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Numerical solution of the general form linear Fredholm–Volterra integro-differential equations by the Tau method with an error estimation
Applied Mathematics and Computation 167 (2005) 1418–1429
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Numerical solution of the general form linear Fredholm–Volterra integrodifferential equations by the Tau method with an error estimation S. Shahmorad Department of Mathematical Science, Tabriz University, Tabriz, Iran
Abstract The Tau method, produces approximate polynomial solutions of differential, integral and integro-differential equations. In this paper extension of the Tau method has been done for the numerical solution of the general form of linear Fredholm–Volterra integro-differential equations. An efficient error estimation for the Tau method is also introduced. Details of the method are presented and some numerical results along with estimated errors are given to clarify the method and its error estimator. 2004 Elsevier Inc. All rights reserved. Keywords: Tau method; Fredholm and Volterra integral and integro-differential equations
1. Introduction In 1981, Ortiz and Samara [1] proposed an operational technique for the numerical solution of nonlinear ordinary differential equations with some
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S. Shahmorad / Appl. Math. Comput. 167 (2005) 1418–1429
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supplementary conditions based on the Tau method [2]. During the recent years considerable work has been done both in the development of the technique, its theoretical analysis and numerical applications. The same technique has been described in a series of papers [3–7] for the case of linear ordinary differential eigenvalue problems, in [8–12] for the case of partial differential equations and their related eigenvalue problems and in [13] for the iterated solutions of linear operator equations. The object of this paper is to present developments of the operational approach to the Tau method for the numerical solution of linear Fredholm–Volterra integro-differential equations of the second kind together with mixed supplementary conditions. Special cases of this method solves Fredholm and Volterra integral and integro-differential equations separately. An other special case of this method solves differential equations with mixed supplementary conditions. In [15] Yalcinbas and Sezer, have introduced the Taylor series approximation for the solution of such problems which is a particular case of the Tau method. In this paper, we consider Ortiz and SamaraÕs operational approach to the Tau method for differential part of the equation, which leads to algorithms of remarkable simplicity, while retaining the accuracy of results. The organization of this paper is as follows: In Section 2, we introduce the considered problem. In part (a) of this section we recall the Tau method to obtain a matrix form of differential part. In parts (b), (c) and (d), converting other parts of equation to a matrix form is shown. At the end this section corresponding system of linear algebraic equations is given. In Section 3 we recall efficient Tau error estimator. In Section 4, some numerical results are given to clarify the method, where we have computed the numerical results by Maple programming and finally Section 5 contains conclusions. Remark 1.1. It should be noted that existence and uniqueness of solution of equations is not investigated, in this paper.
2. Fredholm–Volterra integro-differential equations Consider the following Fredholm–Volterra integro-differential equation together with the given mixed supplementary conditions: DyðsÞ k1
Z
b
K 1 ðs; tÞyðtÞ dt k2
a
Z
s
K 2 ðs; tÞyðtÞ dt ¼ f ðsÞ;
s 2 ½a; b;
a
ð2:1Þ nd X k¼1
ð1Þ
ð2Þ
ð3Þ
½cjk y ðk1Þ ðaÞ þ cjk y ðk1Þ ðbÞ þ cjk y ðk1Þ ðcÞ ¼ d j ;
j ¼ 1; . . . ; nd ;
ð2:2Þ
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S. Shahmorad / Appl. Math. Comput. 167 (2005) 1418–1429
a < c < b; where D is a linear differential operator of order nd with polynomial coefficients pi(s) of degree ai: D¼
nd X
pi ðsÞ
i¼0
di ; dsi
pi ðsÞ ¼
ai X
pij sj :
ð2:3Þ
j¼0
If f(s) and Ki(s,t), (i = 1, 2) in (2.1) are not polynomials, they can be approximated by polynomials to any degree of accuracy (by interpolation or Taylor series or other suitable methods). Unless otherwise stated, s will always be the independent variable of the functions which appear throughout this paper and will be defined in a finite interval. Moreover suppose that yn(s) be the Tau method approximation of degree n for y(s), so we can write: pi ðsÞ ¼
ai X
pij sj ¼ pi s;
ð2:4Þ
fj sj ¼ f s;
ð2:5Þ
j¼0
f ðsÞ ¼
n X j¼0
K i ðs; tÞ ¼
n X n X
p q k ðiÞ pq s t ;
i ¼ 1; 2;
ð2:6Þ
p¼0 q¼0
y n ðsÞ ¼
n X
aj sj ¼ an s;
ð2:7Þ
j¼0
where pi = [pi0, . . . , pi,ai, 0, 0, 0, . . .], f = [f0, . . . , fn, 0, 0, 0, . . .], an = [a0, . . . ,an, 0, 0, 0, . . .] and s = [1, s, s2, . . .]T are respectively coefficients vectors of pi(s), right-hand side of Eq. (2.1), unknown coefficients vector and the basis vector. Without lossÕof generality we have taken all polynomials of degree n, because if f(s), K1(s, t), K2(s, t), and yn(s) are respectively of different degrees nf, (n1s, n1t), (n2s,n2t) and ny then we can set n ¼ maxfnf ; n1s ; n1t ; n2s ; n2t ; ny g: 2.1. Matrix representation for different parts of (2.1),(2.2) (a) Matrix representation for Dy(s) The effect of differentiation or shifting (multiplication by the current variable s) on the coefficients an = [a0, a1, . . . ,an, 0, 0, 0, . . .] of a polynomial yn(s) = ans is the same as that of post-multiplication of an by either the matrix g or the matrix l:
S. Shahmorad / Appl. Math. Comput. 167 (2005) 1418–1429
d y ðsÞ ¼ an gs; ds n
and
1421
sy n ðsÞ ¼ an ls
where 2
0
6 61 6 g¼6 6 4
0 2 ::
3
2
7 7 7 7; 07 5 ::
6 6 6 l¼6 6 4
0
3
1 0
1 0 ::
7 7 7 7: 17 5 ::
Lemma 2.1. The effect of r repeated differentiations or q shifts on the coefficients of a polynomial yn(s) is equivalent to the post-multiplication of an respectively by gr or lq. The proof follows immediately by induction. Theorem 2.2. If the operator D and the polynomial yn(s) are of the form (2.3) and (2.7) then Dyn(s) = anPs where P¼
nd X
gi pi ðlÞ
ð2:8Þ
i¼0
see [1]. (b) Matrix representation for the Fredholm integral term It can be seen that Z b K 1 ðs; tÞy n ðtÞ dt ¼ an Kf s; a
where
2 Pn
ð1Þ q¼0 k 0q vqþ1
6 6 .. 6 . 6 Kf ¼ 6 Pn ð1Þ 6 6 q¼0 k 0q vqþnþ1 4 .. .
.. .
Pn
ð1Þ q¼0 k n;q vqþ1
Pn
.. .
.. .
ð1Þ q¼0 k n;q vqþnþ1
.. .
0 .. . 0 .. .
3
7 7 7 7 7 7 7 5
and so Kf lm ¼
n X i¼0
ð1Þ
k mi viþlþ1 ;
viþlþ1 ¼
biþlþ1 aiþlþ1 ; iþlþ1
l; m ¼ 0; . . . ; n:
ð2:9Þ
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S. Shahmorad / Appl. Math. Comput. 167 (2005) 1418–1429
(c) Matrix representation for the Volterra integral term Here the columns of the matrix Kv associated with Volterra integral term are computed as following: 3 2 P n ð2Þ 1 k 0q aqþ1 qþ1 7 6 q¼0 7 6 7 6 P n ð2Þ 6 1 ð2:10Þ ðKvÞ1 ¼ 6 k aqþ2 7 7 qþ2 0q 7 6 q¼0 5 4 .. . and, for m = 2, 3, 2 P
ð2Þ m2 1 q¼0 qþ1 k mq2;q
n P
ð2Þ 1 k aqþ1 qþ1 m1;q
3
7 6 q¼0 7 6 7 6P n P 6 m2 1 ð2Þ ð2Þ 1 qþ2 7 k a 7 6 q¼1 qþ1 k mq2;q1 qþ2 m1;q 7 6 q¼0 7 6 7 6 .. 7 6 . 7 6 ðKvÞm ¼ 6 7: n P 1 ð2Þ 7 6 2 1 qþm1 k k a 7 6 m1 00 qþm1 m1;q 7 6 q¼0 7 6 7 6 n P ð2Þ 7 6 1 qþm k a 7 6 qþm m1;q 7 6 q¼0 5 4 .. .
ð2:11Þ
If a = 0 then the columns of Kv are converted to the following simple forms: 3 2 Pm2 ð2Þ 1 q¼0 qþ1 k mq2;q 7 6P 7 6 m2 1 k ð2Þ 6 q¼1 qþ1 mq2;q1 7 7 6 2 3 7 6 .. 0 7 6 . 7 6 6 7 7; m ¼ 2; 3; . . . : ðKvÞ1 ¼ 4 0 5; ðKvÞm ¼ 6 ð2Þ 1 7 6 k m1 00 .. 7 6 7 6 . 0 7 6 7 6 7 6 0 5 4 .. . ð2:12Þ (d) Matrix representation for the supplementary conditions Replacing yn(s) from (2.7) into left-hand side of (2.2) we have: nd X k¼1
ð1Þ
ð2Þ
ð3Þ
½cjk y ðk1Þ ðaÞ þ cjk y ðk1Þ ðbÞ þ cjk y nðk1Þ ðcÞ ¼ an Bj ; n n
S. Shahmorad / Appl. Math. Comput. 167 (2005) 1418–1429
where for j = 1, . . . ,nd, 2
Bj ¼
1423
3
ð2Þ ð3Þ 0! ð1Þ ½c þ cj1 þ cj1 0! j1 6 7 ð2Þ ð3Þ ð1Þ ð2Þ ð3Þ 1! ð1Þ 6 7 ½c a þ cj1 b þ cj1 c þ 1! ½c þ cj2 þ cj2 1! j1 0! j2 6 7 6 ðnd 1Þ! ð1Þ nd 1 7 ð2Þ nd 1 ð3Þ nd 1 ð2Þ ð3Þ ðnd 1Þ! ð1Þ 6 ðnd 1Þ! ½cj1 a þ cj1 b þ cj1 c þ þ 0! ½cj;nd 1 þ cj;nd 1 þ cj;nd 1 7: 6 7 ð2Þ nd ð3Þ nd ð2Þ ð3Þ 6 7 ðnd Þ! ð1Þ nd ðnd Þ! ð1Þ ½c a þ c b þ c c þ þ ½c a þ c b þ c c 6 7 j;n j;n j;n j1 j1 j1 ðnd Þ! 1! d d d
4
5
.. .
ð2:13Þ
We refer to B as the matrix representation of the supplementary conditions and Bj as its jth column. The following relations for computing the elements of the matrix B can be deduced from (2.13): bij ¼
i X ði 1Þ! ð1Þ ðikÞ ð2Þ ð3Þ ½cjk a þ cjk bðikÞ þ cjk cðikÞ ; ði kÞ! k¼1
bij ¼
i X ði 1Þ! ð1Þ ðikÞ ð2Þ ð3Þ ½cjk a þ cjk bðikÞ þ cjk cðikÞ ; ði kÞ! k¼1
i; j ¼ 1; . . . ; nd :
and
i ¼ nd þ 1; nd þ 2; . . . ; j ¼ 1; . . . ; nd : Hence (2.2) can be written as an B ¼ d;
ð2:14Þ
where d = [d1, . . . ,dnd] is the vector obtained from the right-hand side of (2.2). Consequently, using theorem (2.2) and the results of (b), (c) and (d) parts, we can write the following system of linear equations instead of (2.1) and (2.2): an ðP k1 Kf k2 KvÞ ¼ f ; ð2:15Þ aB ¼ d: Now setting ^ ¼ P k1 Kf k2 Kv P
ð2:16Þ
^ 1; . . . ; P ^ nþ1n Gn ¼ ½B1 ; . . . ; Bnd ; P d
ð2:17Þ
gn ¼ ½d 1 ; . . . ; d nd ; f0 ; . . . ; fnnd ;
ð2:18Þ
and ^ the system of Eq. (2.15) then can be ^ i denotes the ith column of P), (where P written as an Gn ¼ gn ; which must be solved for the unknown coefficients a0, . . . ,an.
ð2:19Þ
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S. Shahmorad / Appl. Math. Comput. 167 (2005) 1418–1429
Remark 2.3. For k1 = 0 Eq. (2.1) is transformed into a Fredholm integrodifferential equation and k2 = 0, it is transformed into a Volterra integrodifferential equation. For k1 = k2 = 0 the equation is transformed into a differential equation. For nd = 0 and p0(s) = 1, it is transformed into a, Fredholm–Volterra integral equation. 3. Error estimation of the Tau method In this section an error estimator for the Tau approximate solution of a Fredholm–Volterra integro-differential equation is obtained. Let us call en(s) = y(s) yn(s) as the error function of the Tau approximation yn(s) to y(s), where y(s) is the exact solution of (2.1) and (2.2). Hence, yn(s) satisfies the following problem: Z b Z s Dy n ðsÞ k1 K 1 ðs; tÞy n ðtÞ dt k2 K 2 ðs; tÞy n ðtÞ dt a
a
¼ f ðsÞ þ H n ðsÞ; nd X
ð1Þ
s 2 ½a; b; ð2Þ
ð3:1Þ ð3Þ
½cjk y ðk1Þ ðaÞ þ cjk y ðk1Þ ðbÞ þ cjk y nðk1Þ ðcÞ ¼ d j ; n n
j ¼ 1; . . . ; nd ;
ð3:2Þ
k¼1
a < c < b; where Hn(s) is a perturbation term associated with yn(s) and can be obtained by substituting yn(s) into the equation Z b Z s H n ðsÞ ¼ Dy n ðsÞ k1 K 1 ðs; tÞy n ðtÞ dt k2 K 2 ðs; tÞy n ðtÞ dt f ðsÞ: a
a
ð3:3Þ We proceed to find an approximation en,N(s) to the en(s) in the same way as we did before for the solution (2.1) and (2.2). Subtracting (3.1) and (3.2) from (2.1) and (2.2), respectively, the error function en(s) satisfies the equation Den ðsÞ k1
Z
b
K 1 ðs; tÞen ðtÞ dt k2
Z
a
¼ H n ðsÞ;
s
K 2 ðs; tÞen ðtÞ dt
a
s 2 ½a; b
ð3:4Þ
with the homogeneous conditions nd X k¼1
ð1Þ
ð2Þ
ð3Þ
½cjk enðk1Þ ðaÞ þ cjk enðk1Þ ðbÞ þ cjk eðk1Þ ðcÞ ¼ 0 n
j ¼ 1; . . . ; nd :
ð3:5Þ
S. Shahmorad / Appl. Math. Comput. 167 (2005) 1418–1429
1425
Solving this problem in the same way as Section 2, we get the approximation en;N ðsÞ. It should be noted that in order to construct the Tau approximation en;N ðsÞ to en(s), only the right-hand side of (2.19) needs to be recomputed, the structure of the coefficient matrix Gn remains the same.
4. Numerical examples In this section, we report on numerical results of some examples, selected through integral and integro-differential equations, solved by the Tau method described in this paper. We calculated with 15 and 20 digits of accuracy with Maple programming. For Examples 2–4, we have reported, in Tables 1–3, the values of exact solution y(s), Tau approximate solution yn(s), absolute error jy(s)yn(s)j and estimation error (denoted by exact, Tau, Tau-err and Esti.-err, respectively) at selected points of the given interval. Remark 4.1. It should be noted that, for Examples 2 and 4 we have used a two variate Taylor expansion to approximate their kernels.
Table 1 Numerical results of Example 2 S
Exact
Tau
Tau-err
Esti.-err
n = 10 0.00 0.20 0.40 0.60 0.80 1.00
1 1.04081077 1.17351087 1.43332941 1.89648088 2.71828183
1 1.04081077 1.17351085 1.43332623 1.89637596 2.71666667
0 5.72156e12 2.38451e08 3.18608e06 1.04921e04 1.61516e03
0 5.68889e12 2.33017e08 3.02331e06 9.54437e05 1.38889e03
n = 15 0.00 0.20 0.40 0.60 0.80 1.00
1 1.04081077 1.17351087 1.43332941 1.89648088 2.71828183
1 1.04081077 1.17351087 1.43332941 1.89648013 2.71825397
0 1.63300e16 1.08446e11 7.28709e09 7.51118e07 2.78602e05
0 1.62540e16 1.06522e11 6.99680e09 6.98103e07 2.48016e05
n = 20 0.00 0.20 0.40 0.60 0.80 1.00
1 1.04081077 1.17351087 1.43332941 1.89648088 2.71828183
1 1.04081077 1.17351087 1.43332941 1.89648088 2.71828180
0 1.00000e19 4.46000e17 3.39914e13 1.95219e10 2.73127e08
0 3.60219e22 4.40759e17 3.29740e13 1.84852e10 2.50521e08
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S. Shahmorad / Appl. Math. Comput. 167 (2005) 1418–1429
Table 2 Numerical results of Example 3 S
Exact
Tau
Tau-err
Esti.-err
n=5 0.00 0.20 0.40 0.60 0.80 1.00
0 0.19866933 0.38941834 0.56464247 0.71735609 0.84147098
0 0.19866933 0.38941867 0.56464800 0.71739733 0.84166667
0 2.60e09 3.24e07 5.53e06 4.12e05 1.96e04
0 2.54e09 3.25e07 5.55e06 4.16e05 1.98e04
n = 10 0.00 0.20 0.40 0.60 0.80 1.00
0 0.19866933 0.38941834 0.56464247 0.71735609 0.84147098
0 0.19866933 0.38941834 0.56464247 0.71735609 0.84147101
0 1.00e10 1.00e10 1.00e10 2.20e09 2.49e08
0 5.33e13 5.28e12 1.05e10 2.18e09 2.51e08
n = 15 0.00 0.20 0.40 0.60 0.80 1.00
0 0.19866933 0.38941834 0.56464247 0.71735609 0.84147098
0 0.19866933 0.38941834 0.56464247 0.71735609 0.84147098
0 1.00e10 1.00e10 0 0 0
0 5.32e13 4.23e12 1.41e11 3.30e11 6.33e11
Z
Z
Example 1 [15, Example 1]. sy 00 ðsÞ sy 0 ðsÞ þ 2yðsÞ
1
ðs þ tÞyðtÞ dt 0
¼
1 4 1 3 1 2 13 17 s s s sþ ; 12 6 2 6 12
s
ðs tÞyðtÞ dt 0
0 6 s 6 1;
yð0Þ ¼ 1; y 0 ð0Þ 2yð1Þ þ 2yð0Þ ¼ 1; where nd = 2, a = 0, b = 1, p0(s) = 2, p1(s) = s, p2(s) = s, k1 = k2 = 1, K1(s,t) = s + t, K2(s,t) = s t and f ðsÞ ¼ 121 s4 16 s3 12 s2 136 s þ 17 . Here c 12 can be taken any number between a and b. We approximate the solution y(s) by the Tau approximate y 2 ðsÞ ¼ a0 þ a1 s þ a2 s2
ð4:1Þ
with n = 2. Then using relations, stated at the parts (a), (b), (c) and (d) of Section 2, for a (n + 1) · (n + 1) matrix, we obtain the following matrices
S. Shahmorad / Appl. Math. Comput. 167 (2005) 1418–1429
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Table 3 Numerical results of Example 4 s
Exact
Tau
Tau-err
Esti.-err
n=5 1.00 0.80 0.60 0.40 0.20 0.00 0.20 0.40 0.60 0.80 1.00
0.36787944 0.44932896 0.54881164 0.67032005 0.81873075 1 1.22140276 1.49182470 1.82211880 2.22554093 2.71828183
0.36657770 0.44908424 0.54884907 0.67037659 0.81875183 1 1.22144606 1.49207429 1.82288780 2.22726812 2.72133477
1.30e03 2.45e04 3.74e05 5.65e05 2.11e05 0 4.33e05 2.50e04 7.69e04 1.73e03 3.05e03
1.28e03 2.42e04 3.78e05 5.67e05 2,12e05 0 4.37e05 2.52e04 7.82e04 1.77e03 3.19e03
n = 10 1.00 0.80 0.60 0.40 0.20 0.00 0.20 0.40 0.60 0.80 1.00
0.36787944 0.44932896 0.54881164 0.67032005 0.81873075 1 1.22140276 1.49182470 1.82211880 2.22554093 2.71828183
0.36787946 0.44932897 0.54881164 0.67032005 0.81873075 1 1.22140276 1.49182468 1.82211868 2.22554039 2.71827972
2.29e08 7.31e09 2.26e09 1.43e10 2.71e102.71e10 0 2e09 2.02e08 1.18e07 5.41e07 2.11e06
2.27e08 7.28e09 2.25e09 1.39e10 0 1.99e09 2.02e08 1.18e07 5.39e07 2.10e06
n = 15 1.00 0.80 0.60 0.40 0.20 0.00 0.20 0.40 0.60 0.80 1.00
0.36787944 0.44932896 0.54881164 0.67032005 0.81873075 1 1.22140276 1.49182470 1.82211880 2.22554093 2.71828183
0.36787944 0.44932896 0.54881164 0.67032005 0.81873075 1 1.22140276 1.49182470 1.82211880 2.22554093 2.71828183
3.72e13 2.23e13 1.00e13 3.20e14 3.00e15 0 0 1.70e13 1.51e12 1.17e11 7.52e11
3.67e13 2.19e13 9.75e14 2.90e14 3.06e15 0 1.10e14 1.60e13 1.51e12 1.17e11 7.50e11
2
2
6 P ¼ 40 0 2
1 4 B¼ 0 0
0 0
7 1 0 5; 2 0 3 0 1 5 2
21
3 Kf ¼
2 61 43 1 4
1 2 1 2 1 3
0
3
7 0 5; 0
2
0 6 Kv ¼ 4 0 0
3 0 12 7 0 0 5; 0 0
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S. Shahmorad / Appl. Math. Comput. 167 (2005) 1418–1429
with f ¼ ½ f0
f1
f2 ¼
d ¼ ½ d1
d2 ¼ ½ 1
17
136
12
12 ;
1 :
Now using relations (2.16)–(2.18), we find the system of Eq. (2.19) as follows: 2 3 3 1 0 2 6 7 ½ a0 a1 a2 4 0 1 13 5 ¼ 1; 1; 17 12 0 2 14 and its solution a2 ¼ ½ a0
a1
a2 ¼ ½ 1
1
1 :
Substituting the elements of this vector into (4.1) we obtain the solution y 2 ðsÞ ¼ 1 þ s s2 which is the exact solution.
Example 2 [14, Example (b)]. Z s 0 y ðsÞ ¼ 1 þ 2s yðsÞ þ sð1 þ 2sÞetðstÞ yðtÞ dt;
yð0Þ ¼ 1:
06s61
0 2
The exact solution is y(s) = es . For numerical results see Table 1. Example 3 [15, Example 3]. Z s y 0 ðsÞ þ yðtÞ dt ¼ 1; 0 6 s 6 1 0
yð0Þ ¼ 0: The exact solution is y(s) = sin(s). For numerical results see Table 2. Example 4 [15, Example 4]. y 00 ðsÞ þ sy 0 ðsÞ syðsÞ ¼ es 2 sinðsÞ þ
Z
1
sinðsÞet yðtÞ dt;
1 6 s 6 1
1
yð0Þ ¼ 1;
y 0 ð0Þ ¼ 1:
The exact solution is y(s) = es. For numerical results see Table 3.
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5. Conclusions Most integro-differential equations are usually difficult to solve analytically. In many cases, it is required to obtain the approximate solutions. For this purpose, the Tau method presented in this paper can be applied. Advantages of the method are the solution is expressed as a polynomial, the error estimator is available and this method is an exceedingly good approximating method in the sense that its error for problems with reasonable solutions, decays exponentially as the degree of approximation increases [16]. References [1] E.L. Ortiz, H. Samara, An operational approach to the Tau method for the numerical solution of nonlinear differential equations, Computing 27 (1981) 15–25. [2] E.L. Ortiz, The Tau method, Siam. J. Numer. Anal. 6 (1969) 480–492. [3] K.M. Liu, E.L. Ortiz, Eigenvalue problems for singularly perturbed differential equations, in: J.J.H. Miller (Ed.), Proceedings of the BAIL II Conference, Boole Press, Dublen, 1982, pp. 324–329. [4] K.M. Liu, E.L. Ortiz, Approximation of eigenvalues defined by ordinary differential equations with the Tau method, in: B. Kagestrm, A. Ruhe (Eds.), Matrix Pencils, Springer-Verlag, Berlin, 1983, pp. 90–102. [5] K.M. Liu, E.L. Ortiz, Tau method approximation of differential eigenvalue problems where the spectral parameter enters nonlinearly, J. Comput. Phys. 72 (1987) 299–310. [6] K.M. Liu, E.L. Ortiz, Numerical solution of ordinary and partial function-differential eigenvalue problems with the Tau method, Computing (wien) 41 (1989) 205–217. [7] E.L. Ortiz, H. Samara, Numerical solution of differential eigenvalue problems with an operational approach to the Tau method, Computing 31 (1983) 95–103. [8] K.M. Liu, E.L. Ortiz, Numerical solution of eigenvalue problems for partial differential equations with the Tau-lines Method, Comput. Math. Appl. 12B (5/6) (1986) 1153–1168. [9] K.M. Liu, E.L. Ortiz, K.S. Pun, Numerical solution of SteklovÕs partial differential equation eigenvalue problem, in: J.J.H. Miller (Ed.), Computational and asymptotic methods for boundary and interior layers (III), Boole Press, Dublin, 1984, pp. 244–249. [10] E.L. Ortiz, K.S. Pun, Numerical solution of nonlinear partial differential equations with Tau method, J. Comp. Appl. Math. 12/13 (1985) 511–516. [11] E.L. Ortiz, K.S. Pun, A bi-dimensional Tail-Elements method for the numerical solution of nonlinear partial differential equations with an application to BurgersÕ equation, Comput. Math. Appl. 12B (5/6) (1986) 1225–1240. [12] E.L. Ortiz, H. Samara, Numerical solution of partial differential equations with variable coefficients with an operational approach to the Tau method, Comput. Math. Appl. 10 (1) (1984) 5–13. [13] M.K. EL-Daou, H.G. Khajah, Iterated solutions of linear operator equations with the Tau method, Math. Comput. 66 (217) (1997) 207–213. [14] A. Makroglou, Convergence of a block-by-block method for nonlinear Volterra integrodifferential equations, Math. Comp. 35 (1980) 783–796. [15] S. Yalcinbas, M. Sezer, The approximate solution of high-order linear Volterra–Fredholm integro-differential equations in term of Taylor polynomials, Appl. Math. Comput. 112 (2000) 291–308. [16] D. Gottlieb, S.A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, SIAM, Philadelphia, PA, 1986.
www.elsevier.com/locate/amc
Numerical solution of the general form linear Fredholm–Volterra integrodifferential equations by the Tau method with an error estimation S. Shahmorad Department of Mathematical Science, Tabriz University, Tabriz, Iran
Abstract The Tau method, produces approximate polynomial solutions of differential, integral and integro-differential equations. In this paper extension of the Tau method has been done for the numerical solution of the general form of linear Fredholm–Volterra integro-differential equations. An efficient error estimation for the Tau method is also introduced. Details of the method are presented and some numerical results along with estimated errors are given to clarify the method and its error estimator. 2004 Elsevier Inc. All rights reserved. Keywords: Tau method; Fredholm and Volterra integral and integro-differential equations
1. Introduction In 1981, Ortiz and Samara [1] proposed an operational technique for the numerical solution of nonlinear ordinary differential equations with some
E-mail address: [email protected] 0096-3003/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.08.045
S. Shahmorad / Appl. Math. Comput. 167 (2005) 1418–1429
1419
supplementary conditions based on the Tau method [2]. During the recent years considerable work has been done both in the development of the technique, its theoretical analysis and numerical applications. The same technique has been described in a series of papers [3–7] for the case of linear ordinary differential eigenvalue problems, in [8–12] for the case of partial differential equations and their related eigenvalue problems and in [13] for the iterated solutions of linear operator equations. The object of this paper is to present developments of the operational approach to the Tau method for the numerical solution of linear Fredholm–Volterra integro-differential equations of the second kind together with mixed supplementary conditions. Special cases of this method solves Fredholm and Volterra integral and integro-differential equations separately. An other special case of this method solves differential equations with mixed supplementary conditions. In [15] Yalcinbas and Sezer, have introduced the Taylor series approximation for the solution of such problems which is a particular case of the Tau method. In this paper, we consider Ortiz and SamaraÕs operational approach to the Tau method for differential part of the equation, which leads to algorithms of remarkable simplicity, while retaining the accuracy of results. The organization of this paper is as follows: In Section 2, we introduce the considered problem. In part (a) of this section we recall the Tau method to obtain a matrix form of differential part. In parts (b), (c) and (d), converting other parts of equation to a matrix form is shown. At the end this section corresponding system of linear algebraic equations is given. In Section 3 we recall efficient Tau error estimator. In Section 4, some numerical results are given to clarify the method, where we have computed the numerical results by Maple programming and finally Section 5 contains conclusions. Remark 1.1. It should be noted that existence and uniqueness of solution of equations is not investigated, in this paper.
2. Fredholm–Volterra integro-differential equations Consider the following Fredholm–Volterra integro-differential equation together with the given mixed supplementary conditions: DyðsÞ k1
Z
b
K 1 ðs; tÞyðtÞ dt k2
a
Z
s
K 2 ðs; tÞyðtÞ dt ¼ f ðsÞ;
s 2 ½a; b;
a
ð2:1Þ nd X k¼1
ð1Þ
ð2Þ
ð3Þ
½cjk y ðk1Þ ðaÞ þ cjk y ðk1Þ ðbÞ þ cjk y ðk1Þ ðcÞ ¼ d j ;
j ¼ 1; . . . ; nd ;
ð2:2Þ
1420
S. Shahmorad / Appl. Math. Comput. 167 (2005) 1418–1429
a < c < b; where D is a linear differential operator of order nd with polynomial coefficients pi(s) of degree ai: D¼
nd X
pi ðsÞ
i¼0
di ; dsi
pi ðsÞ ¼
ai X
pij sj :
ð2:3Þ
j¼0
If f(s) and Ki(s,t), (i = 1, 2) in (2.1) are not polynomials, they can be approximated by polynomials to any degree of accuracy (by interpolation or Taylor series or other suitable methods). Unless otherwise stated, s will always be the independent variable of the functions which appear throughout this paper and will be defined in a finite interval. Moreover suppose that yn(s) be the Tau method approximation of degree n for y(s), so we can write: pi ðsÞ ¼
ai X
pij sj ¼ pi s;
ð2:4Þ
fj sj ¼ f s;
ð2:5Þ
j¼0
f ðsÞ ¼
n X j¼0
K i ðs; tÞ ¼
n X n X
p q k ðiÞ pq s t ;
i ¼ 1; 2;
ð2:6Þ
p¼0 q¼0
y n ðsÞ ¼
n X
aj sj ¼ an s;
ð2:7Þ
j¼0
where pi = [pi0, . . . , pi,ai, 0, 0, 0, . . .], f = [f0, . . . , fn, 0, 0, 0, . . .], an = [a0, . . . ,an, 0, 0, 0, . . .] and s = [1, s, s2, . . .]T are respectively coefficients vectors of pi(s), right-hand side of Eq. (2.1), unknown coefficients vector and the basis vector. Without lossÕof generality we have taken all polynomials of degree n, because if f(s), K1(s, t), K2(s, t), and yn(s) are respectively of different degrees nf, (n1s, n1t), (n2s,n2t) and ny then we can set n ¼ maxfnf ; n1s ; n1t ; n2s ; n2t ; ny g: 2.1. Matrix representation for different parts of (2.1),(2.2) (a) Matrix representation for Dy(s) The effect of differentiation or shifting (multiplication by the current variable s) on the coefficients an = [a0, a1, . . . ,an, 0, 0, 0, . . .] of a polynomial yn(s) = ans is the same as that of post-multiplication of an by either the matrix g or the matrix l:
S. Shahmorad / Appl. Math. Comput. 167 (2005) 1418–1429
d y ðsÞ ¼ an gs; ds n
and
1421
sy n ðsÞ ¼ an ls
where 2
0
6 61 6 g¼6 6 4
0 2 ::
3
2
7 7 7 7; 07 5 ::
6 6 6 l¼6 6 4
0
3
1 0
1 0 ::
7 7 7 7: 17 5 ::
Lemma 2.1. The effect of r repeated differentiations or q shifts on the coefficients of a polynomial yn(s) is equivalent to the post-multiplication of an respectively by gr or lq. The proof follows immediately by induction. Theorem 2.2. If the operator D and the polynomial yn(s) are of the form (2.3) and (2.7) then Dyn(s) = anPs where P¼
nd X
gi pi ðlÞ
ð2:8Þ
i¼0
see [1]. (b) Matrix representation for the Fredholm integral term It can be seen that Z b K 1 ðs; tÞy n ðtÞ dt ¼ an Kf s; a
where
2 Pn
ð1Þ q¼0 k 0q vqþ1
6 6 .. 6 . 6 Kf ¼ 6 Pn ð1Þ 6 6 q¼0 k 0q vqþnþ1 4 .. .
.. .
Pn
ð1Þ q¼0 k n;q vqþ1
Pn
.. .
.. .
ð1Þ q¼0 k n;q vqþnþ1
.. .
0 .. . 0 .. .
3
7 7 7 7 7 7 7 5
and so Kf lm ¼
n X i¼0
ð1Þ
k mi viþlþ1 ;
viþlþ1 ¼
biþlþ1 aiþlþ1 ; iþlþ1
l; m ¼ 0; . . . ; n:
ð2:9Þ
1422
S. Shahmorad / Appl. Math. Comput. 167 (2005) 1418–1429
(c) Matrix representation for the Volterra integral term Here the columns of the matrix Kv associated with Volterra integral term are computed as following: 3 2 P n ð2Þ 1 k 0q aqþ1 qþ1 7 6 q¼0 7 6 7 6 P n ð2Þ 6 1 ð2:10Þ ðKvÞ1 ¼ 6 k aqþ2 7 7 qþ2 0q 7 6 q¼0 5 4 .. . and, for m = 2, 3, 2 P
ð2Þ m2 1 q¼0 qþ1 k mq2;q
n P
ð2Þ 1 k aqþ1 qþ1 m1;q
3
7 6 q¼0 7 6 7 6P n P 6 m2 1 ð2Þ ð2Þ 1 qþ2 7 k a 7 6 q¼1 qþ1 k mq2;q1 qþ2 m1;q 7 6 q¼0 7 6 7 6 .. 7 6 . 7 6 ðKvÞm ¼ 6 7: n P 1 ð2Þ 7 6 2 1 qþm1 k k a 7 6 m1 00 qþm1 m1;q 7 6 q¼0 7 6 7 6 n P ð2Þ 7 6 1 qþm k a 7 6 qþm m1;q 7 6 q¼0 5 4 .. .
ð2:11Þ
If a = 0 then the columns of Kv are converted to the following simple forms: 3 2 Pm2 ð2Þ 1 q¼0 qþ1 k mq2;q 7 6P 7 6 m2 1 k ð2Þ 6 q¼1 qþ1 mq2;q1 7 7 6 2 3 7 6 .. 0 7 6 . 7 6 6 7 7; m ¼ 2; 3; . . . : ðKvÞ1 ¼ 4 0 5; ðKvÞm ¼ 6 ð2Þ 1 7 6 k m1 00 .. 7 6 7 6 . 0 7 6 7 6 7 6 0 5 4 .. . ð2:12Þ (d) Matrix representation for the supplementary conditions Replacing yn(s) from (2.7) into left-hand side of (2.2) we have: nd X k¼1
ð1Þ
ð2Þ
ð3Þ
½cjk y ðk1Þ ðaÞ þ cjk y ðk1Þ ðbÞ þ cjk y nðk1Þ ðcÞ ¼ an Bj ; n n
S. Shahmorad / Appl. Math. Comput. 167 (2005) 1418–1429
where for j = 1, . . . ,nd, 2
Bj ¼
1423
3
ð2Þ ð3Þ 0! ð1Þ ½c þ cj1 þ cj1 0! j1 6 7 ð2Þ ð3Þ ð1Þ ð2Þ ð3Þ 1! ð1Þ 6 7 ½c a þ cj1 b þ cj1 c þ 1! ½c þ cj2 þ cj2 1! j1 0! j2 6 7 6 ðnd 1Þ! ð1Þ nd 1 7 ð2Þ nd 1 ð3Þ nd 1 ð2Þ ð3Þ ðnd 1Þ! ð1Þ 6 ðnd 1Þ! ½cj1 a þ cj1 b þ cj1 c þ þ 0! ½cj;nd 1 þ cj;nd 1 þ cj;nd 1 7: 6 7 ð2Þ nd ð3Þ nd ð2Þ ð3Þ 6 7 ðnd Þ! ð1Þ nd ðnd Þ! ð1Þ ½c a þ c b þ c c þ þ ½c a þ c b þ c c 6 7 j;n j;n j;n j1 j1 j1 ðnd Þ! 1! d d d
4
5
.. .
ð2:13Þ
We refer to B as the matrix representation of the supplementary conditions and Bj as its jth column. The following relations for computing the elements of the matrix B can be deduced from (2.13): bij ¼
i X ði 1Þ! ð1Þ ðikÞ ð2Þ ð3Þ ½cjk a þ cjk bðikÞ þ cjk cðikÞ ; ði kÞ! k¼1
bij ¼
i X ði 1Þ! ð1Þ ðikÞ ð2Þ ð3Þ ½cjk a þ cjk bðikÞ þ cjk cðikÞ ; ði kÞ! k¼1
i; j ¼ 1; . . . ; nd :
and
i ¼ nd þ 1; nd þ 2; . . . ; j ¼ 1; . . . ; nd : Hence (2.2) can be written as an B ¼ d;
ð2:14Þ
where d = [d1, . . . ,dnd] is the vector obtained from the right-hand side of (2.2). Consequently, using theorem (2.2) and the results of (b), (c) and (d) parts, we can write the following system of linear equations instead of (2.1) and (2.2): an ðP k1 Kf k2 KvÞ ¼ f ; ð2:15Þ aB ¼ d: Now setting ^ ¼ P k1 Kf k2 Kv P
ð2:16Þ
^ 1; . . . ; P ^ nþ1n Gn ¼ ½B1 ; . . . ; Bnd ; P d
ð2:17Þ
gn ¼ ½d 1 ; . . . ; d nd ; f0 ; . . . ; fnnd ;
ð2:18Þ
and ^ the system of Eq. (2.15) then can be ^ i denotes the ith column of P), (where P written as an Gn ¼ gn ; which must be solved for the unknown coefficients a0, . . . ,an.
ð2:19Þ
1424
S. Shahmorad / Appl. Math. Comput. 167 (2005) 1418–1429
Remark 2.3. For k1 = 0 Eq. (2.1) is transformed into a Fredholm integrodifferential equation and k2 = 0, it is transformed into a Volterra integrodifferential equation. For k1 = k2 = 0 the equation is transformed into a differential equation. For nd = 0 and p0(s) = 1, it is transformed into a, Fredholm–Volterra integral equation. 3. Error estimation of the Tau method In this section an error estimator for the Tau approximate solution of a Fredholm–Volterra integro-differential equation is obtained. Let us call en(s) = y(s) yn(s) as the error function of the Tau approximation yn(s) to y(s), where y(s) is the exact solution of (2.1) and (2.2). Hence, yn(s) satisfies the following problem: Z b Z s Dy n ðsÞ k1 K 1 ðs; tÞy n ðtÞ dt k2 K 2 ðs; tÞy n ðtÞ dt a
a
¼ f ðsÞ þ H n ðsÞ; nd X
ð1Þ
s 2 ½a; b; ð2Þ
ð3:1Þ ð3Þ
½cjk y ðk1Þ ðaÞ þ cjk y ðk1Þ ðbÞ þ cjk y nðk1Þ ðcÞ ¼ d j ; n n
j ¼ 1; . . . ; nd ;
ð3:2Þ
k¼1
a < c < b; where Hn(s) is a perturbation term associated with yn(s) and can be obtained by substituting yn(s) into the equation Z b Z s H n ðsÞ ¼ Dy n ðsÞ k1 K 1 ðs; tÞy n ðtÞ dt k2 K 2 ðs; tÞy n ðtÞ dt f ðsÞ: a
a
ð3:3Þ We proceed to find an approximation en,N(s) to the en(s) in the same way as we did before for the solution (2.1) and (2.2). Subtracting (3.1) and (3.2) from (2.1) and (2.2), respectively, the error function en(s) satisfies the equation Den ðsÞ k1
Z
b
K 1 ðs; tÞen ðtÞ dt k2
Z
a
¼ H n ðsÞ;
s
K 2 ðs; tÞen ðtÞ dt
a
s 2 ½a; b
ð3:4Þ
with the homogeneous conditions nd X k¼1
ð1Þ
ð2Þ
ð3Þ
½cjk enðk1Þ ðaÞ þ cjk enðk1Þ ðbÞ þ cjk eðk1Þ ðcÞ ¼ 0 n
j ¼ 1; . . . ; nd :
ð3:5Þ
S. Shahmorad / Appl. Math. Comput. 167 (2005) 1418–1429
1425
Solving this problem in the same way as Section 2, we get the approximation en;N ðsÞ. It should be noted that in order to construct the Tau approximation en;N ðsÞ to en(s), only the right-hand side of (2.19) needs to be recomputed, the structure of the coefficient matrix Gn remains the same.
4. Numerical examples In this section, we report on numerical results of some examples, selected through integral and integro-differential equations, solved by the Tau method described in this paper. We calculated with 15 and 20 digits of accuracy with Maple programming. For Examples 2–4, we have reported, in Tables 1–3, the values of exact solution y(s), Tau approximate solution yn(s), absolute error jy(s)yn(s)j and estimation error (denoted by exact, Tau, Tau-err and Esti.-err, respectively) at selected points of the given interval. Remark 4.1. It should be noted that, for Examples 2 and 4 we have used a two variate Taylor expansion to approximate their kernels.
Table 1 Numerical results of Example 2 S
Exact
Tau
Tau-err
Esti.-err
n = 10 0.00 0.20 0.40 0.60 0.80 1.00
1 1.04081077 1.17351087 1.43332941 1.89648088 2.71828183
1 1.04081077 1.17351085 1.43332623 1.89637596 2.71666667
0 5.72156e12 2.38451e08 3.18608e06 1.04921e04 1.61516e03
0 5.68889e12 2.33017e08 3.02331e06 9.54437e05 1.38889e03
n = 15 0.00 0.20 0.40 0.60 0.80 1.00
1 1.04081077 1.17351087 1.43332941 1.89648088 2.71828183
1 1.04081077 1.17351087 1.43332941 1.89648013 2.71825397
0 1.63300e16 1.08446e11 7.28709e09 7.51118e07 2.78602e05
0 1.62540e16 1.06522e11 6.99680e09 6.98103e07 2.48016e05
n = 20 0.00 0.20 0.40 0.60 0.80 1.00
1 1.04081077 1.17351087 1.43332941 1.89648088 2.71828183
1 1.04081077 1.17351087 1.43332941 1.89648088 2.71828180
0 1.00000e19 4.46000e17 3.39914e13 1.95219e10 2.73127e08
0 3.60219e22 4.40759e17 3.29740e13 1.84852e10 2.50521e08
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S. Shahmorad / Appl. Math. Comput. 167 (2005) 1418–1429
Table 2 Numerical results of Example 3 S
Exact
Tau
Tau-err
Esti.-err
n=5 0.00 0.20 0.40 0.60 0.80 1.00
0 0.19866933 0.38941834 0.56464247 0.71735609 0.84147098
0 0.19866933 0.38941867 0.56464800 0.71739733 0.84166667
0 2.60e09 3.24e07 5.53e06 4.12e05 1.96e04
0 2.54e09 3.25e07 5.55e06 4.16e05 1.98e04
n = 10 0.00 0.20 0.40 0.60 0.80 1.00
0 0.19866933 0.38941834 0.56464247 0.71735609 0.84147098
0 0.19866933 0.38941834 0.56464247 0.71735609 0.84147101
0 1.00e10 1.00e10 1.00e10 2.20e09 2.49e08
0 5.33e13 5.28e12 1.05e10 2.18e09 2.51e08
n = 15 0.00 0.20 0.40 0.60 0.80 1.00
0 0.19866933 0.38941834 0.56464247 0.71735609 0.84147098
0 0.19866933 0.38941834 0.56464247 0.71735609 0.84147098
0 1.00e10 1.00e10 0 0 0
0 5.32e13 4.23e12 1.41e11 3.30e11 6.33e11
Z
Z
Example 1 [15, Example 1]. sy 00 ðsÞ sy 0 ðsÞ þ 2yðsÞ
1
ðs þ tÞyðtÞ dt 0
¼
1 4 1 3 1 2 13 17 s s s sþ ; 12 6 2 6 12
s
ðs tÞyðtÞ dt 0
0 6 s 6 1;
yð0Þ ¼ 1; y 0 ð0Þ 2yð1Þ þ 2yð0Þ ¼ 1; where nd = 2, a = 0, b = 1, p0(s) = 2, p1(s) = s, p2(s) = s, k1 = k2 = 1, K1(s,t) = s + t, K2(s,t) = s t and f ðsÞ ¼ 121 s4 16 s3 12 s2 136 s þ 17 . Here c 12 can be taken any number between a and b. We approximate the solution y(s) by the Tau approximate y 2 ðsÞ ¼ a0 þ a1 s þ a2 s2
ð4:1Þ
with n = 2. Then using relations, stated at the parts (a), (b), (c) and (d) of Section 2, for a (n + 1) · (n + 1) matrix, we obtain the following matrices
S. Shahmorad / Appl. Math. Comput. 167 (2005) 1418–1429
1427
Table 3 Numerical results of Example 4 s
Exact
Tau
Tau-err
Esti.-err
n=5 1.00 0.80 0.60 0.40 0.20 0.00 0.20 0.40 0.60 0.80 1.00
0.36787944 0.44932896 0.54881164 0.67032005 0.81873075 1 1.22140276 1.49182470 1.82211880 2.22554093 2.71828183
0.36657770 0.44908424 0.54884907 0.67037659 0.81875183 1 1.22144606 1.49207429 1.82288780 2.22726812 2.72133477
1.30e03 2.45e04 3.74e05 5.65e05 2.11e05 0 4.33e05 2.50e04 7.69e04 1.73e03 3.05e03
1.28e03 2.42e04 3.78e05 5.67e05 2,12e05 0 4.37e05 2.52e04 7.82e04 1.77e03 3.19e03
n = 10 1.00 0.80 0.60 0.40 0.20 0.00 0.20 0.40 0.60 0.80 1.00
0.36787944 0.44932896 0.54881164 0.67032005 0.81873075 1 1.22140276 1.49182470 1.82211880 2.22554093 2.71828183
0.36787946 0.44932897 0.54881164 0.67032005 0.81873075 1 1.22140276 1.49182468 1.82211868 2.22554039 2.71827972
2.29e08 7.31e09 2.26e09 1.43e10 2.71e102.71e10 0 2e09 2.02e08 1.18e07 5.41e07 2.11e06
2.27e08 7.28e09 2.25e09 1.39e10 0 1.99e09 2.02e08 1.18e07 5.39e07 2.10e06
n = 15 1.00 0.80 0.60 0.40 0.20 0.00 0.20 0.40 0.60 0.80 1.00
0.36787944 0.44932896 0.54881164 0.67032005 0.81873075 1 1.22140276 1.49182470 1.82211880 2.22554093 2.71828183
0.36787944 0.44932896 0.54881164 0.67032005 0.81873075 1 1.22140276 1.49182470 1.82211880 2.22554093 2.71828183
3.72e13 2.23e13 1.00e13 3.20e14 3.00e15 0 0 1.70e13 1.51e12 1.17e11 7.52e11
3.67e13 2.19e13 9.75e14 2.90e14 3.06e15 0 1.10e14 1.60e13 1.51e12 1.17e11 7.50e11
2
2
6 P ¼ 40 0 2
1 4 B¼ 0 0
0 0
7 1 0 5; 2 0 3 0 1 5 2
21
3 Kf ¼
2 61 43 1 4
1 2 1 2 1 3
0
3
7 0 5; 0
2
0 6 Kv ¼ 4 0 0
3 0 12 7 0 0 5; 0 0
1428
S. Shahmorad / Appl. Math. Comput. 167 (2005) 1418–1429
with f ¼ ½ f0
f1
f2 ¼
d ¼ ½ d1
d2 ¼ ½ 1
17
136
12
12 ;
1 :
Now using relations (2.16)–(2.18), we find the system of Eq. (2.19) as follows: 2 3 3 1 0 2 6 7 ½ a0 a1 a2 4 0 1 13 5 ¼ 1; 1; 17 12 0 2 14 and its solution a2 ¼ ½ a0
a1
a2 ¼ ½ 1
1
1 :
Substituting the elements of this vector into (4.1) we obtain the solution y 2 ðsÞ ¼ 1 þ s s2 which is the exact solution.
Example 2 [14, Example (b)]. Z s 0 y ðsÞ ¼ 1 þ 2s yðsÞ þ sð1 þ 2sÞetðstÞ yðtÞ dt;
yð0Þ ¼ 1:
06s61
0 2
The exact solution is y(s) = es . For numerical results see Table 1. Example 3 [15, Example 3]. Z s y 0 ðsÞ þ yðtÞ dt ¼ 1; 0 6 s 6 1 0
yð0Þ ¼ 0: The exact solution is y(s) = sin(s). For numerical results see Table 2. Example 4 [15, Example 4]. y 00 ðsÞ þ sy 0 ðsÞ syðsÞ ¼ es 2 sinðsÞ þ
Z
1
sinðsÞet yðtÞ dt;
1 6 s 6 1
1
yð0Þ ¼ 1;
y 0 ð0Þ ¼ 1:
The exact solution is y(s) = es. For numerical results see Table 3.
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5. Conclusions Most integro-differential equations are usually difficult to solve analytically. In many cases, it is required to obtain the approximate solutions. For this purpose, the Tau method presented in this paper can be applied. Advantages of the method are the solution is expressed as a polynomial, the error estimator is available and this method is an exceedingly good approximating method in the sense that its error for problems with reasonable solutions, decays exponentially as the degree of approximation increases [16]. References [1] E.L. Ortiz, H. Samara, An operational approach to the Tau method for the numerical solution of nonlinear differential equations, Computing 27 (1981) 15–25. [2] E.L. Ortiz, The Tau method, Siam. J. Numer. Anal. 6 (1969) 480–492. [3] K.M. Liu, E.L. Ortiz, Eigenvalue problems for singularly perturbed differential equations, in: J.J.H. Miller (Ed.), Proceedings of the BAIL II Conference, Boole Press, Dublen, 1982, pp. 324–329. [4] K.M. Liu, E.L. Ortiz, Approximation of eigenvalues defined by ordinary differential equations with the Tau method, in: B. Kagestrm, A. Ruhe (Eds.), Matrix Pencils, Springer-Verlag, Berlin, 1983, pp. 90–102. [5] K.M. Liu, E.L. Ortiz, Tau method approximation of differential eigenvalue problems where the spectral parameter enters nonlinearly, J. Comput. Phys. 72 (1987) 299–310. [6] K.M. Liu, E.L. Ortiz, Numerical solution of ordinary and partial function-differential eigenvalue problems with the Tau method, Computing (wien) 41 (1989) 205–217. [7] E.L. Ortiz, H. Samara, Numerical solution of differential eigenvalue problems with an operational approach to the Tau method, Computing 31 (1983) 95–103. [8] K.M. Liu, E.L. Ortiz, Numerical solution of eigenvalue problems for partial differential equations with the Tau-lines Method, Comput. Math. Appl. 12B (5/6) (1986) 1153–1168. [9] K.M. Liu, E.L. Ortiz, K.S. Pun, Numerical solution of SteklovÕs partial differential equation eigenvalue problem, in: J.J.H. Miller (Ed.), Computational and asymptotic methods for boundary and interior layers (III), Boole Press, Dublin, 1984, pp. 244–249. [10] E.L. Ortiz, K.S. Pun, Numerical solution of nonlinear partial differential equations with Tau method, J. Comp. Appl. Math. 12/13 (1985) 511–516. [11] E.L. Ortiz, K.S. Pun, A bi-dimensional Tail-Elements method for the numerical solution of nonlinear partial differential equations with an application to BurgersÕ equation, Comput. Math. Appl. 12B (5/6) (1986) 1225–1240. [12] E.L. Ortiz, H. Samara, Numerical solution of partial differential equations with variable coefficients with an operational approach to the Tau method, Comput. Math. Appl. 10 (1) (1984) 5–13. [13] M.K. EL-Daou, H.G. Khajah, Iterated solutions of linear operator equations with the Tau method, Math. Comput. 66 (217) (1997) 207–213. [14] A. Makroglou, Convergence of a block-by-block method for nonlinear Volterra integrodifferential equations, Math. Comp. 35 (1980) 783–796. [15] S. Yalcinbas, M. Sezer, The approximate solution of high-order linear Volterra–Fredholm integro-differential equations in term of Taylor polynomials, Appl. Math. Comput. 112 (2000) 291–308. [16] D. Gottlieb, S.A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, SIAM, Philadelphia, PA, 1986.