximation [29], A Taylor method [30], Chebyshev series method [31] and the modified variational iteration method [32]. Recently, Taylor, Chebyshev, Legendre, Bernstein and Bessel matrixcollocation methods have been used [33-47] to solve some types of differential, integral, integro-differential-difference equations. Also, Sheikholeslami et al. [51-61] have studied on solutions of various differential equations and model problems. In this paper, by means of collocation points and the matrix relations between the Bessel functions of the first kind, we develope the Bessel collocation method [38,39] for solving the m th-order nonlinear differential equation m

n

m

m

 Pk ,r ( x)y r ( x) y (k ) ( x)   Qk ,r ( x)y (r ) ( x) y (k ) ( x)  g ( x) , 0  a  x  b k 0 r 0

(1)

k 1 r 1

with the intial and boundary conditions m-1

 (a k 0

jk

y ( k ) (a)  b jk y ( k ) (b))   j , j  0,1,..., m  1

(2)

where y (0) ( x)  y( x) , y 0 ( x)  1 and y ( x) is an unknown function, the known functions Pk ,r ( x) , Qk ,r ( x) and g ( x) are defined on interval a  x  b , and a jk , b jk and  j are real or complex constants. Our aim is to find an approximate solution of (1) expressed in the truncated Bessel series form N

y ( x)   an J n ( x) ,

a xb .

(3)

n 0

Here, an , n  0,1, 2,..., N are the unknown Bessel coefficients. Here, N is chosen any positive integer such that N  m , and J n ( x) , n  0,1, 2, , N are the Bessel functions of first kind defined by

J n ( x) 

N n 2

 k 0

(1)k  x    k !(k  n)!  2 

2k n

, n , 0  x   .

2. Main matrix relations required for solution method Firstly we can write the Bessel polynomials J n ( x) in the matrix form as follows [38,39]; JT ( x)  DXT ( x)  J( x)  X( x)DT where

(4)

J( x)   J 0 ( x) J1 ( x)

J N ( x) , X( x)  1 x x 2

x N  ;

and if N is odd and even, respectively,

D

D

   1  0   0!0!2       0          0           0     0      1  0   0!0!2       0          0           0     0  

1

0

1!1!2

1 1

( 1)

0

0

1

( 1)

0!1!2

0

0!2!2

0

2

0

( 1)

N 3 2

0

 N  3   N  1  N 1   !  !2  2  2 

0!( N  1)!2

0

N 1 2

 N 1  N 1   !   2  2 

1

0

0

N 1 2

 N  1   N  1  N 1   !  !2  2  2 

2

0

N 1

1

0

0! N !2 1

0

1!1!2

1 1

2

( 1)

0

0

1 0!2!2

0

0

0

0

  

N 2 2

 N  2   N  N 1   !  !2  2   2

0!1!2

2

0

N  

1 0!( N  1)!2 0

     N N N  ! !2   2 2     0       N 2  . 2  ( 1)   2 N 2 N  ! !2   2 2            N  0! N !2 ( N 1)( N 1)

( 1)

0

N 1

N

             N  !2     ,                   ( N 1)( N 1)

     

     

N 2

  

  

0

1

We consider the desired solution y ( x) of Eq. (1) defined by the truncated Bessel series (3). Then the function defined in relation (3) can be written in the matrix form T aN  [ y( x)]  J( x)A ; A   a0 a1 or from Eq. (4) (5) [ y( x)]  X( x)DT A . On the other hand, it is clearly seen from [38] that the relation between the matrix X( x) and its derivatives X( k ) ( x) is (6) X( k ) ( x)  X( k 1) ( x)BT  X( x)(BT )k where 0 0  BT    0 0

1 0 0 2 0 0 0 0

0 0  .   N 0 

Using the relations (5) and (6) we have the matrix relation as follows y ( k ) ( x)  X( x)(BT )k DT A , k  0,1, 2,, m . By using collocation points xi defined by ba xi  a  i , i  0,1, , N N in Eq. (5), we have the system of matrix equations y ( k ) ( xi )  X( xi )(BT )k DT A or compact form Y( k )  X(BT )k DT A

(7)

(8) (9) (10)

where  X( x0 )   y ( k ) ( x0 )   X( x )   (k )  y ( x1 )  . 1  and X Y(k )        (k )     y ( xN )   X( xN ) 

Similarly, substituting the collocation points (8) into the y r ( x) y ( k ) ( x) [42] and y ( r ) ( x) y ( k ) ( x) , and using the relations (7), we obtain the matrix representation  y r ( x0 ) y ( k ) ( x0 )   y r ( x0 ) 0  r   (k ) r y ( x1 )  y ( x1 ) y ( x1 )    0     r   (k ) 0  y ( xN ) y ( xN )   0 (r ) (k ) (r )  y ( x0 ) y ( x0 )   y ( x0 ) 0  (r )   (k ) (r ) y ( x1 )  y ( x1 ) y ( x1 )    0     (r )   (k ) 0  y ( xN ) y ( xN )   0

where

Y = XDA and  Y 

 X( x0 ) 0  0 X( x1 ) X   0  0

(r )

  y ( k ) ( x0 )    (k )    y ( x1 )    Y r Y ( k ) ,      y r ( xN )   y ( k ) ( xN )  0   y ( k ) ( x0 )    0   y ( k ) ( x1 )  (r )   Y  Y(k )     y ( r ) ( xN )   y ( k ) ( xN )  0 0

 X  B  DA r

(11)

(12)

0   BT 0   T 0  , B 0 B     X( x N )  0 0

 DT 0   0 0 , D    T B  0

A 0 0 0 A  0 , A    T D  0 0

0 DT 0

0 0  .   A

3. Method of solution To obtain a Bessel polynomial solution of Eq. (1) under the mixed conditions (2), the following matrix method is used. This method is based on computing the Bessel coefficients by means of collocation points. Firstly, the collocation points are substituted in Eq. (1) m

n

m

m

 Pk ,r ( xi )y r ( xi ) y (k ) ( xi )   Qk ,r ( xi )y (r ) ( xi ) y (k ) ( xi )  g ( xi ) k 0 r 0

k 1 r 1

and then this system can be written in the matrix form m

n

 P  Y  k 0 r 0

k ,r

r

m

n

Y( k )   Qk ,r  Y  Y ( k )  G (r )

(13)

k 1 r 1

where 0  Pk ,r ( x0 )  0 Pk ,r ( x1 ) Pk ,r =    0  0

 0 Qk ,r ( x0 )   0  Qk ,r ( x1 ) , Q k ,r =  0     Pk ,r ( xN )  0  0 0

  g ( x0 )     0  and G =  g ( x1 )  .       Qk ,r ( xN )   g ( xN )  0

Let us substitute the relations (10) and (12) into Eq. (13). Thus we have the fundamental matrix equation m n k r r m n  T k T P XDA X B D  Qk ,r X  B  DAX  BT  DT  A  G . (14)     k ,r   k 1 r 1  k 0 r 0  Briefly, Eq. (14) can also be written in the form (15) WA  G or [ W; G] where

W =  Pk ,r  XDA  X  BT  DT   Qk ,r X  B  DAX  BT  DT . m

n

m

k

r

k 0 r 0

n

r

k

k  r 1

Here, Eq. (15) corresponds to a system of the ( N  1) nonlinear algebraic equations with the unknown Bessel coefficients an , n  0,1, 2,..., N . Now, let us find a matrix representation of the mixed conditions (2). Using the relation (7) at points a and b , the matrix representation of mixed conditions (2) which depends on the Bessel coefficients matrix A becomes  m1 T (k ) T    a jk X(a)  b jk X(b) (B ) D  A =  j , j  0,1, 2,..., m  1  k 0  or briefly (16) U j A  [ j ] or [U j ;  j ] ; j  0,1, 2,..., m  1 where m 1

U j    a jk X(a)  b jk X(b) (BT )( k ) DT k 0

 u j 0

u j N  .

uj 1 uj 2

Consequently, by replacing the row matrices (16) by the m rows of the augmented matrix (15), we have the new augmented matrix [ W; G] or WA  G which is a nonlinear algebraic system. For convenience, if last rows of the matrix are replaced, the new augmented matrix of the above system is as follows:  w0,0  w  1,0  w2,0    wN  m ,0 [ W; G ]    u0,0  u  1,0  u2,0    um 1,0

w0,1

w0,2



w0, N

;

w1,1 w2,1

w1,2 w2,2

 

w1, N w2, N

; ;

wN  m ,1

wN  m ,2

 wN  m , N

;

u0,1 u1,1 u2,1

u0,2 u1,2 u2,2

  

u0, N u1, N u2, N

; ; ;

um 1,1

um 1,2



um 1, N

;

g ( x0 )  g ( x1 )  g ( x2 )  .  g ( xN  m )   0  1   2     m 1 

(17)

However, we do not have to replace the last rows. For example, if the matrix W is singular, then the rows that have the same factor or all zero are replaced. Thus, by solving nonlinear system (17), the unknown Bessel coefficients an , n  0,1, 2,..., N are determined and substtuted in (3); thus we get the Bessel polynomial solution. 4. Accuracy of solution We can easily check the accuracy of the method. Since the truncated Bessel series (3) is an approximate solution of Eq. (1), when the function yN ( x) and its derivatives are substituted in Eq. (1), the resulting equation must be satisfied approximately; that is, for x  xq [a, b] , q  0,1, 2, , E ( xq ) 

m

n

m

m

 Pk ,r ( xq )y r ( xq ) y (k ) ( xq )   Qk ,r ( xq )y (r ) ( xq ) y (k ) ( xq )  g ( xq )  0 k 0 r 0

k 1 r 1

or

E ( xq )  10

 kq

( k q is any positive integer).

k

If max 10 q  10 k ( k is any positive integer) is prescribed, then the truncation limit N is increased until the difference E ( xq ) at each of the points becomes smaller than the prescribed 10 k , see [38-40, 43-44]. On the other hand, the error can be estimated by the function m

n

m

m

EN ( x)   Pk ,r ( x)yNr ( x) yN( k ) ( x)   Qk ,r ( x)yN( r ) ( x) yN( k ) ( x)  g ( x) . k 0 r 0

k 1 r 1

5. Numerical examples The method of this study is useful in finding the numerical solution of the nonlinear ordinary differential equations in terms of Bessel polynomials and also the error analysis of the solutions. In this regard, we have reported in Tables and Figures, the values of exact solution y ( x) , the polynomial approximate solution yN ( x) , the error function m

n

m

m

EN ( x)   Pk ,r ( x)yNr ( x) yN( k ) ( x)   Qk ,r ( x)yN( r ) ( x) yN( k ) ( x)  g ( x) and the absolute error k 0 r 0

k 1 r 1

function eN ( x)  y( x)  yN ( x) at the selected points of the given interval. All the numerical computations have been done using Maple and MATLAB. We now illustrate the numerical solution with the following examples. Example 1. [9] Let us first consider the Abel differential equation y( x) y '( x)  xy( x)  y 2 ( x)  x 2 y3 ( x)  xe x  x 2e3 x , 0  x  1 with initial condition

(18)

y(0)  1

and the approximate the solution y( x) by Bessel polynomial N

y ( x)   an J n ( x) n 0

where N  4, P0,0 (x)  x, P0,1(x)  1, P0,2 (x)  x 2 , P1,1(x)  1 and g ( x)  xe x  x 2e3 x . Hence, the set of collocation points (8) for N  4 is computed as 1 2 3    x0  0, x1  , x2  , x3  , x4  1 4 4 4   and from Eq. (14), the fundamental matrix equation of the problem is

P

0,0

where



XDT  P0,1XDAXDT  P0,2  XDA  XDT  P1,1XDAXBT DT A  G 2

W = P0,0 XDT  P0,1XDAXDT  P0,2  XDA  XDT  P1,1XDAXBT DT 2

1 0 0 0 0  0 0 1 0 0 1 16 0 1 0 0 0  0  0 1 0 0  , P0,2  0 0 1 4 0   0 0 1 0 0 9 16 0 0   0 0 0 1 0 0 0 0

0 0  0 , P0,0  0 1  0 0 0 0   1  X(0)  1   0  X(1/4)  1 1/4 1/16 1/64 1/256  1/2 0 0 0       T X   X(1/2)   1 1/2 1/4 0 1/8 0 0 , 1/8 1/16  , D   1/4       1/16 0 1/48 0   0  X(3/4)  1 3/4 9/16 27/64 81/256  1/64  X(1)  1 1 0 1/96 0 1/384  1 1 1  0 0 1 0 0 1 4 0 0   0 0 1 2 0  0 34 0 0 0 0 0 0

0 0  0  0 P  P  1,1 0  , 0,1   0 0 0  1 0 0 0 0

0 0 0 0   X(0)  DT  0   X(1/4) 0 0 0   0 X 0 0 X(1/2) 0 0 , D   0    0 0 X(3/4) 0   0 0 0  0 0 0 0 X(1)  

0 0  BT   0  0 0

1 0 0 0 A 0 0 2 0 0   0 0 3 0 , A   0   0 0 0 4 0  0 0 0 0 0 

0 DT

0 0

0 0

0 0 0

DT 0 0

0 DT 0

0  0 0 ,  0 DT 

0   0 772/3443   A 0 0 0   0 A 0 0  , G   377/1050  .    0 0 A 0  555/1342   383/917  0 0 0 A  0

0

0

From Eq. (15), the matrix form for initial condition is  U0 ; 0   1 0 0 0 ; 1 . Thus, the new augmented matrix [ W; G] for the problem is gained. Solving this system, the Bessel coefficients matrix are found as T A  1 2 5.98953683691 13.6663755372 29.1457627923 and the approximate solution for N  4 in terms of the Bessel polynomials is obtained as y4 ( x)  1  x  0.498692104614 x2  0.159716157025x3  (0.291344152205e  1) x4 . Table 1 shows the numerical results of the exact solution, the approximate solutions and absolute error functions obtained by the present method and Taylor matrix method [9]. Figure 1-(a) displays the exact solution and the approximate solutions obtained by the present method and Taylor matrix method [9]. Figure 1-(b) shows the absolute error functions obtained by the present method and Taylor matrix method [9]. It is seen from Table 1 and Figure 1 that the results obtained by the present method are better than that obtained by the Taylor matrix method. Table 1Numerical results of of the solutions and the absolute error functions for the xi values of Eq. (18) Exact solution  xi

xi

y ( xi )  e

0 0.2 0.4 0.6 0.8 1

1 0.818730753078 0.670320046036 0.548811636094 0.449328964117 0.367879441171

Taylor matrix method [9]

Present method

N  5 , y5 ( xi )

e5 ( xi )

N  4 , y4 ( xi )

e4 ( xi )

1 0.818730666670 0.670314666670 0.548752000000 0.449002666670 0.366666666670

0 0.86408e-7 0.5379366e-5 0.59636094e-4 0.326297447e-3 0.1212774501e-2

1 0.818730753078 0.670320046036 0.548811636094 0.449328964117 0.367879441171

0 0.14183094e-4 0.5302316e-5 0.5348141e-5 0.7233094e-5 0.230921634e-3

Figure 1. For N  4 , 5 of Eq. (18), (a) Comparison of the solutions y ( x ) and

(b) Comparison of the absolute error functions

eN ( x) .

Example 2. Consider the fourth order nonlinear differential equation y (4) ( x)  y 2 ( x) y (3) ( x)  sin( x) y (1) ( x) y (2) ( x)  cos( x) y( x)  sin( x)(1  cos( x)) , 0  x  1

(19)

with initial conditions y(0)  0 , y (1) (0)  1 , y (2) (0)  0 , y (3) (0)  1 and the exact solution y( x)  sin( x) . Here, P0,0 (x)  cos( x), P3,2 (x)  1, P4,0 (x)  1, Q2,1(x)  sin( x) and g ( x)  sin( x)(1  cos( x)) . From Eq. (14), the fundamental matrix equation of the problem is

P

0,0



XDT  P3,2  XDA  X  BT  DT  P4,0 X  BT  DT  Q2,1XBDAX  BT  DT A  G. 2

3

4

2

As in previous example, we find the approximate solutions by Bessel polynomials of the problem for N  5 , 8 and 11 , respectively, y5 ( x)  x  0.166666666667x3  (0.827782992292e-2) x5 , y8 ( x)  x  0.166666666667x3 +(0.833293560410e-2) x5  (0.221285182398e-5) x 6  (0.203739225263e-3) x 7  (0.612261224506e-5) x8 ,

y11 ( x)  x  0.166666666667x3 +(0.833333334324e-2) x5  (0.976299890517e-10) x 6  (0.198412220176e-3) x 7  (0.138432558003e-8) x8  (0.275817178334e-5) x9  (0.249393772211e-8) x10  (0.238477766383e-7) x11. Table 2 and Table 3 show numerical results of the exact solution, the approximate solutions, the error functions and the absolute error funtions of Eq. (19) by presented method for N  5, 8 and 11 . We display the exact and approximate solutions in Figure 2-(a). The

error functions EN ( x) for N  5, 8, 11 are compared in Figure 2-(b), and Figure 2-(c) shows the zoomed in Figure 2-(b) in the interval 0.2  x  1 . The absolute error functions eN ( x) for N  5, 8, 11 are compared in Figure 2-(d), and Figure 2-(e) shows the zoomed in Figure 2-(d) in the interval 0.2  x  1 . It is seen from Table 2-3 and Figure 2 that the numerical results of the absolute errors is better than the numerical results of the error functions. Also, Table 3 and Figure 2 show that as N increases, the errors decrease more rapidly.

Figure 2. For N  5 , 8 , 11 of Eq. (19), (a) Comparison of the solutions y ( x ) , (b) Comparison of the error functions E N ( x ) , (c) The zoomed in figure of (b) in the interval 0.2  x  1 , y ( x ) , (d) Comparison of the absolute error functions

eN ( x)

and (e) The zoomed in figure of (d) in the interval 0.2  x  1 .

Table 2 Numerical results of the solutions of Eq. (19) Exact solution Pesent metod y ( xi )  sin( xi )

xi 0 0.2 0.4 0.6 0.8 1

0 0.198669330795061 0.389418342308651 0.564642473395035 0.717356090899523 0.841470984807897

N  5, y ( xi ) 5

N  8, y ( xi ) 8

N  11, y ( xi ) 11

0 0.198669315572239 0.389418098311723 0.564643684054734 0.717379145975638 0.84161116325592

0 0.198669330755492 0.389418341863908 0.564642471757316 0.717356083082583 0.841470865175906

0 0.198669330795059 0.389418342308635 0.564642473394983 0.717356090899385 0.841470984804177

The numerical results of the errors fonctions E ( xi )

eN ( x) of Eq. (19) The numerical results of the absolute errors e ( xi ) N

E ( xi ) 5

E ( xi ) 8

E ( xi ) 11

e5 ( xi )

e8 ( xi )

e11 ( xi )

0 3.7104e-013 7.8422e-003 3.0223e-002 7.0787e-002 1.2832e-001

0 3.1885e-007 3.2999e-007 1.7527e-005 2.8419e-004 1.4540e-003

0 1.8065e-012 1.1354e-012 4.0131e-011 1.2926e-008 3.2605e-007

0 1.5223e-008 2.4400e-007 1.2107e-006 2.3055e-005 1.4018e-004

0 3.9570e-011 4.4474e-010 1.6377e-009 7.8169e-009 1.1963e-007

0 2.1392e-015 1.5959e-014 5.2335e-014 1.3770e-013 3.7191e-012

Table 3 Numerical results of the error functions E N ( x ) and the absolute error functions N

xi 0 0.2 0.4 0.6 0.8 1

Example 3. [48] We consider the fifth order nonlinear differential equation y (5) ( x)  e x y 2 ( x) , 0  x  1

(20)

with boundary conditions y(0)  y (1) (0)  y (2) (0)  1 , y(1)  y (1) (1)  e and exact solution y( x)  e x .Here, P0,1(x)  e x , P5,0 (x)  1 and g ( x)  0 . From Eq. (14), the fundamental matrix equation of the problem is

P XDAXD  P T

0,1

5,0



X  BT  DT A  G. 5

By following the method given in Section 3, we comput the approximate solutions by Bessel polynomials of the problem for N  6 , 9 and 12 , respectively, y6 ( x)  1  x  0.5x2  0.166201435154x3  (0.422357517440e-1)x4  (0.83333333333e-2)x5  (0.151130822653e-2)x6 , y9 ( x)  1  x  0.5x 2  0.166666476627x3  (0.416668763037e-1)x 4  (0.83333333333e-2)x5  (0.138883783840e-2)x6  (0.198683817987e-3)x7 +(0.241719348816e-4)x8 +(0.34486066002e-5)x9

and y12 ( x)  1  x  0.5x 2  0.166666666650x3  (0.416666666841e-1)x 4  (0.83333333334e-2)x5  (0.138888889483e-2)x6  (0.198412646428e-3)x7 +(0.248018216818e-4)x8 +(0.275508571596e-5)x9  (0.27669700688e-6)x10 +(0.238543486750e-7)x11 +(0.279135952441e-8)x12 . Table 4 shows numerical results of the exact solution and the approximate solutions obtained by the present method of Eq. (20) for N  6, 9 and 12 . Figure 3-(a) displays the exact solution and the approximate solutions obtained by the present method of Eq. (20) for N  6, 9 and 12 . The numerical results of the absolute error functions obtained by the present method for N  6, 9 and 12 , Variational iteration method [48] and Decomposition method [49] are compared in Table 5. Figure 3-(b) displays the absolute error functions obained by the present method for N  6, 9 and 12 , variational iteration method [48] and decomposition method [49]. Figure 3-(c) shows the zoomed in Figure 3-(b) in the interval 0.2  x  1 . It is seen from Table 5 and Figure 3-(b)-(c) that the results obtained by the present method are better than that obtained by the other methods. Also, as can be seen from the Table 4 and Figure 3-(a), the result obtained by the Bessel polynomial method for N  12 is almost the same as the results of the exact solution.

Figure 3. For N  6 , 9 and 12 of Eq. (20), (a) Comparison of the solutions y ( x ) , (b) Comparison of the absolute error functions and (c) The zoomed in figure of (b) in the interval 0.2  x  1 .

Table 4 Numerical results of the solutions of Eq. (20) Exact solution Present method

xi 0 0.2 0.4 0.6 0.8 1

x y ( xi )  e i

N  6, y ( xi ) 6

N  9, y ( xi ) 9

N  12, y ( xi ) 12

1 1.22140275816 1.49182469764 1.82211880039 2.22554092849 2.71828182846

1 1.22139995207 1.49180965075 1.82209177502 2.22552174576 2.71828182846

1 1.22140275697 1.49182469082 1.82211878636 2.22554091600 2.71828182846

1 1.22140275816 1.49182469764 1.82211880039 2.22554092849 2.71828182846

Table 5 numerical the absolute error functions for the xi value of Eq. (20)

xi 0 0.2 0.4 0.6 0.8 1

e( xi )

The results of

Absolute errors with Pesent method for N  6, 9 and 12

Absolute errors e( xi )

Absolute errors

with variational iteration method [48]

with decomposition method [49]

e6 ( xi )

e9 ( xi )

e12 ( xi )

0 1.0e-05 1.0e-04 3.6e-04 3.1e-04 9.9e-05

0 2.0e-09 2.0e-08 3.7e-08 3.1e-08 0

0 2.8061e-006 1.5047e-005 2.7025e-005 1.9183e-005 1.2157e-012

0 1.1860e-009 6.8217e-009 1.4027e-008 1.2494e-008 2.8233e-012

0 1.0536e-013 6.1807e-013 1.3265e-012 1.3479e-012 1.7478e-013

Example 4. [20] Consider the Riccati equation y '( x)  2 y( x)  y 2 ( x)  1 , 0  x  1

(21)

  2  1  1 with initial condition y(0)  0 and exact solution y ( x)  1  2 tanh  2 x  log    2   2  1   where P1,0 (x)  1, P0,0 (x)  2, P0,1(x)  1, and g( x)  1. This problem was solved by homotopy perturbation method [20] and standard Adomian’s decomposition method [50] and the approximate the solutions were found as follows respectively 1 1 7 7 53 7 221 8 y ( x)  x  x 2  1 x3  2 x 4  2 x5 . yh ( x)  x  x 2  x3  x 4  x5  x 6  x  x, a 3 3 15 3 3 15 45 315 1260

From Eq. (14), the fundamental matrix equation of the problem is

P XB D  P T

T

1,0

0,0



XDT  P0,1  XDA  XDT A  G. 1

Hence, we gain the approximate solutions by the Bessel polynomials of the problem for N  5 and 8 , respectively, y5 ( x)  x  0.924115434628x2  0.806003058025x3  1.40146850400x4  0.353711696628x5 , y8 ( x)  x  0.998333887865x 2  0.357213815946x3  0.488624978309x 4  (0.958889747694e-1) x5  1.34728883879x6  1.58027829220x7  0.506481123628x8 .

For the numerical results of Eq (22), see Table 6-7, we display the exact solution and the approximate solutions obtained by the present method, Adomian method [50] and homotopy method [20] of Eq. (21) in Figure 4-(a). Table 7 show the numerical results of the the absolute error functions obained the present method, Adomian method [50] and homotopy method [20]. Figure 4-(c) displays the zoomed in Figure 4-(b) in the interval 0.2  x  1 . It is seen from Table 6-7 and Figure 4 that our results are better than that obtained by the other methods

Table 6 Numerical results of the solutions of Eq. (21) Adomian Homotopy Exact solution method [50] method [20]

xi

y ( xi )

Present method

N  5, y ( xi ) 5

N  8, y ( xi ) 8

N  5, y ( xi ) 5

N  8, y ( xi ) 8

0

0

0

0

0

0

0.2

0.241976799621

0.241642666667

0.241972339810

0.241283479986

0.241972655034

0.4

0.567812166293

0.565632000000

0.567223458540

0.567187079325

0.567806915337

0.6

0.953566216472

0.955968000000

0.943490358857

0.952652520411

0.953560545713

0.8

1.34636365537

1.38129066667

1.27457933410

1.34597019338

1.34635853077

1

1.68949839159

1.80000000000

1.38492063492

1.68236168528

1.68932003005

Table 7 Numerical results of the absolute error functions of Eq. (21) Adomian method Homotopy Present method [50] method [20]

xi

e5 ( xi )

e8 ( xi )

e5 ( xi )

e8 ( xi )

0

0

0

0

0

0.2

3.2999e-004

3.1522e-007

6.9332e-004

4.1446e-006

0.4

2.1749e-003

5.8346e-004

6.2509e-004

5.2510e-006

0.6

2.4075e-003

1.0070e-002

9.1370e-004

5.6708e-006

0.8

3.4932e-002

7.1779e-002

3.9346e-004

5.1246e-006

1

1.1068e-001

3.0440e-001

7.1367e-003

1.7836e-004

Figure 4. For N  5, 8 of Eq. (21), (a) Comparison of the solutions y ( x ) , (b) Comparison of the absolute error functions

eN ( x)

, and (c) The zoomed in figure of (b) in the interval 0.2  x  1 .

Example 5. Finally, consider the fourth order nonlinear differential equation y(4) ( x)  xy 2 ( x)  y3 ( x) y(2) ( x)  xy(1) ( x) y(2) ( x)  x2 y(4) ( x) y(3) ( x)  12x14  x9  48x6  576x3  24 , 0  x  1

with initial condition y(0)  y (1) (0)  y (2) (0)  y (3) (0)  0 so that P4,0 (x)  1, P0,1(x)  x, P2,3 (x)  1,

Q2,1(x)   x, Q3,4 (x)   x 2 and g ( x)  12 x14  x9  48x6  576 x3  24 . From Eq. (14), the fundamental matrix equation of the problem is P X  BT 4 DT  P XDAXDT  P  XDA 3 X  BT 2 DT  Q XBDAX  BT 2 DT  0,1 2,3 2,1  4,0    A  G. 3 4 Q3,4 X  B  DAX  BT  DT    Thereby, we find the approximate solutions by the Bessel polynomials of the problem for N  4 is the same as y( x)  x 4 , the exact solution. 6. Conclusions Nonlinear ordinary differential equations are usually difficult to solve analytically. In many cases, it is required to obtain approximate solutions. For this purpose, a numerical approach by using the Bessel functions of the first kind is presented for a class of nonlinear differential equations. The method is explained by the numerical examples. In Figures and Tables of the examples, we give the values of the solutions and error functions. It is observed that the results are effective. Also, our results are compared with the other methods and the exact solution. From them, it is seen that the present method gives good results. The numerical

results show that the accuracies of the solutions improve when N is increased. It can be seen from Tables and Figures. On the other hand, one of the considerable advantages of the method is that the Bessel coefficients of the solution are found very easily by using the computer pragrams. Acknowledgements The author is supported by the Scientific Research Project Administration of Akdeniz University. The author would like to thank anonymous referees for their valuable comments that help improve this manuscript.

References [1] J. P. M. Lebrun, On two coupled Abel-type differential equations arising in a magnetostatic problem, Il Nuovo Cimento,1990, 103A, 1369-1379. [2] F. Borghero and A. Melis, On Szebehely's problem for holonomic systems involving generalized potential functions, Celestial Mechanics and Dynamical Astronomy, 1990, 49, 273-284. [3] S. S. Bayin, Solutions of Einstein's field equations for static fluid spheres, Phys. Rev. 1978, D18, 2745-2751. [4] A. Garcia, A. Macias and E.W. Mielke, Stewart-Lyth second order approach as an Abel equation for reconstructing inflationary dynamics, Phys. Lett. 1997, A229, 32-36. [5] V. R. Gavrilov, V.D. Ivashchuk and V. N. Melnikov, Multidimensional integrable vacuum cosmology with two curvatures, Class. Quantum Gray. 1996, 13, 3039-3056. [6] G. Haager and M. Mars, A self-similar inhomogeneous dust cosmology, Glass. Quantum Gray. 1998, 15, 1567-1580. [7] M. K. Mak and T. Harko, Full causal bulk-viscous cosmological models, J. Math. Phys. 1998, 39, 54585476. [8] M. K. Mak, T. Harko, Addendum to "Exact causal viscous cosmologies", Gen. Rel. Gray. 1999, 31, 273-274. [9] C. Guler, A new numerical algorithm for the Abel equation of the second kind, Int. Journal of Comput. Math. 2007, 84, 109-119. [10] M.P. Markakis, Closed-form solutions of certain Abel equations of the first kind, Appl. Math. Letters, 2009, 22, 1401-1405. [11] B. Chen, R. Garcı´a-Bolo´s, L. Jo´dar, M.D. Rosello´ , Chebyshev polynomial approximations for nonlinear differential initial value problems, Nonlinear Anal, 2005, 63, e629–e637. [12] M. Merdan, On the solutions of nonlinear fractional Klein–Gordon equation with modified Riemann– Liouville derivative, Applied Mathematics and Computation, 2014, 242, 877-888. [13] T.Geyikli, S.B.G. Karakoç, Subdomain finite element method with quartic B-Splines for the modified equal width wave equation, Comput. Math. Math.Phy. 2015, 3, 410-421. [14] R. Jiwari, S. Pandit, R C Mittal, Numerical simulation of two-dimensional sine-Gordon solitons by differential quadrature method, Comput. Phys. Commun., 2012, 183, 600-616. [15] R.C. Mittal, R. Jiwari, A higher order numerical Scheme for some nonlinear differential equations models in biology, Int. J. for Comput. Methods in Eng. Science and Mech., 2011, 12(3), 134-140. [16] M. Dehghan, F. Shakeri, Approximate solution of a differential equation arising in astrophysics using the variational iteration method, New Astronomy 2008, 13, 53–59. [17] F. Genga, Y. Lin, M. Cui, A piecewise variational iteration method for Riccati differential equations, Comput. Math. Appl. 2009, 58, 2518–2522. [18] S. Abbasbandy, A new application of He’s variational iteration method for quadratic Riccati differential equation by using Adomian’s polynomials, J. Comput. Appl. Math. 2007, 207, 59–63. [19] M. Lakestani, Mehdi Dehghan, Numerical solution of Riccati equation using the cubic B-spline scaling functions and Chebyshev cardinal functions, Comput. Phys. Commun. 2010, 181, 957–966. [20] S. Abbasbandy, Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian’s decomposition method, Appl. Math. Comput. 2006, 172, 485–490. [21] J. H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons & Fractals 2005, 26, 695–700. [22] M. A. Noor, S. T. Mohyud-Din and A. Waheed, Variation of parameters method for solving fifth-order boundary value problems, App. Math. Inform.Sci., 2(2) (2008), 135-141. [23] A.M. Wazwaz, S. M. El-Sayed, A new modification of the Adomian decomposition method for linear and nonlinear operators, Appl. Math. Comput. 2001, 122, 393–404. [24] M.A. El-Tawil, A.A. Bahnasawi, A. Abdel-Naby, Solving Riccati differential equation using Adomian’s decomposition method, Appl. Math. Comput. 2004, 157, 503–514. [25] A.Başhan, S. B.G. Karakoç, Turabi Geyikli, Approximation of the KdVB equation by the quintic B-spline differential quadrature method, Kuwait Journal of Science, 42 2015, 67-92. [26] S.K. Vanani, A Aminataei, On the numerical solution of differential equations of Lane-Emden type, Comput. Math. Appl., 2010, 59(8), 2815-2820. [27] M. Dehghan, R. Salehi, The use of variational iteration method and Adomian decomposition method to solve the Eikonal equation and its application in the reconstruction problem, Comm. Numer. Methods in Engrg. 2011, 27, 524–540. [28] S.K. Vanani, A Aminataei, On the numerical solution of nonlinear delay differential equations, J.Conc. Appl. Math., 2010, 8(4) p568. [29] S. R. Alavizadeh. F. M. Maalek Ghaini, Numerical Solution of Higher-Order Linear and Nonlinear Ordinary Differential Equations with Orthogonal Rational Legendre Functions, J.Math.Extension, 8(4) (2014) 109-130.

[30] B.Bülbül, M.Sezer, A numerical approach for solving generalized Abel-type nonlinear differential equations, App. Mat. Comput. 2015, 262 169-177. [31] A.Akyüz-Daşcıoğlu, H. Çerdik-Yaslan, The solution of high-order nonlinear ordinary differential equations by Chebyshev Series, Appl. Math. Comput., 2011, 217, 5658-5666 [32] M. A.Noor and S.T. Mohyud-Din, Solution of singular and nonsingular initial and boundary value Problems by modified variational iteration method, 2008 (2008), Article ID 917407, 23 pages. [33] S. Yalçınbaş, M. Sezer, H. H Sorkun, Legendre polynomial solutions of high–order linear Fredholm integro-differential equations, Appl. Math. Comput. 2009, 210, 334-349. [34] N. Kurt, M. Sezer, Polynomial solution of high-order linear Fredholm integro-differential equations with constant coefficients, J. Franklin Ins. 2008, 345, 839-850. [35] S. Yalçınbaş, M. Sezer, The approximate solution of high-order linear Volterra-Fredholm integrodifferential equations in terms of Taylor polynomials, Appl. Math. Comput. 2000, 112, 291-308. [37] Ş. Yüzbaşı, Laguerre approach for solving pantograph-type Volterra integro-differential equations, App. Math. Comput., 2014, 232, 1183-1199. [38] Ş. Yüzbaşı, N. Şahin, M. Sezer, Numerical solutions of systems of linear Fredholm integro-differential equations with Bessel polynomial bases, Comput. Math. Appl. 2011, 61, 3079–3096. [39] Ş Yüzbaşı, A numerical approach to solve the model for HIV infection of CD4 +T cells, App.Math. Model. 2012, 36, 5876-5890. [40] A. Akyüz-Daşcıoğlu, H. Ç. Yaslan, An approximation method for solution of nonlinear integral equations, Appl. Math. Comput. 2006, 174, 619-629. [41] Ş Yüzbaşı, A numerical scheme for solutions of the Chen system, Math. Meth. App. Sci., 2012, 35, 885– 893. [42] Ş Yüzbaşı, N.Şahin, On the solutions of a class of nonlinear ordinary differential equations by the Bessel polynomials, J. Numer. Math., 20(1) (2012) 55-79. [43] K. Maleknejad, Y. Mahmoudi, Taylor polynomial solutions of high–order nonlinear Volterra-Fredholm integro-differential equation, Appl. Math. Comput. 2003, 145, 641-653. [44] S. Yalçinbaş, Taylor polynomial solution of nonlinear Volterra-Fredholm integral equations, Appl. Math. Comput. 2002, 127, 196-206. [45] Ş. Yüzbaşı, Numerical solutions of hyperbolic telegraph equation by using the Bessel functions of first kind and residual correction. Appl. Math. Comput. 287-288 (2016) 83-93. [46] Ş. Yüzbaşı, A collocation method based on Bernstein polynomials to solve nonlinear Fredholm-Volterra integro-differential equations. Appl. Math. Comput. 273 (2016) 142-154. [47] Ş. Yüzbaşı, A collocation method based on the Bessel functions of the first kind for singular perturbated differential equations and residual correction. Math. Methods Appl. Sci. 38(14) (2015) 3033-3042. [48] J. Zhang, The numerical solution of fifth-order boundary value problems by the variational iteration method, Comput. Math. Appl. 2009, 58, 2347-2350. [49] A.M. Wazwaz, The numerical solution of fifth-order boundary-value problems by Adomian decomposition, J. Comput. Appl. Math. 2001, 136 , 259-270. [50] G. Adomian, Solving frontier problems of fhysics: Decomposition method, Kluwer Academic Publishers, Dordrecth, 1994. [51] M. Sheikholeslami, D. D. Ganji, H. R. Ashorynejad, Houman B. Rokni, Analytical investigation of JefferyHamel flow with high magnetic field and nano particle by Adomian decomposition method, Appl. Math. Mech.Engl. Ed, 33 (1) (2012) 1553-1564. [52] M. Sheikholeslami, D. D. Ganji, H. R. Ashorynejad, Investigation of squeezing unsteady nanofluid flow using ADM, Powder Technology, 239 (2013) 259-265. [53] M.Sheikholeslami, M.Azimi , D.D.Ganji, Application of Differential Transformation Method for Nanofluid Flow in a Semi-Permeable Channel Considering Magnetic Field Effect, Inter. J.Comput.Meth.Eng. Sci. Mech.16(4) (2015) 246-255. [54] M. Sheikholeslami, D.D. Ganji, Magnetohydrodynamic flow in a permeable channel filled with nanofluid, Scientia Iranica B (2014) 21(1), 203-212. [55] M.Sheikholeslami, Hamid Reza Ashorynejad ,Davood Domairry, Ishak Hashim, Investigation of the Laminar Viscous Flow in a Semi-Porous Channel in the Presence of Uniform Magnetic Field using Optimal Homotopy Asymptotic Method, Sains Malaysiana 41(10)(2012): 1177–1229.

[56] M. Sheikholeslami, DD. Ganji, Heat transfer of Cu-water nanofluid flow between parallel plates. Powder Technology2013;235:873–879. [57] M. Sheikholeslami, H. R. Ashorynejad, D. D. Ganji, A. Kolahdooz, Investigation of Rotating MHD Viscous Flow and Heat Transfer between Stretching and Porous Surfaces Using Analytical Method, 2011 (2011), Article ID 258734, 17 pages, http://dx.doi.org/10.1155/2011/258734. [58] M. Sheikholeslami, H.R. Ashorynejad, D.D. Ganji, A. Yıldırım, Homotopy perturbation method for threedimensional problem of condensation film on inclined rotating disk, Scientia Iranica, 19(3) (2012) 437–442 [59] M. Sheikholeslami, M.M. Rashidi, Dhafer M. Al Saad, F. Firouzi, Houman B. Rokni, G. Domairry,Steady nanofluid flow between parallel plates considering thermophoresis and Brownian effects, Journal of King Saud University - Science, doi:10.1016/j.jksus.2015.06.003. [60] M. Sheikholeslami, D.D.Ganji, Nanofluid flow and heat transfer between parallel plates considering Brownian motion using DTM, Comput. Meth. Appl.Mech. Eng.283 (2015) 651–663. [61] D. Domairry, M. Sheikholeslami, H.R. Ashorynejad,R. S. Reddy Gorla, M. Khani, Natural convection flow of a non-Newtonian nanofluid between two vertical flat plates, Proc IMechE Part N:J Nanoengineering and Nanosystems 225(3) 115–122©IMechE 2012. DOI: 10.1177/1740349911433468.