Chebyshev Polynomial Solution of Nonlinear Fredholm-Volterra Integro

Çankaya Üniversitesi Fen-Edebiyat Fakültesi, Journal of Arts and Sciences Say›: 5, May›s 2006

Chebyshev Polynomial Solution of Nonlinear Fredholm-Volterra IntegroDifferential Equations Handan ÇERDİK-YASLAN and Ayşegül AKYÜZ-DAŞCIOĞLU*

Abstract In this paper, a Chebyshev collocation method [1] is developed to find an approximate solution for nonlinear Fredholm-Volterra integro–differential equation. This method transforms the nonlinear Fredholm-Volterra integro–differential equation into the matrix equation with the help of Chebyshev collocation points. The matrix equation corresponds to a system of nonlinear algebraic equations with the unknown Chebyshev coefficients. Finally, some numerical examples are presented to illustrate the accuracy of the method. Keywords: Nonlinear integro-differential equation; Chebyshev series; Colllocation Method. Özet Bu çal›şmada, lineer olmayan Fredholm-Volterra integro–diferansiyel denklemlerin yaklaş›k çözümlerini bulmak için Chebyshev s›ralama yöntemi [1] geliştirilmiştir. Bu yöntem lineer olmayan Fredholm-Volterra integro–diferansiyel denklemini, s›ralama noktalar›n› kullanarak matris denklemine dönüştürür. Bu matris denklemi ise bilinmeyeni Chebyshev katsay›lar› olan lineer olmayan cebirsel denklem sistemine karş›l›k gelir. Çal›şman›n sonunda yöntemin doğruluğunu göstermek için baz› say›sal örnekler sunulmuştur. Anahtar Kelimeler: Lineer Olmayan integro-diferansiyel denklemler; Chebyshev serileri; S›ralama Yöntemi.

*Pamukkale Üniversitesi Fen Edebiyat Fakültesi Matematik Bölümü, K›n›kl›, Denizli. TURKEY. [email protected] 89

Chebyshev Polynomial Solution of Nonlinear Fredholm-Volterra Integro-Differential Equations

1. INTRODUCTION Consider the following nonlinear Fredholm–Volterra integro-differential equation m

∑ P ( x) y k

k= 0

(k )

1

2

x

2

( x ) = g( x ) + λ1 ∫ ∑ Fi ( x, t) y i ( t)dt + λ2 ∫ ∑ K j ( x, t) y j ( t)dt −1 i=1

(1)

−1 j =1

under the mixed conditions

∑ [aij y ( j ) (−1) + bij y( j ) (1) + cij y ( j ) (c)]= µi ,

m −1

i = 0, 1 ,..., m-1 ,-1≤ c ≤1

(2)

j =0

where y (x ) is an unknown function, the functions g( x ), Pk ( x ), Fi ( x, t) and K j ( x, t ) are defined on interval −1 ≤ x, t ≤ 1 and aij , bij , cij , λ1, λ2 , µi are constants. Let us seek the solution of (1) expressed in terms of Chebyshev polynomials as N

y ( x ) = ∑ © arTr ( x ) , −1 ≤ x ≤ 1

(3)

r= 0

where ar , 0 ≤ r ≤ N , are unknown Chebyshev coefficients and N is chosen any ' denotes a sum whose first term is halved, positive integer such that m ≤ N.

∑

Tr(x) denotes the Chebyshev polynomials of the first kind of degree r. The Chebyshev collocation points defined by

 sπ  xs = cos  , s = 0, 1, ..., N N 

(4)

are used in the following sections. 2. FUNDAMENTAL RELATIONS Let us write Eq. (1) in the form

D( x ) = g( x ) + λ1I ( x ) + λ2 J ( x )

(5)

where the differential part m

D( x ) = ∑ Pk ( x ) y ( k ) ( x ) k= 0

Fredholm integral part is

90

(6)

Handan ÇERDİK-YASLAN, Ayşegül AKYÜZ-DAŞCIOĞLU

I ( x) =

1 2

Fi ( x, t ) y (t )dt ∫∑ i =1 i

(7)

−1

and Volterra integral part is

J ( x) =

x 2

K j ( x, t ) y ∫∑ j =1

j

(t )dt

(8)

−1

2.1. Matrix Representation for Differential Part Let us assume that the kth derivative of the function (3) with respect to x has the truncated Chebyshev series expansion by

y

(k )

( x) =

N

∑ ' ar( k )Tr ( x) ,

−1 ≤ x ≤ 1

r =0

where

ar(k )

(k = 0, 1,..., m) are Chebyshev coefficients.

Then the solution expressed by (3) and its derivatives can be written in the matrix forms respectively y(x) = T(x)A

(9)

y ( k ) ( x ) = T ( x ) A( k )

(10)

and It is well known from [6] that the relation between the Chebyshev coefficient matrix A of y(x) and the Chebyshev coefficient matrix A(k) of y(k)(x) is given

A( k ) = 2k M k A Then the expression (10) becomes

y ( k ) ( x ) = 2 k T ( x )M k A

(11)

where

a T ( x) = [T0 ( x ) T1( x) ... TN ( x)] , A =  0 2

a1

 ... a N  

T

91


0 1 / 2 0 0  0 0 M= M M 0 0  0 0

0 3 / 2 0 5 / 2 L N / 2 2 0 4 0 L 0   0 3 0 5 L N   for odd N M M M M O M  0 0 0 0 L N   0 0 0 0 L 0 

0 1 / 2 0 0  0 0 M= M M 0 0  0 0

0 3/ 2 0 5/ 2 L

and

2

0

4

0

L

0

3

0

5

L

M

M

M

M

O

0

0

0

0

L

0

0

0

0

L

0 N  0  for even N M N  0 

Substituting the Chebyshev collocation points into Eq.(6) and using (11), the matrix representation of D(xi) can be given by m

D( xi ) = ∑ 2 k Pk ( xi )T ( xi )M k A

(12)

k =0

2. 2. Matrix Representation for Fredholm Integral Part Let us substitute the Chebyshev collocation points into Eq. (7) to obtain the matrix relation of I(xi) and assume that, for each xs, F0(xs,t) and F1(xs,t) is expanded to the Chebyshev series in the form

Fi ( xs , t ) =

N

∑ ' ' fir ( xs )Tr (t ) ,

i = 0 ,1

r =0

where a summation symbol with double primes denotes a sum with first and last terms halved and Chebyshev coefficients fir(xs) are determined by means of the relation

f ir ( xs ) =

2 N

N

∑ ' ' Fi ( xs , t j )Tr (t j ) j =0

Then the matrix representations of Fi(xs,t) become 92

 jπ   , j = 0, 1, ..., N  N 

, t j = cos


Fi ( x s , t ) = Fi ( x s )T (t )T

(13)

where

 f i ,0 ( x s ) Fi ( x s ) =   2

f i ,1 ( x s ) L

f i , N −1 ( x s )

f i, N ( xs )   2 

Besides, y2(t) function can be written in the matrix form [5],

y 2 (t ) = T (t )B

(14)

in which

 b1 ... b2 N  

b T (t ) = [T0 (t ) T1 (t ) ... T2 N (t )], B =  0 2 

T

and the elements bi of the column matrix B consist of ai and ai = a-i as follows:

 2  a i  i N−   2  2    + ∑  a i  a i  , for even i  −r  + r  bi =  2 r =1  2  2   N − i −1    2  , for odd i  ∑  a i +1  a i −1  −r +r  r =1  2  2  When the relations (9), (13) and (14) are substituted in I(xs), we have

I ( x s ) = F1 ( x s )Z1 A + F2 ( x s )Z 2 B

(15)

where 1

Z1 =

∫ T (t)

T

−1

1

∫ T (t)

Z2 =

−1

[ ]

T ( t) dt = zij , i= 0, 1,..., N, T

T ( t) dt = [ zij ] , i = 0, 1,..., N,

j = 0,1 ,..., N j = 0,1 ,..., 2N

and

1 1  , for even i + j +  2 zij = ∫ Ti (t )T j (t )dt =  1 − (i + j ) 1 − (i − j )2  −1 0 , for odd i + j 1

93


2. 3. Matrix Presentation for Volterra Integral Part Firstly, the Chebyshev collocation points are substituted into (8). Similarly the previous section, it is supposed that the kernel function Kj(xs,t) can be expanded to univarete Chebyshev series with respect to t. Then the matrix form of the kernel function is

K j ( x s , t ) = K j ( x s )T (t ) T , j = 0,1 s = 0,1,..., N

(16)

where

 k j,0 ( x s ) K j ( xs ) =   2

k j ,1 ( x s ) L k j , N −1 ( x s )

k j, N ( xs )   2 

Substituting the relations (9), (14) and (16) in J(xs), the matrix representation of J(xs) is obtained as

J ( x s ) = K 1 ( x s ) Z 1 ( x s ) A + K 2 ( x s ) Z 2 ( x s )B

(17)

where

Z1 ( x s ) =

xs

∫ T (t)

T

−1

Z 2 ( xs ) =

xs

∫ T (t)

−1

T

T ( t) dt = [ zij ( x s )] I = 0, 1,..., N, T ( t) dt = [ zij ( x s )] , I = 0, 1,..., N,

j=0,1 ,..., N

j=0,1 ,..., 2N

and

zij ( x ) =

x

∫ T (t)T (t)dt i

j

−1

 2x 2 − 2 , for i + j = 1 T ( x ) Ti + j −1 ( x ) 1 1  i + j +1 − − + + x2 −1 − j =1 ,, for fori |i-j|=1  i + j +1 i + j −1 i + j +1 i + j −1 1   1 1 =  Ti + j +1 ( x ) Ti− j −1 ( x) T1+i − j ( x) T1−i+ j ( x) + + + + 2 + , for even i + j 4 2 2 i + j +1 i − j −1 1+ i − j 1− i + j 1 − (i + j ) 1 − (i − j )    1 1  Ti + j +1 ( x) Ti − j −1 ( x ) T1+ i− j ( x) T1−i + j ( x)  + + + − + , for odd i + j 2   i + j +1 i − j −1 1+ i − j 1− i + j 2 2 1 − (i + j ) 1 − (i − j )  

94


2. 4. Matrix Representation for the Conditions Using the relation (11), the matrix form of the conditions defined in (2) can be written as m −1

∑ 2 (a T (−1) +b T (1) + c T (c ))M A =µ , j

j

ij

ij

ij

i

i=0, 1,…, N

j= 0

Let us define Ui as m −1

U i = ∑ 2 j ( aijT (−1) +b ijT (1) + c ijT (c ))M j = [ ui 0 j= 0

ui1 ... uiN ]

Thus, the matrix forms of conditions (2) become

U i A = µi

(18)

3. METHOD OF SOLUTION To construct the fundamental matrix equation corresponding to Eq. (1), the Chebyshev collocation points are substituted in (5) and then using the matrix relations (12), (15) and (17), it is obtained for s = 0, 1,…, N m

∑ 2 k Pk ( x s )T ( xs )M k A = g ( xs ) + λ1 (F1 ( x s )Z1 A + F2 ( xs )Z 2 B )

k =0

+ λ 2 (K 1 ( x s ) Z1 ( x s ) A + K 2 ( x s ) Z 2 ( x s )B ) Thereby, the fundamental matrix is gained of the form

m k  ∑ 2 Pk TM k − λ1F1Z1 − λ2K1Z1 A - ( λ1F2 Z2 + λ2K2 Z2 )B = G  k= 0 

(19)

where Pk ( x 0 ) 0  Pk ( x1 )  0 Pk =  M M  0  0

K n ( x 0 ) L 0    L 0 ,  0 Kn =   M O M    0 L Pk ( x N )

L 0   K n ( x1 ) L 0  M O M   0 L K n ( x N ) 0

 Z n (x 0 )  T ( x 0 )   g( x 0 )   Fn ( x 0 )          Fn ( x1 ) ,  Z n (x1 )  for n = 1, 2  T ( x1 ) ,  g( x1 ) ,  Zn =  T= G = F = M  M  M  n  M          Z n (x N ) T ( x N ) Fn ( x N ) g( x N )

95


This equation corresponds to a system of N+1 nonlinear algebraic equations with unknown Chebyshev coefficients a0, a1, …, aN. Finally, to obtain the solution of Eq.(1) under the mixed conditions (2), m equations in nonlinear algebraic system (19) are replaced with m equations in linear algebraic equations system (18). Therefore, Chebyshev coefficients are determined by solving the new nonlinear algebraic system. The method also can be developed for the problem defined on the domain [0,1] m

1 2

x 2

k =0

0 i =1

0 j =1

∑ Pk ( x) y (k ) ( x) = g ( x) + λ1 ∫ ∑ Fi ( x, t ) y i (t )dt + λ 2 ∫ ∑ K j ( x, t ) y j (t )dt (20) The solution of this equation under the mixed conditions is found in terms of *

shifted Chebyshev polynomials Tr ( x ) of the form

y( x) =

N

∑ ' ar*Tr* ( x) ,

0 ≤ x ≤1

r =0

where

Tr* ( x )

= Tr (2 x − 1) .

It is followed the previous procedure using the collocation points defined by

xs =

1  sπ 1 + cos  2 N

  , s = 0,1,..., N 

(21)

and the relation

A*

(k )

= 4 k M k A* , k = 0, 1, ..., N

where T

 a* A = 0  2

a1*

*

L

a *N

  . 

Then we obtain the fundamental matrix equation for (20) as m

∑4 k= 0

k

P kT*M k A* − λ1 [F1* Z1* A* + F2* Z*2B* ] − λ2 [K1* Z1* A* + K*2 Z*2B* ] = G

Moreover, the matrix forms of the conditions become m −1

∑ 4 (a T j

ij

j= 0

96

*

(−1) +b ijT * (1) + c ijT * (c ))M j A* =µ i ,

i=0, 1,…, N

Handan ÇERDİK-YASLAN, Ayşegül AKYÜZ-DAŞCIOĞLU *

It is easily seen that T = T , Z n = 2Z *n and Z n = 2Z *n for n = 0 ,1, because of the properties of the Chebyshev polynomials. 5. NUMERICAL EXPERIMENTATIONS The efficiency of the presented method is shown in following three examples. Results were computed using the program written in Mathcad 2000 Professional. Example1. Let us consider two examples of nonlinear Fredholm- Volterra integro-differential equation. These problems has been solved by Taylor polynomials for N = 4 and N = 5 respectively in [3]. 1

x

−1

−1

2 a) y′′( x) − xy′( x ) + xy ( x) = g ( x) + ∫ xty (t )dt + ∫ (x − 2t )y (t )dt

where g ( x) =

2 6 1 4 23 5 x − x + x 3 − 2 x 2 − x + and y (0) = −1, y′(0) = 0 . 15 3 15 3 1

x

0

0

2 b) y′( x ) + xy ( x) = g ( x) + ∫ (x + t )y (t )dt + ∫ (x − t )y (t )dt

where g ( x) =

(22)

(23)

5 3 −1 6 1 4 and y (0) = −2 . x + x + x3 − 2 x 2 − x + 30 3 3 4

a) Let us take N = 2 for solution of Eq. (22) and seek the solution y(x) as a truncated Chebyshev series

y( x) =

2

∑ ' arTr ( x) ,

−1 ≤ x ≤ 1

r =0

Fundamental matrix equation of this problem defined in Section 3 is

(4TM

2

− 2PTM + P0T - F1Z1 ) A − K2 Z2B = G 1

and condition equations are

T (0) A = −1 and 2T (0)MA = 0 This matrix equation corresponds to nonlinear algebraic system as follows:

− 16 1 2 1 2 14 4 8 2 a0 − a1 + a2 − a0 2 − a12 − a2 2 + a0 a1 + a1a2 + a0 a2 = 2 3 2 3 15 3 15 3 15 1 2 1 2 1 2 2 4 5 4a2 − a0 − a1 − a2 + a0a1 + a1a2 = 4 2 3 3 15 3 −1 8 (24) a0 + a1 − a2 = 0 2 3 and condition equations are 97


1 a0 − a2 = −1 2 a1 = 0

(25)

In system (24), first and second equations are replaced by condition equations in (25) and new linear algebraic system is obtained. This system is solved easily, so we have

y( x) = x 2 − 1 which is the exact solution of Eq.(22). b) Let us consider solution of Eq. (23) for N = 2 and seek the solution y(x) as a truncated Chebyshev series

y( x) =

N

∑ ' ar*Tr* ( x) ,

0 ≤ x ≤1

(26)

r =0

The fundamental matrix equation of this problem defined in Section 3 is

(4PT M + P T *

1

*

0

− F1* Z1* ) A* − K2* Z2*B* = G

and for condition equation is

T * (0)A * = −2 The matrix equation corresponds to nonlinear algebraic system as follows:

− 1 * 17 * 19 * 1 *2 1 *2 7 * 2 1 * * 1 * * 1 * * 103 a0 + a1 + a2 − a0 − a1 − a2 + a0a1 + a1 a2 + a0 a2 = 4 6 2 8 6 30 6 15 6 60 − 1 * 11 * 1 * 1 *2 1 *2 1 *2 1 * * 1 * * 2359 a0 + a1 − a2 − a0 − a1 − a2 + a0a1 + a1 a2 = 4 6 6 32 16 24 12 30 1920

−1 * 11 * 47 * 3 a0 + a1 − a2 = 4 6 6 4

(27)

and condition equation is

1 * a0 − a1* + a2* = −2 2

(28)

When the first equation the system (27) is replaced by Eq. (28), new nonlinear algebraic system is obtained. Taking starting points ai* = 0 ( i = 0, 1, 2) the solution of the system is obtained and we have

98


y ( x ) = −1.625T0* ( x) + 0.5T1* ( x ) + 0.125T2* ( x ) or

y( x) = x 2 − 2 which is the exact solution of Eq.(23). Example 2. Consider the nonlinear Volterra integro-differential equation x

y′( x) = −1 + ∫ y 2 (t )dt , y (0) = 0

(29)

0

Using the method in Section 3, Eq. (29) is solved for N = 6. The solution of this example can be found analytically by reducing to differential equation, but the analytical solution is not represented by the elementary functions. However, it can be represented by hypergeometric functions. The numerical solutions of Eq. (29) were given by Sepehrian-Razzaghi [4] and by Avudainayagam-Vani [2]. A comparison of these solutions with the present solution is given in Table 1. Table 1. Numerical results of Example (2)

0.0000

Wavelet-Galerkin Method 0.0000

Walsh Series Method (m=60) 0.00000

Presented Method N=6 0.00000

Exact Solution 0.00000

0.0625

-0.0625

-0.06250

-0.06250

-0.06250

0.1250

-0.1250

-0.12498

-0.12498

-0.12498

0.1875

-0.1874

-0.18740

-0.18740

-0.18740

0.2500

-0.2497

-0.24967

-0.24967

-0.24967

0.3125

-0.3117

-0.31171

-0.31171

-0.31171

0.3750

-0.3734

-0.37336

-0.37336

-0.37336

0.4375

-0.4345

-0.43446

-0.43446

-0.43446

0.5000

-0.4948

-0.49482

-0.49482

-0.49482

0.5625

-0.5542

-0.55423

-0.55423

-0.55423

0.6250

-0.6124

-0.61243

-0.61243

-0.61243

0.6875

-0.6692

-0.66916

-0.66917

-0.66917

0.7500

-0.7242

-0.72415

-0.72415

-0.72415

0.8125

-0.7771

-0.77709

-0.77709

-0.77709

0.8750

-0.8277

-0.82766

-0.82767

-0.82767

0.9375

-0.8756

-0.87557

-0.87557

-0.87557

1.0000

-0.9205

-0.92047

-0.92047

-0.92048

x

99

Chebyshev Polynomial Solution of Nonlinear Fredholm-Volterra Integro-Differential Equations 1

y′′′( x ) − xy′′( x) + sin xy ( x) = e x (1 − x + sin x )− 2 + ∫ e − 2t y 2 (t )dt −1

with the conditions y (0) = y′(0) = y′′(0) = 1 . Let us suppose that y(x) is approximated by Chebyshev series

y( x) =

7

∑ ' arTr ( x) ,

−1 ≤ x ≤ 1

r =0

Using the procedure in Section 3, we find the approximate solution of this equation. A comparison of the obtained solution with the exact solution at the collocation points is given in Table 2. Table 2. Numerical results of Example (3) x x0 x1 x2 x3 x4 x5 x6 x7

Presented Method

Exact solution ex

2.718281 2.519044 2.028115 1.466214 0.682029 0.493068 0.396976 0.367879

2.718282 2.519044 2.028115 1.466214 0.682029 0.493069 0.396976 0.367879

6. CONCLUSIONS In this work, Chebyshev collocation method has applied to nonlinear integrodifferential equation. The study has showed that solving Fredholm part is easier than Volterra part. An interesting feature of this method is that the analytical solution is obtained for smaller N as shown in the Example 1. Moreover, this method gives better approximate solutions than the other methods as shown in the Example 2. One of the advantages of this method that solution is expressed as a truncated Chebyshev series, then y(x) can be easily evaluated for arbitrary values of x.

100

Handan ÇERDİK-YASLAN, Ayşegül AKYÜZ-DAŞCIOĞLU REFERENCES 1. A. Akyüz, M. Sezer, A Chebyshev collocation method for the solution linear integro differential equations, J. Comput. Math., 72 (1999) 491-507. 2. A. Avudainayagam, C. Vani, Wavelet-Galerkin method for integro-differential equations, Applied Numerical Mathematics, 32 (2000), 247-254. 3. K. Maleknejad, Y. Mahmoudi, Taylor polynomial solution of high-order nonlinear VolterraFredholm integro-differential equations, Appl. Math. Comput., 145 (2003) 641-653. 4. B. Sepehrian, M. Razzaghi, Single-term Walsh series method for the Volterra integro-differential equations, Engineering Analysis with Boundary Elements, 28 (2004) 1315-1319. 5. M. Sezer, S. Doğan, Chebyshev series solution of Fredholm Integral equations, Int. J. Math. Educ. Sci. Technol., 27 (1996) 649-657. 6. M. Sezer, M. Kaynak, Chebyshev polynomial solutions of linear differential equations, Int. Math. Educ. Sci. Technol., 27 (1996), 607-61.

101