Open Journal of Applied Sciences, 2012, 2, 115-121 doi:10.4236/ojapps.2012.22016 Published Online June 2012 (http://www.SciRP.org/journal/ojapps)
Collocation Method for Nonlinear Volterra-Fredholm Integral Equations Jafar Ahmadi Shali1, Parviz Darania2, Ali Asgar Jodayree Akbarfam1 1
Department of Mathematics and Computer Science, University of Tabriz, Tabriz, Iran 2 Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran Email: {j_ahmadishali, pdarania}@tabrizu.ac.ir,
[email protected] Received April 18, 2012; revised May 14, 2012; accepted May 25, 2012
ABSTRACT A fully discrete version of a piecewise polynomial collocation method based on new collocation points, is constructed to solve nonlinear Volterra-Fredholm integral equations. In this paper, we obtain existence and uniqueness results and analyze the convergence properties of the collocation method when used to approximate smooth solutions of VolterraFredholm integral equations. Keywords: Collocation Method; Nonlinear Volterra-Fredholm Integral Equations; Convergence Analysis; Chelyshkov Polynomials
1. Introduction We shall consider the nonlinear Volterra-Fredholm integral equation
y t g t 1 y t 2 Fy t , t I 0, T . (1) The Volterra integral operators given by
: C I C I
y t 0 k1 t , s, y s ds, t
Pmk t
(2)
where k1 C D and D t , s : 0 s t T and Fredholm integral operators given by F :C I C I
Fy t 0 k2 t , s, y s ds, T
(3)
where r , r 1, 2 denotes (real or complex) parameters and k2 C I I and let g C I be a given function. The mentioned equations are characterized by the presence of a linear functional argument and play an important role in explaining many different phenomena. In particular, they turn out to be fundamental when ordinary differential equations based model fail. These equations arise in industrial applications and in studies based on biology, economy, control and electro-dynamic. Collocation method is a widely popular numerical technique in solving integral equations, differential equations, etc. When collocation method is used to solve complicated engineering problems, it has several disadvantages, that is, low efficiency, ill-conditioned, etc. Thus, Copyright © 2012 SciRes.
different types of techniques were proposed to improve the computational performance of collocation method. Recently, Chelyshkov has introduced sequences of polynomials in [1], which are orthogonal over the interval 0,1 with the weight function 1. These polynomials are explicitly defined by mk
1 mj k mm kk 1 j t k j , j
k 0,1, , m. (4)
j 0
The polynomials Pmk t have properties, which are analogous to the properties of the classical orthogonal polynomials. These polynomials can also be connected to , a fixed set of Jacobi polynomials Pm t . Precisely
Pmk t 1
mk
t k Pn k
0,2 k 1
2t 1 .
Investigating more on (4), we deduce that in the family of orthogonal polynomials
m P t k 0 mk
have k multiple
zeros t 0 and m k distinct real zeros in the interval 0,1 . Hence, for every m the polynomial Pm 0 t has exactly m simple roots in 0,1 . Following [1], it can be shown that the sequence of polynomials
P t
m0
m 0
generate a family of orthogonal polynomials on 0,1 which possesses all the properties of other classic orthogonal polynomials e.g. Legendre or Chebyshev polynomials. Therefore, if the roots of Pm 0 t are chosen as collocation points, then we can obtain an accurate numerical quadrature. In the present paper, we further develop the works carried out in [2-6]. OJAppS
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We discuss existence and uniqueness results and analyze the convergence properties of the collocation method when used to approximate smooth solutions of linear Volterra-Fredholm integral equations and finally, some numerical results are presented in the final section, which support the theoretical results obtained in this paper.
2. Existence and Uniqueness Results
max g .
t
Lemma 2.1. Assume H is a nonempty closed set in a Banach space V, and that T : H H is continuous. Suppose T m is a contraction for some positive integer m. Then, T has a unique fixed-point in H. Proof: For proof see [7]. Here, in integral Equation (1), we assume that ki for some constants M i , satisfies a Lipschitz condition with respect to its third argument ki t , s, y1 ki t , s, y2 M i y1 y2 ,
(6)
yi , i 1, 2.
Theorem 2.2. Assume g and ki satisfy the condition (6) and given functions g , ki , are continuous on their domains. Moreover, assume 1
M2 T .
T
we get M t 2 2 1 M 2T y1 y2 T y1 t T y2 t 2! By a mathematical induction, we obtain
T : C I C I ,
Ty t g t 1 y t 2 Fy t ,
(8)
t I 0, T
M t m m 1 M 2T y1 y2 m !
.
(13)
T m y1 t T m y2 t TM m m 1 TM 2 y1 y2 m !
.
(14)
Since, 0 TM 2 1 then lim TM 2 0, m
m
and M 1m 0, m m ! lim
the operator T m is a contraction on C I when m is chosen sufficiently large. By the Lemma 2.1, the operator T has a unique fixed-point in C I .
3. Collocation Method Let tn nh, n 0, , N 1, t N T define a uniform partition for I 0,T , and let N : 0 t0 t1 t N T ,
n : tn , tn 1 0 n N 1 . The mesh N is constrained in the following sense:
0 k1 t , s, y1 s k1 t , s, y2 s ds t
(9)
0 k2 t , s, y1 s k2 t , s, y2 s ds T
Ty1 t Ty2 t M 1 0 y1 s y2 s ds t
M 2 0 y1 s y2 s ds. (10) T
M 1t M 2T y1 y2
h
T N
with a given mesh N we associate the set of its interior points, Z N : tn : n 1, , N 1 . For a fixed N 1 and, for given integers d 1 and m 1, the d piecewise polynomial space Sm d Z N is defined by
Then
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. (12)
Thus
Let us show that for m sufficiently large, the operator T is a contraction on C I . For y1 , y2 C I Ty1 t Ty2 t
T m y1 t T m y2 t
m
Since
2
(7)
Then the integral Equation (1) has a unique solution y t C I . Proof: We define the nonlinear integral operator
(11)
0 k2 t , s, Ty1 s k2 t , s, Ty2 s ds,
(5)
t I
0 s t T,
0 k1 t , s, Ty1 s k1 t , s, Ty2 s ds
2
Let C I denote the Banach space continuous realvalued functions, such that g C I with g
T 2 y1 t T 2 y2 t
.
d
Sm d Z N : u : C d I ; u
n m d , 0
n N 1 ,
where π m d denotes the set of (real) polynomials of a degree not exceeding m d . The dimension of this d space is given by dim Sm d Z N N m d 1. OJAppS
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For integral equation, we have d 1, hence, the 1 Sm 1
collocation space will be
Z N . Let
n 1 m
n, j
1
u Sm 1 Z N , n
l 1
u t u tn sh Lr s u tn cr h , r 1
(15)
n 0,1, , N 1 1
From (15) we see that an element u S m 1 Z N , is well defined when we know the coefficients n
cr h
where 0 c1 cm 1, and define the set
m , N 1 j 1, n 0
of collocation points by
tn, j : tn c j h,
The collocation solution u Sm 1 Z N , will be determined by imposing the condition that u satisfies the integral Equation (1) on the finite set X N u t g t 1 u t 2 Fu t ,
(16)
t XN ,
Thus, for t tn, j tn c j h the collocation Equation (16) assumes the form
u tn, j g tn , j 1 0 k1 tn , j , s, u s ds
17)
2 0 k2 tn, j , s, u s ds. T
From this equation and after some computations, we obtain
n 1
u tn, j g tn , j 1h 0 k1 tn, j , ti sh, u ti sh ds i 0
1
1h k1 tn, j , tn sh, u tn sh ds cj
0
N 1
2 h 0 k2 tn , j , ti sh, u ti sh ds 1
i 0
(18) Now, by using the local Lagrange basis functions L1 s
s cr , l 1, , m, r 1, r l cl cr m
(19)
for approximating the integral terms, we use the Lagrange interpolating polynomial to approximate
k1 tn , j , s, u s
and k2 tn , j , s, u s , we obtain
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cj 0
2 h k2 tn, j , ti c1h, u ti cl h i 0 l 1
L s ds 1
0
1
L j , l s ds
L s ds. 1
l
0
(20) Defining the quadrature weights wl : 0 Ll s ds, l 1, , m, 1
(21)
w j ,l : 0 L j ,l s ds, l , j 1, , m , cj
(22)
the fully discretized collocation equation corresponding to (20)-(22) is thus given by
g t h w k t n 1 m
1
, ti c1h, u ti c1 h
u tn , j
j 1, , m, n 0, , N 1.
tn , j
n, j
and
for all n 0, , N 1. In order to compute these coefficients, we consider the set of collocation parameters
X N : tn, j
i 0 l 1
N 1 m
u t
1
m
m
j
1
1h k1 tn, j , tn c1h, u tn cl h
for all t n we have
c ,
g t h k t u tn, j
n, j
1
l 1
i 0 l 1
n, j
, ti cl h, u ti cl h
m
1h w j ,l k1 tn, j , tn cl h, u tn cl h l 1
N 1 m
(23)
2 h wl k2 tn , j , ti cl h, u ti cl h . i 0 l 1
1 Note that, u S m 1 Z N and Equation (23) represent for each n 0,1, , N 1, a recursive system of m nonlinear algebraic equations with the unknowns u tn, j .
4. Global Convergence Let u S m 1 Z N denote the (exact) collocation solution to (1) defined by (16). In our convergence analysis we examine the linear test equation 1
y t g t 1 0 k1 t , s y s ds t
2 0 k2 t , s y s ds, t I 0, T , T
(24)
where k1 C D , k2 C I I . We will assume that 21 is not in the spectrum F of the Fredholm integral operator F. A comment of the convergence results to the nonlinear Equation (1) can be found at the end of this section. Theorem 4.1. Assume that the given function in (24) satisfy g C m , k1 C m D , k2 C m I I . Then for all sufficiently small h T N the constrained mesh collocation solution u S m 1 Z N 1
to (24), for all
n 0,1, , N 1, satisfies
Cm M m h m ,
(25)
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where Cm are positive constants not depending on h. This estimate holds for all collocation parameters c j with 0 c1 cm 1. Proof: In each interval ti ti 1 , the exact solution y of (24) is m times continuously differentiable. This follows from the smoothness hypotheses we have imposed on g , k1 , k2 and from the expressions for y t . From this it is obvious that both the left and right limits of y t , as t tends to nh , exist and are finite. We will prove the estimate (25) by using the Peano’s Theorem to write
m
y tn sh Lr s y tn cr h h m Rm , n s , r 1
s 0,1.
(26)
k t , t sh L s ds ,
Define the matrices in L m , 1
i 1, n
i
r
0 i n N 1,
i, j 1, 2, , m
(32)
(33)
1 k t , t sh L s ds r , 3, n 0 2 n, j i r , j 1, 2, , m 0 i n N 1, and the vectors in m by i
A1, n i
(27)
0 k1 tn, j , ti sh Rm,i s ds 1
A2, n
m 1 m 1 m 1 s z Lk s ck z , m 1! k 1
A3, n i
(28)
cj
0
k1 tn , j , tn sh Rm, n s ds
k t 1
0
2
n, j
, ti sh Rm,i s ds
Thus, it follows from (15) that the collocation error
with n , j y tn c j h u tn c j h , and it satisfies the equation
T
By substituting the (29) in the (30) and after some computations, we obtain n 1 m
n, j 1h i 0 r 1
k t 1
0 1
n 1
n, j
, ti sh Lr s ds ir
0
i 0
m
r 1
N 1 m
2 h k2 tn , j , ti sh Lr s ds ir i 0 r 1
N 1
1
0
2 h m 1 0 k2 tn , j , ti sh Rm,i s ds. 1
i 0
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n 1
i 0
i 0 N 1
m 1
N 1
A2, n 2 h 3, n i 2 h m 1 A3, n , i 0
i
(39)
i
i 0
N 1
n 1h2, n n 2 h 3,i n i n 1
N 1
1h 1, n i 1h m 1 A1, n 1h m 1 A2, n 2 h m 1 A3, n i
i 0
i
i
i 0
(40)
1h m 1 0 k1 tn , j , tn sh Rm, n s ds cj
(37)
i 0
T
this linear algebraic system may be written more concisely as
i 0
1h 0 k1 tn , j , tn sh Lr s ds nr cj
n 1
1h
n 1
1h m 1 k1 tn, j , ti sh Rm,i s ds 1
,
n 1h 1, in i 1h m 1 A1, in 1h2, n n
(30)
(36)
by substituting the Equations (32)-(38) in Equation (31) we obtain
tn , j
2 0 k2 tn , j , s s ds.
, T
n n,1 n,2 nm , n 0,1, , N 1, (38)
m
tn , j 1 0 k1 tn , j , s s ds
(35)
T
tn sh Lr s n, r h m Rm, n s , s (0,1], (29)
T
j 1, 2, , m, i n,
: y u possesses to the local representation
,
j 1, 2, , m,
z 0,1
r 1
(34)
j 1, 2, , m, i n,
and K m s, z
n, j
c j k t , t sh L s ds r , 0 1 n, j n r , j 1, 2, , m
2, n
Here, we have 1 m Rm , n s : 0 K m s, z y tn zh dz ,
0 1
Now, let (31)
I m 21hB2,0 0 0 0 I m 21hB2,1 0 0 0 0 0 0 I 2 h B m 1 2, N 1
(41) OJAppS
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J. A. SHALI ET AL. 0 1 N 1 I m 22 h3,0 22 h3,0 22 h3,0 Q (42) 0 1 N 1 22 h3, N 1 22 h3, N 1 I m 22 h3, N 1
0 1 N 1
(43)
ameters h 0, h , the uniform bound
Im
1 1
P0 ,
(49)
holds. Here, for B L m , B 1 denotes the matrix (operator) norm induced by the l1 -norm in m . Assume B1, n P1 for 0 i n N 1, and
that 1 and
i
1
1 P2 . 1 From (46) and (48) we have
1 I m n P0 n 1 , 1
Then we have n 1
n 1
i 1, n i
Q 21h 21h m 1 A i 0
i 0
N 1
(44)
21h m 1 A2, n 22 h m 1 A3, n . i
1 2
F : max 0 k2 t , s ds k 2T k 2 , *
(45)
tI
assuming that k2 t , s
i 0
n 1 i 0
i 21h m 1 A2, n 22 h m 1 A3, n i 0 1
(51)
2 P0 P1 1h
i 1 2 P0 1hm1 nmk 1M m K m
1h
mk1 M m K m 2 h m 1 Nmk 2 M m K m ,
m 1
n 1 i 0
and hence
for some matrix norm. This clearly holds whenever h is sufficiently small. In other words, there is an h 0 so that for any mesh N with h h , each matrix n has a uniformly bounded inverse. Therefore, matrix has a uniformly bounded inverse. Also, the invertibility of the N m N m block matrix Q now depends not only on h but also on 2 it is guar1 anteed if 2 F , where 2
* k2
n 1
N 1
Since the kernel K i is continuous on their domains, the elements of the matrixes 2, n , n 0,1, , N 1 are all bounded. By using the Neumann Lemma the inverse of the matrix n I m 21h2,n exists whenever
T
n 1 1 21h 1, in i 21h m 1 A1, in
i 0
1h 2, n
(50)
1
i 1, n
and the elements of the
matrixes Q are all bounded. Thus from (44), we get
Im n
(46)
1Q
(47)
where
n 1
1 0 i 1 1 M m h m 1 ,
(52)
i 0
where
0 2 P0 P1 1h , 1 2 P0 mK m k1 1 n 1 2 N k2 A1, n mk1 K m M m , i n, A2,n mk1 K m M m , i
1
1
i
m A3,n mk2 K m M m , i n, M m : y 1
km : max 0 K m s, z dz , 1
s[0,1]
,
: max 0 k1 s, v dv k1. 1
tI
(53) Also,
n
1
m
N 1
j 1
l 0
j 1
nj
m
lj
(54)
1
Then, from (52) and (54), we have
and
n 1
n 1 0 i 1 1 M m h m 1 , n 0,1, , N 1. i 0
n 1
n 1
i 0
i 0
n 1 21h 1, in i 21h m 1 A1, in +21h
m 1
A2, n 22 h
m 1
N 1
i
i 0
(48)
A3,n .
It is clear that, matrix has a uniformly bounded inverse and the elements of the matrixes are all bounded. Note that, from these assumptions and 1 Q 1 , there exists a constant D0 so that for all mesh diCopyright © 2012 SciRes.
Now, by using the discrete Gronwall inequality, we have n 1 2 M m h m 1 , n 0,1, , N 1, where 2 1 exp n 0 . Now, by using the local error representation (29) this yields, setting Wm : max L j , j
tn sh Wm n 1 h m K m M m Cm M m h m , OJAppS
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uniformly for s 0,1 and 0 n N 1, where
Table 1. Error for example 1.
Cm Wm 2 K m .
The is equivalent to the estimate
m Cm y
hm .
We conclude this section with a comment regarding the extension of the results of Theorem 1 to the nonlinear Equation (1). Under the assumption of the existence of a (unique) solution y t on I, the nonlinear analogue of the error Equation (30) is tn , j tn , j 1 0 k1 tn, j , s, y s k1 tn, j , s, u s ds
2 0 k2 tn , j , s, y s k2 tn, j , s, u s ds. T
(55) ki , i 1, 2 are continuous y and bounded on D D1 with
If the partial derivatives
D1 : y : y y s M , s I ,
for some M , and if h 0 is sufficiently small, then (55) may again be written in the form (30). The roles of ki are now assumed by H i t , s :
ki t , s, zi s y
, i 1, 2
where zi s : i y s 1 i u s , 0 i s 1. Hence, the above proof is easily adapted to deal with the nonlinear case (1), and so the convergence results of Theorem 1 remain valid for nonlinear Volterra-Fredholm integral equations.
5. Presentation of Results In this section, we report on the numerical result of test problem solved by the proposed method of this article. Typical forms of collocation parameters c j are: Gauss points: Zeros of Pm 2t 1 ; Radou I points: Zeros of Pm 2t 1 Pm 1 2t 1 ;
m
N
Guass e
Radau e
Radau e
Chelyshkov e
3
2
2 × 10–30
2 × 10–30
2 × 10–30
1 × 10–30
3
4
2 × 10–30
2 × 10–30
3 × 10–30
1 × 10–30
3
8
2 × 10–30
2 × 10–30
2 × 10–30
2 × 10–30
has the following analytical solution y t t therefore, provides an example to verify the accuracy of this method. Table 1 shows the maximum errors involved pre1 1 1 sented method with h , , , along with the exact 2 4 8 solution. For computational purposes, in the test problem different forms of kernels are considered. All the computations were carried out with Maple. In each cases of Example the obtained nonlinear equations was solved by the Newton’s method. The result for collocation points c j are presented in Table 1 which indicates that the numerical solutions ob1 1 1 and tained from (56) and step sizes equal to , 8 2 4 are nearly identical. These results indicate that, if we use the Chelyshkov points, then we obtain the numerical solutions of minimum error.
6. Conclusion We have shown that the collocation method yields an efficient and very accurate numerical method for the approximation of solutions to Volterra-Fredholm integral equations. Also we have shown that, if the roots of Pm 0 t are chosen as collocation points, then we can obtain an accurate numerical quadrature.
7. Acknowledgements The authors truly appreciate the comments made by referees.
Radou II points: Zeros of Pm 2t 1 Pm 1 2t 1 ;
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