Ain Shams Engineering Journal (2012) 3, 417–422
Ain Shams University
Ain Shams Engineering Journal www.elsevier.com/locate/asej www.sciencedirect.com
ENGINEERING PHYSICS AND MATHEMATICS
Legendre approximation solution for a class of higher-order Volterra integro-differential equations S.G. Venkatesh *, S.K. Ayyaswamy, S. Raja Balachandar Department of Mathematics, School of Humanities and Sciences, SASTRA University, Thanjavur 613 401, Tamil Nadu, India Received 10 August 2011; revised 13 January 2012; accepted 9 April 2012 Available online 24 August 2012
KEYWORDS Legendre polynomials; Legendre wavelets; Integro-differential equations; Gaussian integration; Legendre wavelet method
Abstract The aim of this work is to study the Legendre wavelets for the solution of boundary value problems for a class of higher order Volterra integro-differential equations using function approximation. The properties of Legendre wavelets together with the Gaussian integration method are used to reduce the problem to the solution of nonlinear algebraic equations. Also a reliable approach for convergence of the Legendre wavelet method when applied to a class of nonlinear Volterra equations is discussed. Illustrative examples have been discussed to demonstrate the validity and applicability of the technique and the results obtained by Legendre wavelet method is very nearest to the exact solution. The results demonstrate reliability and efficiency of the proposed method. 2012 Ain Shams University. Production and hosting by Elsevier B.V. All rights reserved.
1. Introduction Integro-differential equation (IDE) is an equation that the unknown function appears under the sign of integration and it also contains the derivatives of the unknown function. Mathematical modeling of real-life problems usually results in functional equations, e.g. partial differential equations, inte* Corresponding author. Tel.: +91 4362 264101x108; fax: +91 4362 264120. E-mail addresses:
[email protected] (S.G. Venkatesh),
[email protected] (S.K. Ayyaswamy),
[email protected] (S. Raja Balachandar). Peer review under responsibility of Ain Shams University.
Production and hosting by Elsevier
gral and integro-differential equations, stochastic equations and others. Many mathematical formulations of physical phenomena contain integro-differential equations, these equations arise in fluid dynamics, biological models and chemical kinetics. In the past several decades, many effective methods for obtaining approximation/numerical solutions of linear/nonlinear differential equations have been presented, such as Adomian decomposition method [1], variational iteration method [2,3], homotopy perturbation method [4–6], He’s homotopy perturbation method [7–10], homotopy analysis method [11], and wavelet methods [12–14,4,5]. The literature of numerical analysis contains little on the solution of the boundary value problems for higher-order integro-differential equations. The boundary value problems for higher-order integro-differential equations had been investigated by Morchalo [16,17] and Agarwal [18] among others. Agarwal [18] discussed the existence and uniqueness of the solutions for these problems. The following gives the details
2090-4479 2012 Ain Shams University. Production and hosting by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.asej.2012.04.007
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S.G. Venkatesh et al.
of literature survey for the integro-differential equations of lower order. Ghasemi et al. [10] presented He’s homotopy perturbation method for solving nonlinear integro-differential equations. Zhao and Corless [19] adopted finite difference method for integro-differential equations. Yusufoglu (Agadjanov) [20] had solved initial value problem for Fredholm type linear integro-differential equation system. Seyed Alizadeh et al. [21] discussed an Approximation of the analytical solution of the linear and nonlinear integro-differential equations by Homotopy Perturbation Method. Wazwaz [22] gave a reliable algorithm for solving boundary value problems for higher-order integro-differential equations. Lepik [23] had solved the nonlinear integro-differential equations using Haar wavelet method. Ghasemi et al. [4] discussed the comparison between wavelet Galerkin method and homotopy perturbation method for the nonlinear integro-differential equations. Ghasemi et al. [15,5] established numerical solution of linear integrodifferential equations by using sine–cosine wavelet method and they have also compared with homotopy perturbation method. In recent years, wavelets have found their way into many different fields of science and engineering. Many researchers started using various wavelets [12–14,4,5,13] for analyzing problems of greater computational complexity and proved wavelets to be powerful tools to explore new direction in solving differential equations. Legendre wavelet based approximate solution of Lane-Emden type was studied by Yousefi [24] recently. In the present article, we apply Legendre wavelet method (LWM) to find the approximate solution of mth order integro-differential equation [22] of the form Z x yðmÞ ðxÞ ¼ fðxÞ þ Kðx; tÞFðyðtÞÞdt; 0 < x < b ð1Þ 0
2. Properties of Legendre wavelets 2.1. Wavelets and Legendre wavelets Wavelets constitute a family of functions constructed from dilation and translation of a single function called the mother wavelet. When the dilation parameter a and the translation parameter b vary continuously, we have the following family of continuous wavelets as: tb 12 wa;b ðtÞ ¼ jaj w ; a; b 2 R; a–0: a If we restrict the parameters a and b to discrete values as k a ¼ ak 0 ; b ¼ nb0 a0 ; a0 > 1; b0 > 0, n and k are positive integers, we have the following family of discrete wavelets: 1 wk;n ðtÞ ¼ jaj2 w ak0 t nb0 where wk,n(t) form a basis of L2(R). In particular, when a0 = 2 and b0 = 1 then wk,n(t) forms an orthonormal basis. Legendre wavelets wnm ðtÞ ¼ wðk; n^; m; tÞ have four arguments: n^ ¼ 2n 1, n ¼ 1; 2; 3 . . . ; 2k1 , k can assume any positive integer, m is the order of Legendre polynomials and t is the normalized time. They are defined on the interval [0, 1) as ( qffiffiffiffiffiffiffiffiffiffiffiffi k n^þ1 m þ 1222 Pm ð2k t n^Þ; for n^21 ; k 6 t 6 2k wnm ðtÞ ¼ ð2Þ 0; otherwise k1 whereqmffiffiffiffiffiffiffiffiffiffiffi = 0, ffi 1, 2, . . ., M 1, n = 1, 2, 3, . . ., 2 . The coeffi1 cient m þ 2 is for orthonormality, the dilation parameter is a = 2k and translation parameter is b ¼ n^2k . Here Pm(t) are well-known Legendre polynomials of order m which are defined on the interval [1, 1], and can be determined with the aid of the following recurrence formulae:
P0 ðtÞ ¼ 1;
With the boundary conditions ðjÞ
y ð0Þ ¼ aj ; j ¼ 0; 1; 2; . . . ; ðr 1Þ yðjÞ ðbÞ ¼ bj ; j ¼ r; ðr þ 1Þ; . . . ; ðm 1Þ
Pmþ1 ðtÞ ¼
P1 ðtÞ ¼ t 2m þ 1 m tPm ðtÞ Pm1 ðtÞ; mþ1 mþ1
m ¼ 1; 2; 3; . . .
(m)
where y (x) indicates the mth derivative of y(x), and F(y(x)) is a nonlinear function. In addition, the kernel K(x, t) and f(x) are given in advance. It is of interest to point out that y(x) and f(x) are assumed real and as many times differentiable as required for x e [0, b], and aj, 0 6 j 6 ðr 1Þ and bj ; r 6 j 6 ðm 1Þ are real finite constants. The Legendre wavelet method (LWM) consists of conversion of integro-differential equations into integral equations and expanding the solution by Legendre wavelets with unknown coefficients. The properties of Legendre wavelets together with the Gaussian integration formula are then utilized to evaluate the unknown coefficients and find an approximate solution to Eq. (1). The organization of the paper is as follows: In Section 2, we describe the basic formulation of wavelets and Legendre wavelets required for our subsequent development. Section 3 is devoted to the solution of Eq. (1) by using integral operator and Legendre wavelets. Convergence analysis and the error estimation for the proposed method have been discussed in Section 4. In Section 5, we report our numerical finding and demonstrate the accuracy of the proposed scheme by considering numerical examples with error calculations. Concluding remarks are given in the final section.
2.2. Function approximation A function f(t) defined over [0, 1) may be expanded as fðtÞ ¼
1 X 1 X cnm wnm ðtÞ
ð3Þ
n¼1 m¼0
where cnm = Æf(t), wnm(t)æ, in which Æ., .æ denotes the inner product. If the infinite series in Eq. (3) is truncated, then Eq. (3) can be written as fðtÞ ffi
2k1 M 1 X X cnm wnm ðtÞ ¼ CT wðtÞ; n¼1 m¼0
where C and w(t) are 2k1M · 1 matrices given by C ¼ ½c10 ; c11 ; . . . c1M1 ; c20 ; . . . ; c2M1 ; . . . ; c2k1 0 ; . . . ; c2k1 M1 T ;
ð4Þ
wðtÞ ¼½w10 ðtÞ; w11 ðtÞ; . . . ; w1M1 ðtÞ; w20 ðtÞ; . . . ; w2M1 ðtÞ; . . . ; w2k1 0 ðtÞ; . . . ; w2k1 M1 ðtÞT :
ð5Þ
Legendre approximation solution for a class of higher-order Volterra integro-differential equations 4. Convergence analysis
3. Legendre wavelet scheme for boundary value problems of higher-order integro-differential equations
In this section, we provide the theorem on convergence analysis and error estimation of our proposed method.
Consider the integro differential equation given in Eq. (1) Z xZ x Z x dm Let us define L ¼ m ; L1 ¼ ... dxdx . . . dx |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} dx 0 0 0
Theorem. The series solution Eq. (3) of problem (1) using LWM converges towards u(x).
m times
Operating with L1 on both sides of Eq. (1) Z x Kðx:tÞFðyðtÞÞdt L1 ½ym ðxÞ ¼ L1 ½fðxÞ þ L1 0
yðxÞ ¼ gðxÞ þ hðxÞ þ L1
Z
x
419
Kðx:tÞFðyðtÞÞdt 0
1
where h(x) = L [f(x)] and g(x) is a function of x along with constants. Z x yðxÞ ¼ NðxÞ þ ðx tÞm Kðx; tÞFðyðtÞÞdt ð6Þ 0
Proof. Let 1 L2(R) be the Hilbert space and let wk;n ðtÞ ¼ jaj2 wðak0 t nb0 Þwhere wk,n(t) form a basis of L2(R). In particular, when a0 = 2 and b0 = 1, wk,n(t) forms an orthonormal basis. P Let uðxÞ ¼ M1 i¼1 C1i w1i ðxÞ where C1i = Æu(x), w1i(x)æ for K = 1 and Æ., .æ represents an inner product. uðxÞ ¼
n X huðxÞ; w1i ðxÞiw1i ðxÞ i¼1
Let us denote w1i(x) as w(x).
where N(x) = g(x) + h(x) Let yðxÞ ¼ CT wðxÞ:
ð7Þ
Therefore we have Z x ðx tÞm Kðx; tÞFðCT wðtÞÞdt: CT wðxÞ ¼ NðxÞ þ
ð8Þ
Let aj = Æu(x), w(x)æ Define the sequence of partial sums {Sn} of (ajw(xj)); let Sn and Sm be arbitrary partial sums with n P m. We are going to prove that {Sn} is a Cauchy sequence in Hilbert space.
0
We now collocate Eq. (8) at 2k1M points xi as Z x ðx tÞm Kðx; tÞFðCT wðxi ÞÞdx: CT wðxi Þ ¼ NðxÞ þ
Let Sn ¼
j¼1
ð9Þ
0
Suitable collocation points are zeros of Chebyshev polynomials xi ¼ cosðð2i þ 1Þp=2k MÞ;
i ¼ 1; 2; . . . ; 2k1 M:
huðxÞ; Sn i ¼ huðxÞ;
n n n X X X aj wðxj Þi ¼ aj huðxÞ; wðxj Þi ¼ aj aj j1
j¼1
j1
n X ¼ jaj j2 j1
In order to use the Gaussian integration formula for Eq. (9), we transfer the intervals [0, xi] into the interval [1, 1] by means of the transformation 2 s ¼ t 1: xi Eq. (9) may then be written as Z m x xi 1
xi i CT wðxi Þ ¼Nðxi Þ þ x ðs þ 1Þ K x; ðs þ 1Þ 2 1 2 2
x i T F C w ðs þ 1Þ ds 2
We will claim that kSn Sm k2 ¼
Pn
j¼mþ1 jaj j
2
forn > m. Now
2 * + X n n n X X a wðxj Þ ¼ ai wðxi Þ; aj wðxj Þ j¼mþ1 j i¼mþ1 j¼mþ1 ¼
n n X X
ai aj hwðxi Þ; wðxj Þi
i¼mþ1j¼mþ1
¼
n X
aj aj ¼
j¼mþ1
n X
jaj j2
j¼mþ1
i:e:kSn Sm k2 ¼
By using the Gaussian integration formula, we get s
m xi X xi wj x ðsj þ 1Þ 2 j¼1 2
x
x i i ðsj þ 1Þ ds K x; ðsj þ 1Þ F CT w 2 2
n X aj wðxj Þ
n X
jaj j2 for n > m:
j¼mþ1
CT wðxi Þ ¼Nðxi Þ þ
ð10Þ
where s is s zeros of Legendre polynomials Ps+1 and wj are the corresponding weights. The idea behind the above approximation is the exactness of the Gaussian integration formula for polynomials of degree not exceeding 2s + 1. Here the weight wj can with the help of the formula R 1 Qbe identified ss wj ¼ 1 sj¼0;j–i ðsi sjj Þds. Eq. (10) gives 2k1M nonlinear equations which can be solved for the elements of C in Eq. (7) using Newton’s iterative method.
P 2 From Bessel’s inequality, we have 1 j¼1 jaj j is convergent and 2 hence kSn Smk fi 0 as m, n fi 1, i.e. kSn Smk fi 0 and {Sn} is a Cauchy sequence and it converges to say ‘s’. We assert that u(x) = s. Now hs uðxÞ; wðxj Þi ¼ hs; wðxj Þi huðxÞ; wðxj Þi ¼ hLtn!1 Sn ; wðxj Þi aj ¼ Ltn!1 hSn ; wðxj Þi aj ¼ aj aj ) hS uðxÞ; wðxj Þi ¼ 0 P Hence u(x) = s and nj¼1 aj wðxj Þ converges to u(x). Hence the theorem is proved.
h
420
S.G. Venkatesh et al. subject to the boundary conditions
4.1. Error estimation
yð0Þ ¼ 1; y0 ð0Þ ¼ 1; yð1Þ ¼ 1 þ e; y0 ð1Þ ¼ 2e In this part, an error estimation for the approximate solution of Eq. (6) is discussed and this approach is based on the technique presented in [25]. Let us consider en ðxÞ ¼ uðxÞ uðxÞ as the error function of the approximate solution uðxÞ for u(x), where u(x) is the exact solution of Eq. (6). Z x uðxÞ ¼ NðxÞ þ ðx tÞm Kðx; tÞFðyðtÞÞdt þ Hn ðxÞ 0
where Hn(x) is the perturbation term. Z x ðx tÞm Kðx; tÞFðyðtÞÞdt: Hn ðxÞ ¼ uðxÞ NðxÞ
ð14Þ
We apply the method presented in this paper and solve Eq. (13) with k = 1 and M = 4. Applying L1 on both sides of Eq. (13), we get 0000
L1 ½y ðxÞ ¼ L1 ½xð1 þ ex Þ þ L1 ½3ex þ L1 ½yðxÞ L1
Z
x
yðtÞdt 0
x2 x3 x5 yðxÞ ¼ 1 þ x þ A þ B þ þ L1 ½xex þ 3ex þ L1 ½yðxÞ 2! 3! 5! Z x 1 yðtÞdt L 0
ð11Þ where A = y00 (0); B ¼ y000 (0).
0
We proceed to find an approximation en ðxÞ to the error function en(x) in the same way as we did before for the solution of the problem. Subtracting Eq. (11) from Eq. (6), the error function en(t) satisfies the problem Z x en ðxÞ þ ðx tÞm Kðx; tÞFðyðtÞÞdt ¼ Hn ðxÞ ð12Þ
x2 x3 x5 þ B þ þ 12x2 ex 42xex 2x3 15x2 2! 3! 5! Z x Z x 3 x þ 6e 6 þ ðx tÞ yðtÞdt ðx tÞ4 yðtÞdt
yðxÞ ¼ 1 þ x þ A
0
0
T
Replacing y(x) by C w(x), we get
0
It should be noted that in order to construct the approximate en ðxÞ to en(x), only Eq. (12) needs to be recalculated in the same way as we did before for the solution of Eq. (7). We ensure the stability of LWM through this convergence and error estimation.
x2 x3 x5 þ B þ þ 12x2 ex 42xex 2! 3! 5! Z x 3 2 x 2x 15x þ 6e 6 þ ðx 0 Z x tÞ3 CT wðtÞdt ðx tÞ4 CT wðtÞdt
CT wðxÞ ¼ 1 þ x þ A
5. Illustrative examples
ð15Þ
0
On solving Eq. (15), we get In the examples that follow, the Legendre wavelet method will be tested by discussing three boundary-value problems of fourth-order integro-differential equations presented in [22]. Example 1. We first consider the linear boundary value problem for the integro-differential equation Z x 0000 yðtÞdt; 0 < x < 1 ð13Þ y ðxÞ ¼ xð1 þ ex Þ þ 3ex þ yðxÞ 0
Table 1
B 7ð2A þ BÞ 4 ð2A þ BÞ pffiffiffi ; C11 ¼ ; C12 24 4 3 2A þ B B pffiffiffi ; C13 ¼ pffiffiffi ¼ 24 5 120 7
C10 ¼
By applying the boundary conditions, we find that A = 2; B = 3. Using Eq. (7), we get y(x) = c10w10 + c11w11 + c12w12 + c13w13.
The errors for Example 1 at M = 4, 10, 15. x
Exact
LWM solution
4
0.0 0.2 0.4 0.6 0.8 1.0
1 1.2442805 1.5967298 2.0932712 2.7804327 3.7182818
1 1.244 1.592 2.068 2.696 3.5
10
0.0 0.2 0.4 0.6 0.8 1.0
1 1.2442805 1.5967298 2.0932712 2.7804327 3.7182818
1 1.2443 1.5967 2.0933 2.7804 3.7183
0 0.0000195 3e10 1.772e8 3.214e7 0.0000182
15
0.0 0.2 0.4 0.6 0.8 1.0
1 1.2442805 1.5967298 2.0932712 2.7804327 3.7182818
1 1.2443 1.5967 2.0933 2.7804 3.7183
0 0.0000195 0.0000298 0.0000288 0.0000327 0.00001182
M
LWM solution-error 0 2.8055e04 4.7299e04 0.02527 0.08443 0.21828
Legendre approximation solution for a class of higher-order Volterra integro-differential equations
421
2
Hence yðxÞ ¼ 1 þ xð1 þ x1 þ x2! Þ.
Table 2
Therefore, we have y(x) = 1 + xex which is the exact solution. Table 1 depicts the error for Example 1 for different values of M with exact solution.
x
Exact
LWM
4
0.0 0.2 0.4 0.6 0.8 1.0
1 1.2214027 1.4918246 1.8221188 2.2255409 2.7182818
1 1.2213 1.4907 1.8160 2.2053 2.6667
0 0.0001027 0.0011246 0.0061188 0.0202409 0.0512818
10
0.0 0.2 0.4 0.6 0.8 1.0
1 1.2214027 1.4918246 1.8221188 2.2255409 2.7182818
1 1.2214 1.4918 1.8221 2.2255 2.7183
0 0.0000027 0.0000246 0.0000188 0.0000409 0.0000182
15
0.0 0.2 0.4 0.6 0.8 1.0
1 1.2214027 1.4918246 1.8221188 2.2255409 2.7182818
1 1.2214 1.4918 1.8221 2.2255 2.7183
0 0.0000027 0.0000246 0.0000188 0.0000409 0.0000182
M
Example 2. We next consider the nonlinear boundary value problem for the integro-differential equation 0000
y ðxÞ ¼ 1 þ
Z
ex y2 dx;
ð16Þ
0
