On the solution of the Abel equation of the second kind by the shifted Chebyshev polynomials

Applied Mathematics and Computation 217 (2011) 4827–4833

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On the solution of the Abel equation of the second kind by the shifted Chebyshev polynomials Mustafa Gülsu ⇑, Yalçın Öztürk, Mehmet Sezer Department of Mathematics, Faculty of Science, Mugla University, Mugla, Turkey

a r t i c l e

i n f o

a b s t r a c t

Keywords: Abel equation Shifted Chebyshev polynomials and series Chebyshev polynomial solutions Approximation method

This paper presents a new approximate method of Abel differential equation. By using the shifted Chebyshev expansion of the unknown function, Abel differential equation is approximately transformed to a system of nonlinear equations for the unknown coefficients. A desired solution can be determined by solving the resulting nonlinear system. This method gives a simple and closed form of approximate solution of Abel differential equation. The solution is calculated in the form of a series with easily computable components. The numerical results show the effectiveness of the method for this type of equation. Comparing the methodology with some known techniques shows that the present approach is relatively easy and highly accurate. Ó 2010 Elsevier Inc. All rights reserved.

1. Introduction In this paper we consider the nonlinear Abel differential equation of the second type [1]

½A1 ðtÞ þ A2 ðtÞyðtÞy0 ðtÞ þ BðtÞyðtÞ þ CðtÞy2 ðtÞ þ DðtÞy3 ðtÞ ¼ EðtÞ

ð1Þ

with the mixed condition

a1 yðaÞ þ b1 yðbÞ þ c1 yðcÞ þ a2 y2 ðaÞ þ b2 y2 ðbÞ þ c2 y2 ðcÞc2 y2 ðcÞ þ a3 y3 ðaÞ þ b3 y3 ðcÞ ¼ k

ð2Þ

and the solution is expressed in the form

yðtÞ ¼

N X

0

an T n ðtÞ;

T n ðtÞ ¼ cosðnhÞ;

2t  1 ¼ cos h;

0 6 t 6 1;

ð3Þ

n¼0

P where T n ðtÞ denotes the shifted Chebyshev polynomials of the first kind, 0 denotes a sum whose first term is halved, an (0 6 n 6 N) are unknown Chebyshev coefficients and N is chosen any positive integer such that N P m. The nonlinear differential equations are essential tools for modelling many physical situations: chemical reactions, spring-mass systems, bending of beams and so forth. These equations have also demonstrated their usefulness in ecology and economics. Thus, the solution methods for these equations are of great importance to engineers and scientists. Although many important differential equations can be solved by well known analytical techniques, a greater number of physically significant differential equations cannot be solved. So far, many approaches have been proposed for determining the numerical solution to Abel equation [2]. For example, Fettis [3] proposed a numerical form of the solution to Abel equation by using the Gauss–Jacobi quadrature rule. Piessens and Verbaeten [4] and Piessens [5] developed an approximate solution to Abel ⇑ Corresponding author. E-mail address: [email protected] (M. Gülsu). 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.11.044

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equation by means of the integral transformations. Furthermore, Garza et al. [6] and Hall et al. [7] used the wavelet method to solve Abel equation. However, very few references have been found in technical literature dealing with Abel differential equations [8–18]. In this study, we present a new, simple approach for solving the approximate solution to Abel differential equations. By expanding the unknown function to be determined as a shifted Chebyshev polynomial, we can convert Abel differential equation to a system of nonlinear equations for the unknown function. An error analysis of approximate Abel differential equation is given. Finally, several examples are given to show the effectiveness of the present method. The rest of this paper is organized as follows. Analysis of shifted Chebyshev collocation method and fundamental relations are presented in Section 2. The new scheme are based on shifted Chebyshev collocation method. Section 3 is devoted to the solution of Eqs. (1) and (2). In Section 4, we report our numerical finding and demonstrate the accuracy of the proposed numerical scheme by considering numerical example. Section 5 concludes this article with a brief summary. 2. Analysis of shifted Chebyshev collocation method Let us consider the Abel differential Eq. (1) and find the truncated shifted Chebyshev series expansions of each term in expression (1) and their matrix representations. We first consider the desired solution y(t) and its derivatives have truncated shifted Chebyshev series expansion of the form respectively,

yðtÞ ¼

N X

0

an T n ðtÞ;

ð4Þ

n¼0

yð1Þ ðtÞ ¼

N X

0 ð1Þ  an T n ðtÞ;

t 2 ½0; 1:

ð5Þ

r¼0

Then the function defined in relation (4) can be written in the matrix form

yðtÞ ¼ TðtÞA

ð6Þ

where

  TðtÞ ¼ T 0 ðtÞ T 1 ðtÞ    T N ðtÞ A ¼ ½a0

a1    aN T :

ð7Þ

Similarly, the matrix representation of function (5) becomes

yð1Þ ðtÞ ¼ TðtÞAð1Þ :

ð8Þ (k)

Using the relation between the Chebyshev coefficient matrix A of y(t) and the Chebyshev coefficient matrix A we find the relation

Að1Þ ¼ 4 MA;

(k)

of y (t) [14],

ð9Þ

where for odd N,

2

0 1=2 0

3=2 0 5=2   

N 2

3

0

2 0

4 0

0 .. .

0 .. .

3 .. .

0 .. .

0

0

0

0 0

7  0 7 7  N7 7 . . .. 7 7 . . 7 7 7  N5

0 0

0

0

0 0

 0

6 60 6 60 6 M¼6 6 .. 6. 6 6 40

5 .. .

ð10Þ

ðNþ1ÞxðNþ1Þ

and for even N,

2

0 6 60 6 60 6 M¼6 6 .. 6. 6 60 4

1=2 0

3=2 0 5=2    0

0

2 0

4 0

0 .. .

0 .. .

3 .. .

0 .. .

0

0

0

0 0

0 0

0

0

0 0

5 .. .

3

7  N7 7  0 7 7 . . .. 7 7 . . 7 7  N7 5  0

:

ð11Þ

ðNþ1ÞxðNþ1Þ

Thus expression (9) becomes

yð1Þ ðtÞ ¼ TðtÞAð1Þ ¼ 4TðtÞMA: And after substituting the Chebyshev collocation points defined by

ð12Þ

M. Gülsu et al. / Applied Mathematics and Computation 217 (2011) 4827–4833

ti ¼

   1 ip 1 þ cos ; 2 N

i ¼ 0; 1; 2; . . . ; N;

4829

ð13Þ

into Eq. (12), we have

yð1Þ ðt i Þ ¼ 4Tðti ÞMA

ð14Þ

or the compact form

Yð1Þ ¼ 4TMA;

ð15Þ

where

2

Tðt0 Þ

3

6 Tðt 1 Þ 7 7 6 7 T¼6 6 .. 7; 4 . 5

2

yð1Þ ðt 0 Þ

3

6 yð1Þ ðt Þ 7 1 7 6 7; Yð1Þ ¼ 6 .. 7 6 5 4 .

2

3

6 yðt 1 Þ 7 7 6 7 Y¼6 6 .. 7: 4 . 5

yð1Þ ðtN Þ

Tðt0 Þ

yðt0 Þ

ð16Þ

yðt N Þ

Similarly, substituting the Chebyshev collocation points into the yr(t) and using the relation (5), it is obtained the matrix representation

Yr ¼ ðYÞr1 Y;

r ¼ 2; 3;

ð17Þ

where

2

yr ðtÞ

3

2

6 yr ðtÞ 7 7 6 7; Yr ¼ 6 7 6 .. 5 4.



0

3

yðt1 Þ    .. ... .

0 .. .

7 7 7 7 5

yðt 0 Þ

6 0 6 Y¼6 6 .. 4 .

yr ðtÞ

0

0

0

ð18Þ

   yðt N Þ

and

Y ¼ TA;

ð19Þ

where

2

0  Tðt 0 Þ 6 0 Þ  Tðt 1 6 T¼6 .. .. 6 .. 4 . . . 0

0

0 0 .. .

3

2

7 7 7; 7 5

3

To obtain matrix form of y , y and yy

60 6 A¼6 6 .. 4. 0

   TðtN Þ 2

A

(1)

0



A  .. . . . . 0

0

3

07 7 .. 7 7: .5

ð20Þ

 A

we construct the following relation by using (8), (18) and (19) respectively,

2

y ðt i Þ ¼ yðti Þyðti Þ ¼ ðTAÞTðti ÞA;

ð21aÞ

y3 ðt i Þ ¼ y2 ðt i Þyðt i Þ ¼ ðTAÞ2 Tðti ÞA;

ð21bÞ

yðt i Þyð1Þ ðt i Þ ¼ 4ðTAÞTðti ÞMA:

ð21cÞ

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

½A1 ðti Þ þ A2 ðt i Þyðt i Þy0 ðti Þ þ Bðt i Þyðt i Þ þ Cðt i Þy2 ðt i Þ þ Dðti Þy3 ðt i Þ ¼ Eðti Þ;

i ¼ 0; 1; . . . ; N

and then this system is written in the matrix form

h where

i 4A1 TM þ 4A2 TATM þ BT þ CðTAÞT þ DðTAÞ2 T A ¼ E;

ð22Þ

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M. Gülsu et al. / Applied Mathematics and Computation 217 (2011) 4827–4833

2 6 6 6 A1 ¼ 6 6 4 2

0

6 6 6 D¼6 6 4



0

A1 ðt 1 Þ   

0

.. .

.. .

..

0

0

   A1 ðtN Þ

Bðt 0 Þ

6 6 0 6 B¼6 . 6 . 4 . 2

0

A1 ðt 0 Þ

0

.. .

.



0

Bðt 1 Þ   

0 .. .

.. .

..

0

0

   BðtN Þ

Dðt0 Þ

0

0

.



0

Dðt1 Þ   

0

2

7 7 7 7; 7 5

6 6 6 A2 ¼ 6 6 4

3

2

7 7 7 7; 7 5

6 6 0 6 C¼6 . 6 . 4 .

.. .

.. .

..

0

0

   DðtN Þ

.. .

.

3

A2 ðt0 Þ 0

7 7 7 7; 7 5

Eðt 0 Þ

0

A2 ðt1 Þ   

0

.. .

..

0

0

   A2 ðt N Þ

0



0

Cðt1 Þ   

0

Cðt 0 Þ

2



.. .

0 3

0

7 7 7 7; 7 5

.. .

..

0

   CðtN Þ

3

7 6 6 Eðt1 Þ 7 7 6 E ¼ 6 . 7; 6 . 7 4 . 5 EðtN Þ

2

Tðt0 Þ

7 7 7 7; 7 5

3

.. .

.

.. .

.

3

3

7 6 6 Tðt 1 Þ 7 7 6 T ¼ 6 . 7: 6 . 7 4 . 5 Tðt N Þ

This is a fundamental matrix equation for the solution of Eq. (1). Briefly, this equation can also be written in the form

WA ¼ E or ½W; E;

W ¼ ½wij ;

i; j ¼ 0; 1; . . . ; N;

ð23Þ

where

W ¼ 4A1 TM þ 4A2 TATM þ BT þ CðTAÞT þ DðTAÞ2 T:

ð24Þ

Now, let us find a matrix representation of the mixed conditions (2). Using the relation (6), the matrix representation of the mixed conditions (2) which depends on the Chebyshev coefficients matrix A is formed,

½yðaÞ ¼ TðaÞA; ½yðbÞ ¼ TðbÞA;

ð25Þ

½yðcÞ ¼ TðcÞA: By using Eq. (21a)

y2 ðt i Þ ¼ yðt i Þyðti Þ ¼ ðTAÞTðt i ÞA; for points a,b and c we obtain

½y2 ðaÞ ¼ yðaÞyðaÞ ¼ ½TATðaÞA; ½y2 ðbÞ ¼ yðbÞyðbÞ ¼ ½TATðbÞA;

ð26Þ

2

½y ðcÞ ¼ yðcÞyðcÞ ¼ ½TATðcÞA: Also, using the Eq. (21b)

y3 ðt i Þ ¼ y2 ðt i Þyðt i Þ ¼ ðTAÞ2 Tðt i ÞA; for points a,b and c we get

½y3 ðaÞ ¼ y2 ðaÞyðaÞ ¼ ½TA2 TðaÞA; ½y3 ðbÞ ¼ y2 ðbÞyðbÞ ¼ ½TA2 TðbÞA; 3

2

ð27Þ

2

½y ðcÞ ¼ y ðcÞyðcÞ ¼ ½TA TðcÞA: Substituting the matrix representation (25)–(27) into Eq. (2), we obtain the matrix forms of conditions (2) become

UA ¼ ½k;

ð28Þ

where

U ¼ a1 TðaÞ þ b1 TðbÞ þ c1 TðcÞ þ a2 ½TATðaÞ þ b2 ½TATðbÞ þ c2 ½TATðcÞ þ a3 ½TA2 TðaÞ þ b3 ½TA2 TðbÞ þ c3 ½TA2 TðcÞ ¼ ½u0 u1 . . . uN : f E ~ or Consequently, replacing the row of the augmented matrix [W; E] by the row of the matrix [U; k], we have ½ W;

ð29Þ

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M. Gülsu et al. / Applied Mathematics and Computation 217 (2011) 4827–4833

f ¼ E; ~ WA

ð30Þ

which is a nonlinear algebraic system. The unknown shifted Chebyshev coefficients an, (n = 0, 1, . . . , N) are determined from P 0 this nonlinear system and substituted in Eq. (3). Thus we obtain the Chebyshev polynomial solution yðtÞ ¼ Nn¼0 an T n ðtÞ. We can easily check the accuracy of this solution as follows: Since the Chebyshev polynomial (3) is an approximate solution of Eq. (1), when the solution y(t) and its derivatives are substituted in Eq. (1), the resulting equation must be satisfied approximately, that is, for t = tr 2 [a, b]

  Eðt r Þ ¼ ½A1 ðtÞ þ A2 ðtÞyðtÞy0 ðtÞ þ BðtÞyðtÞ þ CðtÞy2 ðtÞ þ DðtÞy3 ðtÞ  EðtÞ ffi 0 or

Eðt r Þ 6 10kr

ðkr is any positive integerÞ:

If max ð10kr Þ ¼ 10k (k is any positive integer) is prescribed, then the truncation limit N is increased until the difference jE(tr)j at each of the points becomes smaller than the prescribed 10k. 4. Numerical experiment The equation considered here has the following form

yy0 þ ty þ y2 þ t 2 y3 ¼ eet þ t 2 e3t

ð31Þ

with appropriate condition y(0) = 1. The exact solution of this problem is y(t) = et. Let us seek the solution y(t) as a truncated shifted Chebyshev polynomials

yðtÞ ¼

N X

0

an T n ðtÞ:

n¼0

So that A1(t) = 0, A2(t) = 1, B(t) = t, C(t) = 1, D(t) = t2, E(t) = eet + t2e3t. Then, for N = 5, the collocation points are t0 = 1, t1 = 0.9045084972, t2 = 0.6545084969, t3 = 0.3454915031, t4 = 0.0954915028, t5 = 0 and the fundamental matrix equation of the problem is defined by

½4A1 TM þ 4A2 TATM þ BT þ CðTAÞT þ DðTAÞ2 TA ¼ E; where

2

1

1

1

1

1

3

1

7 6 7 61 0:809016994 0:309016994 0:30901669 0:80901701 1 7 6 7 6 7 61 0:309016994 0:809016995 0:809016998 0:30901700 0:999999997 7 6 T¼6 7; 7 6 1 0:3090169938 0:809016998 0:809016994 0:309016988 0:999999993 7 6 7 6 7 6 1 0:8090169944 0:3090169945 0:3090169939 0:8090169948 1 5 4 1 1 1 1 1 1 2

0 1=2 0 0

0

3

0

0

0

4

0

0

0

0

0

0

0

0

0

0

60 6 6 60 B¼6 60 6 6 40 0

3

0

0

1

5=2

0 7 7 7 5 7 7; 0 7 7 7 5 5

60 6 6 60 M¼6 60 6 6 40 2

3=2 0

2

0

0 0:9045084972 0 0 0 0

4

2

0 0 0 0 0 0

60 6 6 60 A1 ¼ 6 60 6 6 40

3

2

0 0 0 0 07 7 7 0 0 0 0 07 7; 0 0 0 0 07 7 7 0 0 0 0 05

0

0 0

0

0

60 1 0 0 0 6 6 60 0 1 0 0 A2 ¼ 6 60 0 0 1 0 6 6 40 0 0 0 1

0 0 0 0 0 0 0

1

0 0

0 0 0

3

07 7 7 0:6545084969 0 0 07 7; 0 0:3454915031 0 07 7 7 0 0 0:0954915028 0 5 0

0

0

0

0

0

0

0

0

3

07 7 7 07 7; 07 7 7 05

0 1 2

1

0

0

0

0

0

3

60 1 0 0 0 07 7 6 7 6 60 0 1 0 0 07 7 C¼6 6 0 0 0 1 0 0 7; 7 6 7 6 40 0 0 0 1 05 0

0

0

0

0 1

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M. Gülsu et al. / Applied Mathematics and Computation 217 (2011) 4827–4833

Table 1 Error analysis of Example 1 for the t value. t

Taylor matrix met. [19]

Pade aprox. met. [19]

Exact solution

Present Chebyshev method

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1.0000000000 0.9048374167 0.8187306667 0.7408172500 0.6703146667 0.6065104167 0.5487520000 0.4964369167 0.4490026667 0.4059167500 0.3666666667

1.0000000000 0.9048374179 0.8187307455 0.7408181414 0.6703196347 0.6065292096 0.5488076312 0.4965759550 0.4493096647 0.4065333809 0.3678160920

1.0000000000 0.9048374180 0.8187307531 0.7408182207 0.6703200460 0.6065306597 0.5488116361 0.4965853038 0.4493289641 0.4065696597 0.3678794412

1.0000000000 0.9048374178 0.8187307453 0.7408181410 0.6703196344 0.6065292082 0.5488076309 0.4965759540 0.4493096539 0.4065333712 0.3678160915

1 Taylor Matrix Met. Pade Aprox. Met Present Chebyshev Method Exact Solution

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 1. Numerical and exact solution of the test problem.

-4

7

x 10

Taylor Matrix Met. Error Pade Aprox. Met. Error Present Chebyshev Method Error

6

5

4

3

2

1

0 0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

Fig. 2. Error function of the test problem for the different method.

1

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M. Gülsu et al. / Applied Mathematics and Computation 217 (2011) 4827–4833

2

1

0

0

0

0

6 0 0:8181356215 0 0 0 6 6 60 0 0:4283813725 0 0 D¼6 60 0 0 0:1193643787 0 6 6 40 0 0 0 0:009118627107 0

0

0

0

0

0

3

07 7 7 07 7; 07 7 7 05 0

2

0:4176665096

3

6 0:4203360188 7 7 6 7 6 6 0:4002751517 7 7 E¼6 6 0:2869027821 7: 7 6 7 6 4 0:09364195159 5 0

If these matrices are substituted in (23), it is obtained nonlinear algebraic system. This system yields the approximate solution of the problem. The result with N = 5 using the shifted Chebyshev collocation method discussed in Section 2. The Taylor matrix method [19], Pade approximation method [19] and also the exact solution are shown in Table 1. The numerical solution of Eq. (31) is shown in Table 1. We can see from the table that the numerical solution are in good agreement with the exact solutions. Fig. 1 shows the resulting graph of solution of the test problem and it is compared with The Taylor matrix method, Pade approximation method and also the exact solution. In Fig. 2 we plot error function for the test problem. This plot clearly indicates that the numerical solution are in good agreement with the exact solutions and Pade approximation method. 5. Conclusions The aim of present work is to develop an efficient and accurate method for solving Abel differential equation of second kind. In this paper we have presented a suggested method to solve the Abel differential equation using the shifted Chebyshev collocation method. The collocation method avoids the difficulties and massive computational work by determining the analytic solution. The accuracy of the suggested method, Taylor matrix method, Pade approximation method and exact solution are compared in Table 1. It is clearly seen that our numerical solution are good agreement with the exact solutions. Table 1 illustrates the solutions obtained by using the procedure outlined above. The shifted Chebyshev collocation method provides a reliable technique that requires less work and higly accurate results if compared with the traditional techniques and existing numerical methods. A considerable advantage of the method is that shifted Chebyshev coefficients of the solution are found very easily by using the computer programs. We conclude that the present approach will also prove useful for solving more general problems in applied mathematics. Also, in the future, we use the shifted Chebyshev collocation method to solution systems of nonlinear equations. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

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