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The successive approximation method and Padé approximants for solutions the non-linear boundary value problem
Applied Mathematics and Computation 146 (2003) 681–690 www.elsevier.com/locate/amc
The successive approximation method and Pade approximants for solutions the non-linear boundary value problem Hakan S ß imsßek *, Ercan C ß elik € niversitesi, T-025240 Erzurum, Turkey Fen-Edebiyat Fak€ultesi, Matematik B€ol€um€u, Atat€urk U
Abstract In this paper, we suggested an successive approximation method and Pade approximants method for the solution of the non-linear differential equation. First we calculate power series of the given equation system then transform it into Pade (approximants) series form, which give an arbitrary order for solving differential equation numerically. We compare our results with the result obtained by successive method for the non-linear equation. Ó 2002 Elsevier Inc. All rights reserved. Keywords: Successive approximation method; Non-linear equation; Pade approximants
1. Introduction Let us consider the non-linear boundary value problem x00 ðtÞ ¼ aðtÞxðtÞ þ f ðx; tÞ;
xð0Þ ¼ x0 ;
xðT Þ ¼ xT
ð1Þ
In this case, f ðx; tÞ ¼ f ðtÞ which has been investigated by many authors [1–3]. There are many theorems on existence of a unique solution to problem (1) [4– 6]. In this study, we construct the special approximations and Pade approximants for Eq. (1).
*
Corresponding author. E-mail address: [email protected] (H. S ß imsßek).
0096-3003/$ - see front matter Ó 2002 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(02)00612-4
682
H. Sß imsßek, E. C ß elik / Appl. Math. Comput. 146 (2003) 681–690
2. Semi-linear operator equations Definition. Let E be a Banach space with the distance q, and let A be a mapping of E onto itself. Then A is said to be a contracting mapping if there exists a constant a (0 6 a 6 1) such that the inequality qðAx AyÞ 6 aqðx; yÞ holds for every pair of points x; y 2 E. In this section, we will consider the semi-linear operator equation x ¼ A0 x þ A1 ðxÞ
ð2Þ
in the E-Banach spaces, where A0 x is a linear bounded operator defined in E 1 such that ðI A0 Þ is a bounded operator, A1 ðxÞ is a non-linear operator defined in E and I is an identity operator. The successive approximations for Eq. (2) are defined by xn ¼ A0 xn þ A1 ðxn1 Þ;
n ¼ 1; 2; . . .
ð3Þ
for the arbitrary x0 2 E. The approximations (3) can be written as xn ¼ ðI A0 Þ1 A1 ðxn1 Þ;
n ¼ 1; 2; . . .
ð4Þ
First of all we will give the following theorem. 1
Theorem 1. Let us assume that the operator ðI A0 Þ exists and is bounded for linear operator A0 x, which is defined in E-Banach space and the operatorA1 ðxÞ satisfies Lipschitz condition kA1 ðxÞ A1 ðyÞk 6 Lkx yk 1
Also, suppose that b ¼ LkðI A0 Þ k < 1. Therefore Eq. (2) has a unique solution x and this solution is limit of the successive approximations (4). The speed of the convergence is determined by the inequality. Proof can be found [6]. Now, let us apply this theorem to problem (1). It is easy see that problem (1) equivalent to the non-linear Fredholm–Volterra integral equation Z Z t 1 T xðtÞ ¼ uðtÞ ðT sÞaðsÞxðsÞ ds þ ðt sÞaðsÞxðsÞ ds t 0 0 Z t Z T t ðT sÞf ðs; xðsÞÞ ds þ ðt sÞf ðs; xðxÞÞ ds T 0 0
ð5Þ
H. Sß imsßek, E. C ß elik / Appl. Math. Comput. 146 (2003) 681–690
683
or xðtÞ ¼ uðtÞ þ
Z
T
Gðt; sÞaðsÞxðsÞ ds þ
0
Z
t
ðt sÞf ðs; xðsÞÞ ds
ð6Þ
0
where uðtÞ ¼
T t t x0 þ xT T T
and
8 sðT tÞ > < ; T Gðt; sÞ ¼ > : tðT sÞ ; T
06s6t t6s6T
Green function Gðt; sÞ is a negative and continuous function that satisfies the conditions Z T T T2 jGðt; sÞj 6 and jGðt; sÞj 6 4 8 0 On the other hand, we set, Z T F0 x ðT sÞaðsÞxðsÞ ds;
V0 x
0
t F1 ðxÞ T
Z
t
ðt sÞaðsÞxðsÞ ds
0
Z
T
ðT sÞf ðs; xðsÞÞ ds;
V1 ðxÞ
0
F ðxÞ ¼ F1 ðxÞ þ V1 ðxÞ ¼
Z
t
ðt sÞf ðs; xðsÞÞ ds
0
Z
T
GðT ; sÞf ðs; xðsÞÞ ds;
AðxÞ ¼ F ðxÞ þ V0 x
0
Eq. (5) or (6) can be written as xðtÞ ¼ hðtÞ þ
t F0 x þ AðxÞ T
This integral equation is equivalent to problem 1. F0 x Fredholm operator is defined from C½0; T to R. That means that, yðtÞ ¼ h ðtÞ þ
t F0 y T
operator is degenerated kernel-Fredholm operator. If we take K0 ðt; sÞ ¼ Gðt; sÞaðsÞ and K1 ðt; s; xðsÞÞ ¼ Gðt; sÞf ðs; xðsÞÞ, then from Eq. (6) we can write Z T Z T xðtÞ ¼ hðtÞ þ K0 ðt; sÞxðsÞ ds þ K1 ðt; s; xðsÞÞ ds ð7Þ 0
0
Pm where K0 ðt; sÞ degenerated kernel with K0 ðt; sÞ ¼ i¼0 ai ðtÞbi ðsÞ [6] and for continuous K1 ðt; s; xðsÞÞ (0 6 s, t R6 T , 1 < x < 1) functions Rholds the Lipschitz T T condition. If we set, A0 x 0 K0 ðt; sÞxðsÞ ds and A1 ðxÞ 0 K1 ðt; s; xðsÞÞ ds þ f ðtÞ then Eq. (7) can be written as x ¼ A0 x þ A1 ðxÞ.
684
H. Sß imsßek, E. C ß elik / Appl. Math. Comput. 146 (2003) 681–690 1
In order to obtain the operator ðI A0 Þ , we solve x ¼ A0 x þ F ðtÞ or # Z T "X m xðtÞ ¼ ai ðtÞbi ðsÞ ds þ F ðtÞ ð8Þ 0
i¼0
Pm Let us investigate the solution Eq. (8) in following form xðtÞ ¼ i¼0 ai ðtÞci þ F ðtÞ. We write Z T m Z T X bi ðsÞaj ðsÞ ds cj þ bi ðsÞF ðsÞ ds; i ¼ 1; 2; . . . ; m ci ¼ 0
j¼0
0
Suppose that the determinant of the coefficient matrix D 6¼ 0 and Dij strands for the algebraic complement of determinant D with the subscripts ij, then we obtain Z T m 1X ci ¼ Dij bj ðsÞF ðsÞ ds ð9Þ D j¼1 0 Now Eq. (8) can be written as Z T m 1X ai ðtÞDij bj ðsÞF ðsÞ ds þ F ðtÞ xðtÞ ¼ D i;j¼1 0
ð10Þ
and we obtain the inequality
Z T m X
Dij kai k
jbj ðsÞj ds þ 1 kðI A0 Þ1 k 6 D 0 i;j¼1 We can write Eq. (7) as xðtÞ ¼ hðtÞ þ
m X i;j¼1
þ
Z
ai ðtÞ
Dij D
Z
T
bj ðsÞ 0
Z
T
K1 ðt; s; xðsÞÞ ds ds 0
T
K1 ðt; s; xðsÞÞ ds 0
hðtÞ ¼
Z m X Dij T ai ðtÞ bj ðsÞf ðsÞ ds þ f ðtÞ D 0 i;j¼1
considering Eq. (9). Now, it is easily shown that A1 ðxÞ operator holds Lipschitz condition in E ¼ C½0; T for arbitrary xðtÞ; yðtÞ 2 C½0; T . Theorem 2. Suppose that ai ðtÞ, bi ðsÞ, K1 ðt; s; xÞ (0 6 t, s 6 T , 1 < x < 1) are continuous functions and K1 ðt; s; xÞ is uniformly Lipschitz in its their variable with the Lipschitz coefficient L. Also, let
H. Sß imsßek, E. C ß elik / Appl. Math. Comput. 146 (2003) 681–690
685
" #
Z m X Dij T aij b ¼ LT jbj ðsÞ dsj þ 1 < 1
D 0 i;j¼1 Therefore, Eq. (9) has unique solution and this solution is the limit of the successive approximations: Z Z T m X Dij T xn ðtÞ ¼ hðtÞ þ ai ðtÞ bj ðsÞ K1 ðt; s; xn1 ðsÞÞ ds ds D 0 0 i;j¼1 Z T K1 ðt; s; xn1 ðsÞÞ ds þ 0
for n ¼ 1; 2; . . . The speed of the convergence determined by inequality kxn x k 6 bn kx x k.
3. Successive approximations method In this section, we establish the successive approximates for the solution of problem (1). Our aim in this section not to give a theorem concerning the existence of a unique solution to Eq. (1). But we investigate only approximate methods in order to find the solution. Most theorems on the existence of a unique solution to Eq. (1) can be found in Ref. [6]. The following theorem can be proved by using simple approximation method. Theorem 3. Suppose that the function /ðt; xÞ (0 6 t 6 T , 1 < x < 1) is continuous and satisfies Lipschitz condition with respect to x and let q ¼ LT 2 =8 < 1. Then the problem x00 ¼ /ðt; xÞ;
xð0Þ ¼ xt ;
xðT Þ ¼ xT
ð11Þ
has unique solution in ½0; T and this solution is the limit of simple successive approximates Z T Gðt; sÞ/ðs; xn1 ðsÞÞ ds; n ¼ 1; 2; . . . ð12Þ xn ðtÞ ¼ hðtÞ þ 0
Furthermore convergence speed is determined by jxn ðtÞ x ðtÞ 6 jqn kx x0 k
ð13Þ
where x ð0 6 t 6 T Þ is a solution to the problem (11) [6]. If the function /ðt; xÞ satisfies the Lipschitz condition for jxj 6 r then we assume that T 2 M=8 < r, M ¼ max0 6 t 6 T ; jxj
686
H. Sß imsßek, E. C ß elik / Appl. Math. Comput. 146 (2003) 681–690
change the interval ð1; 1Þ, where the non-linear functions satisfy the Lipshitz condition with interval ½r; r [6]. Theorem 4. Suppose that the function /ðt; xÞ (0 6 t 6 T , 1 < x < 1) satisfies x/ðt; xÞs2 P jxj2 þ f ðtÞ ðs ¼ constantÞ where f ðtÞ (0 6 t 6 T ) is a given continuous function. Then, 2
2
jxðtÞj 6 r ¼ jx0 j þ jxT j þ
cosh sT2 1 kf k s2 cosh sT2 1
holds for the solution (if there exists one) to problem (11). Remark. Suppose that the function f ðx; tÞ satisfies 2
x/ðt; xÞ 6 L0 jxj þ f ðtÞ
ðL0 > 0Þ
ð14Þ
then r is 2
cosh
2
r ¼ jx0 j þ jxT j þ
T
pffiffiffiffiffiffiffiffiffiffiffi L0 þkak 2
ðL0 þ kakÞ cosh
T
1 pffiffiffiffiffiffiffiffiffiffiffi kf k L0 þkak 2
ð15Þ
for the solution of Eq. (1). Now we establish the successive approximates for the solution to problem (1) or integral Eqs. (5) and (6) by xn ðtÞ ¼ hðtÞ
t F0 xn ðtÞ þ V0 ðxn1 ðtÞÞ þ F1 ðxn1 ðtÞÞ þ V1 ðxn1 ðtÞÞ T
ð16Þ
xn ðtÞ ¼ hðtÞ þ
t F0 xn ðtÞ þ Aðxn1 ðtÞÞ; T
ð17Þ
or n ¼ 1; 2; . . .
where x0 ðtÞ (0 6 t 6 T ) is an arbitrary continuous function. It can be easily seen from Eqs. (16) and (17) that one needs to solve Eq. (14) to find xn ðtÞ if xn1 is already known. If we let Z T a¼T þ ðT sÞsaðsÞ ds 6¼ 0 ð18Þ 0
then the solution to Eq. (14) is given as [6]. t y ¼ h þ F0 h a
ð19Þ
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687
Namely setting h ¼ hðtÞ þ V0 xn1 þ F1 xn1 ;
hðtÞ ¼ hðtÞ þ Aðxn1 Þ
in Eq. (19) we find from Eqs. (16) and (17) that t t t xn ðtÞ ¼ h þ F1 xn1 þ F0 V0 ðxn1 Þ þ F0 F1 ðxn1 Þ þ F0 V1 ðxn1 Þ a a a þ V0 xn1 þ V1 ðxn1 Þ xn ðtÞ ¼ h ðtÞ þ Aðxn1 Þ þ
t F0 Aðxn1 Þ; T
n ¼ 1; 2; . . .
ð20Þ ð21Þ
where h ðtÞ ¼ hðtÞ þ ðt=aÞF0 hðtÞ. We assume that xn ðtÞ defined by Eq. (21) is an approximate solution of problem (1). As we proved in the introduction if the function f ðt; xÞ satisfies Eq. (13) then the solution to problem (1) also satisfies jxðtÞ 6 rj, where r defined in Eq. (15). Our aim in writing Eq. (16) (or Eq. (1)) is to use is estimating of convergence speed to the solution the Eq. (1) of the sequence fxn ðtÞg defined by Eq. (16). The difference between the iterations made by Eqs. (16) and (17) is to be F ðxÞ in Eq. (17) though Eq. (16) has F1 ðxÞ þ V1 ðxÞ in it. An important advantage of this that the convergence speed can be evaluated in some cases more precisely using the Volterra operator V1 ðxÞ. Therefore, the sequence fxn ðtÞg defined by Eq. (15) converges the solution to problem (1). Now, we are in a position to estimate the convergence speed. If we assume that hðtÞ ¼ hðtÞ þ F ðxÞ and use this in Eq. (19), then we get the integral equation xðtÞ ¼ h ðtÞ þ F ðxÞ þ
t F0 AðxÞ T
ð22Þ
which is equivalent to Eq. (10). We assume that en ðtÞ ¼ xn ðtÞ xðtÞ to get from Eqs. (16) and (17) that en ðtÞ ¼ F en1 þ
t F0 Aðen1 Þ; T
n ¼ 1; 2; . . .
ð23Þ
Let f ðt; xÞ be continuous and satisfy the Lipschitz condition with respect to x; L > 0. From Eq. (23) we have Z T Z T T en ðtÞ jGðt; sÞjjf ðs; xn1 ðsÞÞ f ðs; xðsÞÞj ds þ ðT sÞjaðsÞj jaj 0 0 Z T Z T T jGðs; rÞjjf ðr; xn1 ðrÞÞ f ðr; xðrÞÞj dr ds þ ðT sÞjaðsÞj jaj 0 Z0 s jaðrÞjjxn1 ðrÞ xðrÞj dr ds; 0
688
H. Sß imsßek, E. C ß elik / Appl. Math. Comput. 146 (2003) 681–690
Z T Z T T T4 kakL en ðtÞ 6 L jen1 ðsÞj ds þ jen1 ðsÞj ds 4 8jaj 0 0 Z T T4 2 kak jen1 ðsÞj ds; þ 2jaj 0 jen ðtÞj 6 A
Z
T
jen1 ðsÞj ds;
n
jen ðtÞj 6 ðAT Þ ke0 k
0
where A¼T
L T3 þ 4 2jaj
L þ kak kak 4
ð24Þ
Therefore, n
jxn ðtÞ xðtÞj 6 ðAT Þ kx0 ðtÞ xðtÞk
ð25Þ
is definitely correct. In the case of AT < 1, the sequence fxn ðtÞg defined by Eq. (21) converges to the solution to problem (1). Convergence speed is given by Eq. (25).
4. Pade (approximants) series The power series can be transformed into Pade series easily. Pade series is defined in the following a0 þ a1 x þ a2 x 2 þ ¼
p0 þ p1 x þ þ pM xM 1 þ q1 x þ þ qL x L
ð26Þ
Multiply the both side of (26) by the denominator of right side in (26) and compare the coefficients of both sides in (26). We have al þ
m X
alk qk ¼ pl ;
l ¼ 0; . . . ; M
ð27Þ
alk qk ¼ 0;
l ¼ M þ 1; . . . ; M þ L
ð28Þ
k¼1
al þ
L X k¼1
Solve the linear equation in (28), we have qk (k ¼ 1; . . . ; L). And substitute qk into (27), we have pl (l ¼ 0; . . . ; M). The coefficients in (28) are Toeplitz matrix. Although a fast algorithm developed by Levinson for Toeplitz Matrix [6, p. 43] is known, Gaussian elimination is used to solve above linear equations.
H. Sß imsßek, E. C ß elik / Appl. Math. Comput. 146 (2003) 681–690
689
If M 6 L 6 M þ 2, where M and L is the degree of numerator and denominator in Pade series respectively, then this Pade series gives an A-stable formula for an ordinary differential equation [5,7].
5. An example As an example, let us consider the boundary value problem x00 þ sinð2tÞxðtÞ ¼ 2t sinðxðtÞÞ; xð0Þ ¼ 0;
06t61
ð29Þ
xð1Þ ¼ 2
The solution to problem (29) by using successive approximations and Eqs. (3) and (4) is given in x1 ðtÞ ¼ 1:682941970t þ 0:244191632t2 þ 0:1666666667t4 þ 0:02418301171t5 0:01111111111t6 0:001612200780t7 þ 0:0002976190476t8 þ 0:00004318394946t9 x1 ðtÞ can be transformed into Pade series s ¼ ½4=4 ¼
1:682941970t 0:03813066411t2 þ 0:1017216709t3 þ 0:1641246225t4 : ð1:000000000 0:1677552175t þ 0:08478372620t2 0:01381243155t3 þ 0:004247960733t4 Þ
We show Table 1 for solution of (13) by numerical method. The numerical values in Table 1 obtained above are in full agreement with the exact solutions of Eq. (23). The Pade approximants that often show superior performance over series approximants, provide a promising tool for using in applied fields. Table 1 Values at some points of x1 ðtÞ the interval in ½0; 1 ti
xðti Þ
x1 ðti Þ
jx x1 j
Pade app.
jx Padej
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2
0.1707530105 0.3466297337 0.5282601703 0.7167138036 0.9135063193 1.120596544 1.340373302 1.575632091 1.829541656 2.105600772
0.0292469895 0.0533702663 0.0717398297 0.0832861964 0.0864936807 0.079403456 0.059626698 0.024367909 0.029541656 0.105600772
0.1707530105 0.3466297336 0.5282601706 0.7167138046 0.9135063203 1.120596549 1.340373332 1.575632213 1.829542082 2.105600559
0.0292469895 0.0533702664 0.0717398294 0.0832861954 0.0864936797 0.079403451 0.059626668 0.024367787 0.029542082 0.105600559
690
H. Sß imsßek, E. C ß elik / Appl. Math. Comput. 146 (2003) 681–690
References [1] J.D. Mamedov, G. Berger, H. D€ orner, Eine Methode zur naherrungsweisen einer Randwertaufgabe einer linearen gew€ ohnlichen Differentialgichund 2, Ordnung Zeitschrift f€ ur Analysis und ihre Anwendungen, Bd. 2 (1) (1983) 11–23. [2] J. Casti, R. Kalaba, Imbedding Methods in Applied Mathematics, Adison-Wesley, London, 1973. [3] H. S ß imsßek, An approximate solution of a non-linear boundary value problem, J. Fac. Sci. Ege Univ. 20 (1997) 77–88. [4] W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes, Cambridge University Press, Cambridge, 1988. [5] A.M. Wazwaz, Analytical approximations and Pade approximations for VolterraÕs population model, App. Math. Comput. 100 (1999) 13–25. [6] Y.C. Memmedov, Approximate Methods, Atat€ urk Uni., Erzurum, 1994 (in Turkish). [7] E. C ß elik, M. Bayram, On numerical solution of differential-algebraic equations by Pade series, App. Math. Comput., in press.
The successive approximation method and Pade approximants for solutions the non-linear boundary value problem Hakan S ß imsßek *, Ercan C ß elik € niversitesi, T-025240 Erzurum, Turkey Fen-Edebiyat Fak€ultesi, Matematik B€ol€um€u, Atat€urk U
Abstract In this paper, we suggested an successive approximation method and Pade approximants method for the solution of the non-linear differential equation. First we calculate power series of the given equation system then transform it into Pade (approximants) series form, which give an arbitrary order for solving differential equation numerically. We compare our results with the result obtained by successive method for the non-linear equation. Ó 2002 Elsevier Inc. All rights reserved. Keywords: Successive approximation method; Non-linear equation; Pade approximants
1. Introduction Let us consider the non-linear boundary value problem x00 ðtÞ ¼ aðtÞxðtÞ þ f ðx; tÞ;
xð0Þ ¼ x0 ;
xðT Þ ¼ xT
ð1Þ
In this case, f ðx; tÞ ¼ f ðtÞ which has been investigated by many authors [1–3]. There are many theorems on existence of a unique solution to problem (1) [4– 6]. In this study, we construct the special approximations and Pade approximants for Eq. (1).
*
Corresponding author. E-mail address: [email protected] (H. S ß imsßek).
0096-3003/$ - see front matter Ó 2002 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(02)00612-4
682
H. Sß imsßek, E. C ß elik / Appl. Math. Comput. 146 (2003) 681–690
2. Semi-linear operator equations Definition. Let E be a Banach space with the distance q, and let A be a mapping of E onto itself. Then A is said to be a contracting mapping if there exists a constant a (0 6 a 6 1) such that the inequality qðAx AyÞ 6 aqðx; yÞ holds for every pair of points x; y 2 E. In this section, we will consider the semi-linear operator equation x ¼ A0 x þ A1 ðxÞ
ð2Þ
in the E-Banach spaces, where A0 x is a linear bounded operator defined in E 1 such that ðI A0 Þ is a bounded operator, A1 ðxÞ is a non-linear operator defined in E and I is an identity operator. The successive approximations for Eq. (2) are defined by xn ¼ A0 xn þ A1 ðxn1 Þ;
n ¼ 1; 2; . . .
ð3Þ
for the arbitrary x0 2 E. The approximations (3) can be written as xn ¼ ðI A0 Þ1 A1 ðxn1 Þ;
n ¼ 1; 2; . . .
ð4Þ
First of all we will give the following theorem. 1
Theorem 1. Let us assume that the operator ðI A0 Þ exists and is bounded for linear operator A0 x, which is defined in E-Banach space and the operatorA1 ðxÞ satisfies Lipschitz condition kA1 ðxÞ A1 ðyÞk 6 Lkx yk 1
Also, suppose that b ¼ LkðI A0 Þ k < 1. Therefore Eq. (2) has a unique solution x and this solution is limit of the successive approximations (4). The speed of the convergence is determined by the inequality. Proof can be found [6]. Now, let us apply this theorem to problem (1). It is easy see that problem (1) equivalent to the non-linear Fredholm–Volterra integral equation Z Z t 1 T xðtÞ ¼ uðtÞ ðT sÞaðsÞxðsÞ ds þ ðt sÞaðsÞxðsÞ ds t 0 0 Z t Z T t ðT sÞf ðs; xðsÞÞ ds þ ðt sÞf ðs; xðxÞÞ ds T 0 0
ð5Þ
H. Sß imsßek, E. C ß elik / Appl. Math. Comput. 146 (2003) 681–690
683
or xðtÞ ¼ uðtÞ þ
Z
T
Gðt; sÞaðsÞxðsÞ ds þ
0
Z
t
ðt sÞf ðs; xðsÞÞ ds
ð6Þ
0
where uðtÞ ¼
T t t x0 þ xT T T
and
8 sðT tÞ > < ; T Gðt; sÞ ¼ > : tðT sÞ ; T
06s6t t6s6T
Green function Gðt; sÞ is a negative and continuous function that satisfies the conditions Z T T T2 jGðt; sÞj 6 and jGðt; sÞj 6 4 8 0 On the other hand, we set, Z T F0 x ðT sÞaðsÞxðsÞ ds;
V0 x
0
t F1 ðxÞ T
Z
t
ðt sÞaðsÞxðsÞ ds
0
Z
T
ðT sÞf ðs; xðsÞÞ ds;
V1 ðxÞ
0
F ðxÞ ¼ F1 ðxÞ þ V1 ðxÞ ¼
Z
t
ðt sÞf ðs; xðsÞÞ ds
0
Z
T
GðT ; sÞf ðs; xðsÞÞ ds;
AðxÞ ¼ F ðxÞ þ V0 x
0
Eq. (5) or (6) can be written as xðtÞ ¼ hðtÞ þ
t F0 x þ AðxÞ T
This integral equation is equivalent to problem 1. F0 x Fredholm operator is defined from C½0; T to R. That means that, yðtÞ ¼ h ðtÞ þ
t F0 y T
operator is degenerated kernel-Fredholm operator. If we take K0 ðt; sÞ ¼ Gðt; sÞaðsÞ and K1 ðt; s; xðsÞÞ ¼ Gðt; sÞf ðs; xðsÞÞ, then from Eq. (6) we can write Z T Z T xðtÞ ¼ hðtÞ þ K0 ðt; sÞxðsÞ ds þ K1 ðt; s; xðsÞÞ ds ð7Þ 0
0
Pm where K0 ðt; sÞ degenerated kernel with K0 ðt; sÞ ¼ i¼0 ai ðtÞbi ðsÞ [6] and for continuous K1 ðt; s; xðsÞÞ (0 6 s, t R6 T , 1 < x < 1) functions Rholds the Lipschitz T T condition. If we set, A0 x 0 K0 ðt; sÞxðsÞ ds and A1 ðxÞ 0 K1 ðt; s; xðsÞÞ ds þ f ðtÞ then Eq. (7) can be written as x ¼ A0 x þ A1 ðxÞ.
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In order to obtain the operator ðI A0 Þ , we solve x ¼ A0 x þ F ðtÞ or # Z T "X m xðtÞ ¼ ai ðtÞbi ðsÞ ds þ F ðtÞ ð8Þ 0
i¼0
Pm Let us investigate the solution Eq. (8) in following form xðtÞ ¼ i¼0 ai ðtÞci þ F ðtÞ. We write Z T m Z T X bi ðsÞaj ðsÞ ds cj þ bi ðsÞF ðsÞ ds; i ¼ 1; 2; . . . ; m ci ¼ 0
j¼0
0
Suppose that the determinant of the coefficient matrix D 6¼ 0 and Dij strands for the algebraic complement of determinant D with the subscripts ij, then we obtain Z T m 1X ci ¼ Dij bj ðsÞF ðsÞ ds ð9Þ D j¼1 0 Now Eq. (8) can be written as Z T m 1X ai ðtÞDij bj ðsÞF ðsÞ ds þ F ðtÞ xðtÞ ¼ D i;j¼1 0
ð10Þ
and we obtain the inequality
Z T m X
Dij kai k
jbj ðsÞj ds þ 1 kðI A0 Þ1 k 6 D 0 i;j¼1 We can write Eq. (7) as xðtÞ ¼ hðtÞ þ
m X i;j¼1
þ
Z
ai ðtÞ
Dij D
Z
T
bj ðsÞ 0
Z
T
K1 ðt; s; xðsÞÞ ds ds 0
T
K1 ðt; s; xðsÞÞ ds 0
hðtÞ ¼
Z m X Dij T ai ðtÞ bj ðsÞf ðsÞ ds þ f ðtÞ D 0 i;j¼1
considering Eq. (9). Now, it is easily shown that A1 ðxÞ operator holds Lipschitz condition in E ¼ C½0; T for arbitrary xðtÞ; yðtÞ 2 C½0; T . Theorem 2. Suppose that ai ðtÞ, bi ðsÞ, K1 ðt; s; xÞ (0 6 t, s 6 T , 1 < x < 1) are continuous functions and K1 ðt; s; xÞ is uniformly Lipschitz in its their variable with the Lipschitz coefficient L. Also, let
H. Sß imsßek, E. C ß elik / Appl. Math. Comput. 146 (2003) 681–690
685
" #
Z m X Dij T aij b ¼ LT jbj ðsÞ dsj þ 1 < 1
D 0 i;j¼1 Therefore, Eq. (9) has unique solution and this solution is the limit of the successive approximations: Z Z T m X Dij T xn ðtÞ ¼ hðtÞ þ ai ðtÞ bj ðsÞ K1 ðt; s; xn1 ðsÞÞ ds ds D 0 0 i;j¼1 Z T K1 ðt; s; xn1 ðsÞÞ ds þ 0
for n ¼ 1; 2; . . . The speed of the convergence determined by inequality kxn x k 6 bn kx x k.
3. Successive approximations method In this section, we establish the successive approximates for the solution of problem (1). Our aim in this section not to give a theorem concerning the existence of a unique solution to Eq. (1). But we investigate only approximate methods in order to find the solution. Most theorems on the existence of a unique solution to Eq. (1) can be found in Ref. [6]. The following theorem can be proved by using simple approximation method. Theorem 3. Suppose that the function /ðt; xÞ (0 6 t 6 T , 1 < x < 1) is continuous and satisfies Lipschitz condition with respect to x and let q ¼ LT 2 =8 < 1. Then the problem x00 ¼ /ðt; xÞ;
xð0Þ ¼ xt ;
xðT Þ ¼ xT
ð11Þ
has unique solution in ½0; T and this solution is the limit of simple successive approximates Z T Gðt; sÞ/ðs; xn1 ðsÞÞ ds; n ¼ 1; 2; . . . ð12Þ xn ðtÞ ¼ hðtÞ þ 0
Furthermore convergence speed is determined by jxn ðtÞ x ðtÞ 6 jqn kx x0 k
ð13Þ
where x ð0 6 t 6 T Þ is a solution to the problem (11) [6]. If the function /ðt; xÞ satisfies the Lipschitz condition for jxj 6 r then we assume that T 2 M=8 < r, M ¼ max0 6 t 6 T ; jxj
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change the interval ð1; 1Þ, where the non-linear functions satisfy the Lipshitz condition with interval ½r; r [6]. Theorem 4. Suppose that the function /ðt; xÞ (0 6 t 6 T , 1 < x < 1) satisfies x/ðt; xÞs2 P jxj2 þ f ðtÞ ðs ¼ constantÞ where f ðtÞ (0 6 t 6 T ) is a given continuous function. Then, 2
2
jxðtÞj 6 r ¼ jx0 j þ jxT j þ
cosh sT2 1 kf k s2 cosh sT2 1
holds for the solution (if there exists one) to problem (11). Remark. Suppose that the function f ðx; tÞ satisfies 2
x/ðt; xÞ 6 L0 jxj þ f ðtÞ
ðL0 > 0Þ
ð14Þ
then r is 2
cosh
2
r ¼ jx0 j þ jxT j þ
T
pffiffiffiffiffiffiffiffiffiffiffi L0 þkak 2
ðL0 þ kakÞ cosh
T
1 pffiffiffiffiffiffiffiffiffiffiffi kf k L0 þkak 2
ð15Þ
for the solution of Eq. (1). Now we establish the successive approximates for the solution to problem (1) or integral Eqs. (5) and (6) by xn ðtÞ ¼ hðtÞ
t F0 xn ðtÞ þ V0 ðxn1 ðtÞÞ þ F1 ðxn1 ðtÞÞ þ V1 ðxn1 ðtÞÞ T
ð16Þ
xn ðtÞ ¼ hðtÞ þ
t F0 xn ðtÞ þ Aðxn1 ðtÞÞ; T
ð17Þ
or n ¼ 1; 2; . . .
where x0 ðtÞ (0 6 t 6 T ) is an arbitrary continuous function. It can be easily seen from Eqs. (16) and (17) that one needs to solve Eq. (14) to find xn ðtÞ if xn1 is already known. If we let Z T a¼T þ ðT sÞsaðsÞ ds 6¼ 0 ð18Þ 0
then the solution to Eq. (14) is given as [6]. t y ¼ h þ F0 h a
ð19Þ
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687
Namely setting h ¼ hðtÞ þ V0 xn1 þ F1 xn1 ;
hðtÞ ¼ hðtÞ þ Aðxn1 Þ
in Eq. (19) we find from Eqs. (16) and (17) that t t t xn ðtÞ ¼ h þ F1 xn1 þ F0 V0 ðxn1 Þ þ F0 F1 ðxn1 Þ þ F0 V1 ðxn1 Þ a a a þ V0 xn1 þ V1 ðxn1 Þ xn ðtÞ ¼ h ðtÞ þ Aðxn1 Þ þ
t F0 Aðxn1 Þ; T
n ¼ 1; 2; . . .
ð20Þ ð21Þ
where h ðtÞ ¼ hðtÞ þ ðt=aÞF0 hðtÞ. We assume that xn ðtÞ defined by Eq. (21) is an approximate solution of problem (1). As we proved in the introduction if the function f ðt; xÞ satisfies Eq. (13) then the solution to problem (1) also satisfies jxðtÞ 6 rj, where r defined in Eq. (15). Our aim in writing Eq. (16) (or Eq. (1)) is to use is estimating of convergence speed to the solution the Eq. (1) of the sequence fxn ðtÞg defined by Eq. (16). The difference between the iterations made by Eqs. (16) and (17) is to be F ðxÞ in Eq. (17) though Eq. (16) has F1 ðxÞ þ V1 ðxÞ in it. An important advantage of this that the convergence speed can be evaluated in some cases more precisely using the Volterra operator V1 ðxÞ. Therefore, the sequence fxn ðtÞg defined by Eq. (15) converges the solution to problem (1). Now, we are in a position to estimate the convergence speed. If we assume that hðtÞ ¼ hðtÞ þ F ðxÞ and use this in Eq. (19), then we get the integral equation xðtÞ ¼ h ðtÞ þ F ðxÞ þ
t F0 AðxÞ T
ð22Þ
which is equivalent to Eq. (10). We assume that en ðtÞ ¼ xn ðtÞ xðtÞ to get from Eqs. (16) and (17) that en ðtÞ ¼ F en1 þ
t F0 Aðen1 Þ; T
n ¼ 1; 2; . . .
ð23Þ
Let f ðt; xÞ be continuous and satisfy the Lipschitz condition with respect to x; L > 0. From Eq. (23) we have Z T Z T T en ðtÞ jGðt; sÞjjf ðs; xn1 ðsÞÞ f ðs; xðsÞÞj ds þ ðT sÞjaðsÞj jaj 0 0 Z T Z T T jGðs; rÞjjf ðr; xn1 ðrÞÞ f ðr; xðrÞÞj dr ds þ ðT sÞjaðsÞj jaj 0 Z0 s jaðrÞjjxn1 ðrÞ xðrÞj dr ds; 0
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Z T Z T T T4 kakL en ðtÞ 6 L jen1 ðsÞj ds þ jen1 ðsÞj ds 4 8jaj 0 0 Z T T4 2 kak jen1 ðsÞj ds; þ 2jaj 0 jen ðtÞj 6 A
Z
T
jen1 ðsÞj ds;
n
jen ðtÞj 6 ðAT Þ ke0 k
0
where A¼T
L T3 þ 4 2jaj
L þ kak kak 4
ð24Þ
Therefore, n
jxn ðtÞ xðtÞj 6 ðAT Þ kx0 ðtÞ xðtÞk
ð25Þ
is definitely correct. In the case of AT < 1, the sequence fxn ðtÞg defined by Eq. (21) converges to the solution to problem (1). Convergence speed is given by Eq. (25).
4. Pade (approximants) series The power series can be transformed into Pade series easily. Pade series is defined in the following a0 þ a1 x þ a2 x 2 þ ¼
p0 þ p1 x þ þ pM xM 1 þ q1 x þ þ qL x L
ð26Þ
Multiply the both side of (26) by the denominator of right side in (26) and compare the coefficients of both sides in (26). We have al þ
m X
alk qk ¼ pl ;
l ¼ 0; . . . ; M
ð27Þ
alk qk ¼ 0;
l ¼ M þ 1; . . . ; M þ L
ð28Þ
k¼1
al þ
L X k¼1
Solve the linear equation in (28), we have qk (k ¼ 1; . . . ; L). And substitute qk into (27), we have pl (l ¼ 0; . . . ; M). The coefficients in (28) are Toeplitz matrix. Although a fast algorithm developed by Levinson for Toeplitz Matrix [6, p. 43] is known, Gaussian elimination is used to solve above linear equations.
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689
If M 6 L 6 M þ 2, where M and L is the degree of numerator and denominator in Pade series respectively, then this Pade series gives an A-stable formula for an ordinary differential equation [5,7].
5. An example As an example, let us consider the boundary value problem x00 þ sinð2tÞxðtÞ ¼ 2t sinðxðtÞÞ; xð0Þ ¼ 0;
06t61
ð29Þ
xð1Þ ¼ 2
The solution to problem (29) by using successive approximations and Eqs. (3) and (4) is given in x1 ðtÞ ¼ 1:682941970t þ 0:244191632t2 þ 0:1666666667t4 þ 0:02418301171t5 0:01111111111t6 0:001612200780t7 þ 0:0002976190476t8 þ 0:00004318394946t9 x1 ðtÞ can be transformed into Pade series s ¼ ½4=4 ¼
1:682941970t 0:03813066411t2 þ 0:1017216709t3 þ 0:1641246225t4 : ð1:000000000 0:1677552175t þ 0:08478372620t2 0:01381243155t3 þ 0:004247960733t4 Þ
We show Table 1 for solution of (13) by numerical method. The numerical values in Table 1 obtained above are in full agreement with the exact solutions of Eq. (23). The Pade approximants that often show superior performance over series approximants, provide a promising tool for using in applied fields. Table 1 Values at some points of x1 ðtÞ the interval in ½0; 1 ti
xðti Þ
x1 ðti Þ
jx x1 j
Pade app.
jx Padej
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2
0.1707530105 0.3466297337 0.5282601703 0.7167138036 0.9135063193 1.120596544 1.340373302 1.575632091 1.829541656 2.105600772
0.0292469895 0.0533702663 0.0717398297 0.0832861964 0.0864936807 0.079403456 0.059626698 0.024367909 0.029541656 0.105600772
0.1707530105 0.3466297336 0.5282601706 0.7167138046 0.9135063203 1.120596549 1.340373332 1.575632213 1.829542082 2.105600559
0.0292469895 0.0533702664 0.0717398294 0.0832861954 0.0864936797 0.079403451 0.059626668 0.024367787 0.029542082 0.105600559
690
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References [1] J.D. Mamedov, G. Berger, H. D€ orner, Eine Methode zur naherrungsweisen einer Randwertaufgabe einer linearen gew€ ohnlichen Differentialgichund 2, Ordnung Zeitschrift f€ ur Analysis und ihre Anwendungen, Bd. 2 (1) (1983) 11–23. [2] J. Casti, R. Kalaba, Imbedding Methods in Applied Mathematics, Adison-Wesley, London, 1973. [3] H. S ß imsßek, An approximate solution of a non-linear boundary value problem, J. Fac. Sci. Ege Univ. 20 (1997) 77–88. [4] W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes, Cambridge University Press, Cambridge, 1988. [5] A.M. Wazwaz, Analytical approximations and Pade approximations for VolterraÕs population model, App. Math. Comput. 100 (1999) 13–25. [6] Y.C. Memmedov, Approximate Methods, Atat€ urk Uni., Erzurum, 1994 (in Turkish). [7] E. C ß elik, M. Bayram, On numerical solution of differential-algebraic equations by Pade series, App. Math. Comput., in press.