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Numerical solution of nonlinear Volterra–Fredholm–Hammerstein integral equations via collocation method based on radial basis functions
Applied Mathematics and Computation 218 (2012) 5292–5309
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Numerical solution of nonlinear Volterra–Fredholm–Hammerstein integral equations via collocation method based on radial basis functions K. Parand, J.A. Rad ⇑ Department of Computer Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Evin, Tehran 19839, Iran
a r t i c l e
i n f o
Keywords: Volterra–Fredholm–Hammerstein Collocation method Radial basis functions Legendre polynomials Nonlinear integral equations
a b s t r a c t A numerical technique based on the spectral method is presented for the solution of nonlinear Volterra–Fredholm–Hammerstein integral equations. This method is a combination of collocation method and radial basis functions (RBFs) with the differentiation process (DRBF), using zeros of the shifted Legendre polynomial as the collocation points. Different applications of RBFs are used for this purpose. The integral involved in the formulation of the problems are approximated based on Legendre–Gauss–Lobatto integration rule. The results of numerical experiments are compared with the analytical solution in illustrative examples to confirm the accuracy and efficiency of the presented scheme. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction In recent decades, the so-called meshless methods have been extensively used to find approximate solutions of various types of linear and nonlinear equations [1] such as differential equations (DEs) and integral equations (IEs). Unlike the other methods which were used to mesh the domain of the problem, meshless methods don’t require a structured grid and only make use of a scattered set of collocation points regardless of the connectivity information between the collocation points. For the last years, the radial basis functions (RBFs) method was known as a powerful tool for the scattered data interpolation problem. The main advantage of numerical methods which use radial basis functions is the meshless characteristic of these methods. The use of radial basis functions as a meshless method for the numerical solution of ordinary differential equations (ODEs) and partial differential equations (PDEs) is based on the collocation method. One of the domain-type meshless methods is given in [2] in 1990, which directly collocates radial basis functions, particularly the multiquadric (MQ), to find an approximate solution of linear and nonlinear DEs. Kansa’s method has recently received a great deal of attention from researchers [3–8]. Recently, Kansa’s method was extended to solve various ordinary and partial differential equations including the nonlinear Klein–Gordon equation [7], regularized long wave (RLW) equation [9], high order ordinary differential equations [10], the case of heat transfer equations [11], Hirota–Satsuma coupled KdV equations [12], second-order parabolic equation with nonlocal boundary conditions [13], second-order hyperbolic telegraph equation [14] and so on [15–18]. All of the radial basis functions have global support, and in fact many of them, such as multiquadrics (MQ), do not even have isolated zeros [7,9,19]. The RBFs can be compactly and globally supported, infinitely differentiable, and contain a free parameter c, called the shape parameter [9,19,20]. The interested reader is referred to the recent books and paper by Buhmann [19,21] and Wendland [22] for more basic details about RBFs, compactly and globally supported and convergence rate of the radial basis functions.
⇑ Corresponding author. E-mail addresses: [email protected] (K. Parand), [email protected] (J.A. Rad). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.11.013
K. Parand, J.A. Rad / Applied Mathematics and Computation 218 (2012) 5292–5309
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Despite many research works which are done to finding algorithms for selecting the optimum values of c [23–25], the optimal choice of shape parameter is an open problem which is still under intensive investigation. For example, Carlson and Foley [24] found that the shape parameter is problem dependent. They [24] observed that for rapidly varying functions, a small value of c should be used, but a large value should be used if the function has a large curvature. Tarwater [25] found that by increasing c, the Root-Mean-Square (RMS) of error dropped to a minimum and then increased sharply afterwards. In general, as c increases, the system of equations to be solved becomes ill-conditioned. Cheng et al. [23] showed that when c is very large then the RBFs system error is of exponential convergence. But there is a certain limit for the value c after which the solution breaks down. In general, as the value of the shape parameter c increases, the matrix of the system to be solved becomes highly ill-conditioned and hence the condition number can be used in determining the critical value of the shape parameter for an accurate solution [23]. For some new work on optimal choice of shape parameter, we refer the interested reader to the recent work of [26]. There are two basic approaches for obtaining basis functions from RBFs, namely direct approach (DRBF) based on a differential process [27] and indirect approach (IRBF) based on an integration process [5,10,28]. Both approaches were tested on the solution of second order DEs and the indirect approach was found to be superior to the direct approach [5]. Some of the infinitely smooth RBFs choices are listed in Table 1. The RBFs can be of various types, for example: polynomials of any chosen degree such as linear, cubic, etc., thin plate spline (TPS), multiquadrics (MQ), inverse multiquadrics (IMQ), Gaussian forms (GA), hyperbolic secant (sech) form etc. In the cases of inverse quadratic, inverse multiquadric (IMQ), hyperbolic secant (sech) and Gaussian (GA), the coefficient matrix of RBFs interpolating is positive definite and, for multiquadric (MQ), it has one positive eigenvalue and the remaining ones are all negative [29]. In this paper, we use the multiquadrics direct radial basis function for finding the solution of Volterra–Feredholm–Hemmerstein equations. The MQ was first developed by Hardy [30] in 1971 as a multidimensional scattered interpolation method in modeling of the earth gravitational field. It was not recognized by most of the academic researchers until Franke [31] published a review paper on the evaluation of two-dimensional interpolation methods, whereas MQ was ranked as the best based on its accuracy, visual aspect, sensitivity to parameters, execution time, storage requirements, and ease of implementation. There is a considerable approximation method that discusses approximating the solution of linear and nonlinear mixed Hammerstein integral equations [32–39]. For Fredholm–Hammerstein integral equations, the classical method of successive approximations was introduced in [32]. A variation of the Nystrom method was presented in [40], and a collocation-type method was developed in [33]. Marzban et al. [39] solved the nonlinear Volterra-Fredholm–Hammerstein integral equation by using the Hybrid of block-pulse functions. In [34], Brunner applied a collocation-type method to nonlinear Volterra–Hammerstein integral equations and integro-differential equations and discussed its connection with the iterated collocation method. Han [41] introduced and discussed the asymptotic error expansion of a collocation-type method for Volterra–Hammerstein integral equations. The existence of the solutions to nonlinear Hammerstein integral equations was discussed in [42]. Ezquerro and Hernández applied new fourth-order iterative methods to obtain solution of Hammerstein integral equations [43]. In [44] a numerical procedure based on Sinc basis functions was presented for solving a class of nonlinear integral equations of the Hammerstein-type. For more references about numerical methods for Volterra–Hammerstein integral equations see [75–79]. Maleknejad et al. [45] provided a cubic semiorthogonal compactly supported B-spline wavelets as a basis functions for solution of nonlinear Fredholm–Hammerstein integral equations of the second kind. More recently, Ghoreishi and Hadizadeh [46] gave an extension of the algebraic formulation of the Tau method for the numerical solution of the nonlinear Volterra– Hammerstein integral equations. Also, Maleknejad el al. applied the various numerical methods for solving Hammerstein integral equations [47–50]. For more references about Volterra–Hammerstein integral equations see [51–54]. Several authors considered the nonlinear mixed Volterra–Fredholm integral equation of the form
yðxÞ ¼ f ðxÞ þ k1
Z
x
K 1 ðx; tÞG1 ðt; yðtÞÞdt þ k2
0
Z
1
K 2 ðx; tÞG2 ðt; yðtÞÞdt;
0 6 x 6 1;
ð1Þ
0
where k1 and k2 are constants and f(x) and the kernels K1(x, t) and K2(x, t) are given functions assumed to have nth derivatives on the interval 0 6 x 6 1 and 0 6 t 6 1. Yalcinbas [35] applied Taylor series and Bildik and Inc [36] used the modified decomposition method to find the solution of Eq. (1) for the case G1(t, y(t)) = yp(t) and G2(t, y(t)) = yq(t), where p and q are nonnegative integers. For the case G1(t, y(t)) = F1(y(t)) and G2(t, y(t)) = F2(y(t)), where F1(y(t)) and F2(y(t)) are given continuous functions which are nonlinear with respect to y(t), Yousefi and Razzaghi [37] applied Legendre wavelets to obtain the
Table 1 Some well-known radial basis functions (r = kx xik2 = ri), c > 0. Name of functions
Inverse multiquadrics (IMQ)
Definition pffiffiffiffiffiffiffiffiffiffiffiffiffi r þ c2 pffiffiffiffiffiffiffiffiffiffiffiffiffi 1= r þ c2
Gaussian (GA) Hyperbolic secant (sech)
exp(cr) pffiffiffi sech c r
Multiquadrics (MQ)
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solution, and for the general case, where G1(t, y(t)) and G2(t, y(t)) are given continuous functions which are nonlinear with respect to t and y(t), Ordokhani [38] applied the rationalized Haar functions to get the solution. Also in this case, Marzban et al. [39] solved Eq. (1) by using the Hybrid of block-pulse functions. In this paper we propose a meshless collocation method based on the MQ-RBFs to solve the nonlinear Volterra–Fredholm–Hammerstein integral equations of the form of Eq. (1). For convenience the solution we use RBFs with collocation nodes fxj gNj¼1 which are the zeros of the shifted Legendre polynomial LN(x), 0 6 x 6 1. The shifted Legendre polynomials Li(x) are defined on the interval [0, 1] and satisfy the following formulae [55]:
L0 ðxÞ ¼ 1;
L1 ðxÞ ¼ 2x 1;
2i þ 1 i Liþ1 ðxÞ ¼ ð2x 1ÞLi ðxÞ Li1 ðxÞ; iþ1 iþ1
i ¼ 1; 2; 3; . . . :
ð2Þ
This paper is arranged as follows: in Section 2, we describe the properties of radial basis functions. In Section 3, we introduce the Legendre–Gauss–Lobatto nodes and weights. In Section 4 we implement the problem with the proposed method and in Section 5, we report our numerical finding and demonstrate the accuracy of the proposed methods. In Section 6 we apply a numerical algorithm [9,12] for finding the optimum shape parameter. The conclusions are discussed in the final Section. 2. Radial basis function 2.1. Definition of radial basis function Let Rþ ¼ fx 2 R; x P 0g be the non-negative half-line and let / : Rþ ! R be a continuous function with /(0) P 0. A radial basis functions on Rd is a function of the form
/ðkX X i kÞ; where X; X i 2 Rd and k.k denotes the Euclidean distance between X and Xis. If one chooses N points fX i gNi¼1 in Rd then by custom
sðXÞ ¼
N X
ki /ðkX X i kÞ;
ki 2 R
i¼1
is called a radial basis functions as well [56,57]. The standard radial basis functions are categorized into two major classes [12]: Class 1. Infinitely smooth RBFs [12,58]: These basis functions are infinitely differentiable and heavily depend on the shape parameter c e.g. Hardy multiquadric (MQ), Gaussian (GA), inverse multiquadric (IMQ), and inverse quadric (IQ) (See Table 1). Class 2. Infinitely smooth (except at centers) RBFs [12,58]: The basis functions of this category are not infinitely differentiable. These basis functions are shape parameter free and have comparatively less accuracy than the basis functions discussed in the Class 1. For example, thin plate spline, etc. [12]. 2.2. RBF interpolation The one dimensional function y(x) to be interpolated or approximated can be represented by an RBFs as:
yðxÞ yN ðxÞ ¼
N X
ki /i ðxÞ ¼ UT ðxÞK;
ð3Þ
i¼1
where
/i ðxÞ ¼ uðkx xi kÞ; UT ðxÞ ¼ ½/1 ðxÞ; /2 ðxÞ; . . . ; /N ðxÞ; K ¼ ½k 1 ; k 2 ; . . . ; k N T ; fki gNi¼1
x is the input and are the set of coefficients to be determined [8]. By choosing N interpolate nodes approximate the function y(x).
yj ¼
N X
ki /i ðxj Þ;
j ¼ 1; 2; . . . ; N:
i¼1
To brief discussion on coefficient matrix, we define:
AK ¼ Y;
ð4Þ fxi gNi¼1 ,
we can
K. Parand, J.A. Rad / Applied Mathematics and Computation 218 (2012) 5292–5309
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where
y ¼ ½y1 ; y2 ; . . . ; yN T ; A ¼ ½UT ðx1 Þ; UT ðx2 Þ; . . . ; UT ðxN ÞT ; 3 2 /1 ðx1 Þ /2 ðx1 Þ . . . /N ðx1 Þ 6 / ðx2 Þ / ðx2 Þ . . . / ðx2 Þ 7 2 N 7 6 1 7: ¼6 . .. .. 7 6 .. . 5 4 . . . .
ð5Þ
/1 ðxN Þ /2 ðxN Þ . . . /N ðxN Þ Note that /i(xj) = u(kxi xjk) therefore we have /i(xj) = /j(xi) consequently A = AT. All the infinitely smooth RBFs choices are listed in Table 1 will give coefficient matrices A in Eq. (5) which are symmetric and nonsingular [29], i.e. there is a unique interpolant of the form Eq. (3) no matter how the distinct data points are scattered in any number of space dimensions. In the cases of inverse quadratic, inverse multiquadric (IMQ), hyperbolic secant (sech) and Gaussian (GA) the matrix A is positive definite and, for multiquadric (MQ), it has one positive eigenvalue and the remaining ones are all negative [29]. We have the following theorem about the convergence of RBFs interpolation: Theorem 1 (See [59,60]). Assume fxi gN i¼1 are N nodes in X which is convex, let
h ¼ max min kx xi k2 ; x2X
16i6N
R ^ gÞ < cð1 þ jgjÞð2lþdÞ for any y(x) satisfies ðy ^ gÞdg < 1 we have ^ðgÞÞ2 =/ð when /ð
ða Þ la yN yðaÞ 6 ch ; 1
where /(x) is RBFs and the constant c depends on the RBFs, d is space dimension, l and a are nonnegative integer. It can be seen that not only RBFs itself but also its any order derivative has a good convergence. Proof. A complete proof is given by authors [59,60]. The results of this section can be summarized in the following algorithm. Algorithm. The algorithm works in the following manner: 1. 2. 3. 4. 5.
Choose N center points fxj gNj¼1 from the domain set [a, b]. Approximate y(x) as yN(x) = UT(x)K. Substitute yN(x) into the main problem and create residual function Res(x). Substitute collocation points fxj gNj¼1 into the Res(x) and create the N equations. Solve the N equations with N unknown coefficients of members of K and find the numerical solution.
3. Legendre–Gauss–Lobatto nodes and weights Let HN [1, 1] denote the space of algebraic polynomials of degree 6N
hP i ; Pj i ¼
2 dij : 2j þ 1
ð6Þ
Here, h.,.i represents the usual L2[1, 1] inner product and {Pi}iP0 are the well-known Legendre polynomials of order i which are orthogonal with respect to the weight function w(x) = 1 on the interval [1, 1], and satisfy the following formulae:
P0 ðxÞ ¼ 1; P1 ðxÞ ¼ x; 2i þ 1 i xPi ðxÞ P i1 ðxÞ; Piþ1 ðxÞ ¼ iþ1 iþ1
i ¼ 1; 2; 3; . . . :
Next, we let fxj gNj¼0 as:
_ j Þ ¼ 0; ð1 x2j ÞPðx 1 ¼ x0 < x1 < x2 < . . . < xN ¼ 1;
ð7Þ
_ where PðxÞ is derivative of P(x). No explicit formula for the nodes fxj gN1 j¼1 is known. However, they are computed numerically using the existing subroutines [61,62]. Now, we assume f 2 H2N1 ½1; 1, we have
Z
1
1
f ðxÞdx ’
N X j¼0
xj f ðxj Þ ¼ IG ðf Þ;
ð8Þ
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where xj are the Legendre–Gauss–Lobatto weights given in [55]
xj ¼
2 1 : NðN þ 1Þ ðPN ðxj ÞÞ2
ð9Þ
4. Solution of nonlinear Volterra–Fredholm–Hammerstein integral equations via DRBF Consider the nonlinear Volterra–Fredholm–Hammerstein integral equations given in Eq. (1). In order to use the radial basis functions, we first approximate y(x) as
yðxÞ ’ yN ðxÞ ¼
N X
ki /i ðxÞ ¼ UT ðxÞK;
ð10Þ
i¼1
where UT(x) and K are defined in Eq. (4). Then, from substituting Eq. (10) into Eq. (1), we have
UT ðxÞK ¼ f ðxÞ þ k1
Z
x
K 1 ðx; tÞG1 ðt; UT ðtÞKÞdt þ k2
0
Z
1
K 2 ðx; tÞG2 ðt; UT ðtÞKÞdt:
ð11Þ
0
We now collocate Eq. (11) at points fxi gNi¼1 as
UT ðxi ÞK ¼ f ðxi Þ þ k1
Z
xi
K 1 ðxi ; tÞG1 ðt; UT ðtÞKÞdt þ k2
Z
1
K 2 ðxi ; tÞG2 ðt; UT ðtÞKÞdt:
ð12Þ
0
0
In order to use the Legendre–Gauss–Lobatto integration formula [55] for Eq. (12), we transfer the t-intervals [0, xi] and [0, 1] into the g1 and g2-intervals [1, 1], respectively, by means of the transformations g1 ¼ x2i t 1 and g2 = 2t 1. Let
H1 ðxi ; tÞ ¼ K 1 ðxi ; tÞG1 ðt; UT ðtÞKÞ
ð13Þ
H2 ðxi ; tÞ ¼ K 2 ðxi ; tÞG2 ðt; UT ðtÞKÞ:
ð14Þ
and
Eq. (12) may then be restated as
UT ðxi ÞK ¼ f ðxi Þ þ k1
xi 2
Z
Z x k2 1 1 i H1 xi ; ðg1 þ 1Þ dg1 þ H2 ðxi ; ðg2 þ 1ÞÞdg2 ; 2 2 2 1 1 1
by using the Legendre–Gauss–Lobatto integration formula [55] we have the residual function Res(x)
Resðxi Þ ¼ UT ðxi ÞK þ f ðxi Þ þ k1
r1 r2 x k X xi X 1 2 i w1j H1 xi ; ðg1j þ 1Þ þ w2j H2 xi ; ðg2j þ 1Þ ; 2 2 j¼0 2 2 j¼0
i ¼ 1 . . . N:
ð15Þ
Eq. (15) generates an N set of nonlinear equations which can be solved by mathematical software for the unknowns K. 5. Numerical results In order to illustrate the performance of the RBFs method in solving Hammerstein integral equations and justify the accuracy and efficiency of the method presented in this paper, we consider the following examples. In all examples, we use the multiquadrics (MQ) RBFs. To study the convergence behavior of the MQ-RBFs method, we applied the following laws: (1) Absolute error (Error) between the exact solution and the present solution defined by
Error ¼ yexact ðxÞ yimplicit ðxÞ ; N
x 2 ½0; 1;
(2) The L2 error norm of the solution which is defined by
" # N 2 1=2 X exact implicit implict exact L2 ¼ y ðxÞ yN ðxÞ ¼ y ðxj Þ yN ðxj Þ ; 2
j¼1
fxj gNj¼1
where are interpolate nodes which are the zeros of the shifted Legendre polynomial LN(x), 0 6 x 6 1. (3) The L1 error norm of the solution which is defined by
L1 ¼ yexact ðxÞ yimplicit ðxÞ ¼ maxj yexact ðxj Þ yimplicit ðxj Þ ; N N 1
where 1 6 j 6 N.
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(4) The root mean square (RMS) is defined by
RMS ¼
" # N 2 1=2 X 1 yexact ðxj Þ yimplict ðx Þ : j N N 1 j¼1
(5) The condition number Ks(A) of the coefficient matrix A is given by
K s ðAÞ ¼ kAks kA1 ks ;
s ¼ 2; 1:
The condition number Ks(A) has the following properties: The condition number shows that a small perturbation in initial data may produce a large amount of perturbation in the solution [13]. The condition number of an interpolation matrix gives information on the numerical stability of the interpolation process. The condition number depends more on the separation distance than on the number N of centers [22]. The condition number grows with N for fixed values of the shape parameter c. In practices, the shape parameter must be adjusted with the number of interpolating points in order to produce an interpolation matrix that is well-conditioned enough to be inverted in finite precision arithmetic [20]. The condition number of the system matrix can be used in determining the optimal value of the shape parameter c for an accurate solution [63]. When the number of mesh points increases, the condition number of the matrix becomes very large and the matrix tends to be ill-conditioned [13]. The numerical implementation is carried out in microsoft.maple.13, with hardware configuration: desktop 64-bit Intel Core 2 Duo CPU, 4 GB of RAM, 64-bit Operating System (Windows Vista). 5.1. Problem 1 Consider the following Hammerstein integral equation with the exact solution y(x) = ex [35–37].
1 1 yðxÞ ¼ ex e3x þ þ 3 3
Z
x
y3 ðtÞdt:
ð16Þ
0
We solve Eq. (16) for different values of N. Table 2 shows the absolute error (Error) between the exact and MQ-RBFs solutions, L2-error and L1-error, RMS-error norms, K2(A) and K1(A) in some points of the interval [0, 1] obtained for N = 5, 8, 10, 15. This table indicates that as N increases, the Error decreases more rapidly. From Table 2, it can be observed that the accuracy increases with the increase of number of collocation points. Accuracy of the present approximation is examined in the L2-error and L1-error, RMS-error norms. The results of RMS-error are comparatively better than the L2-error and L1error norms. Fig. 1 represented the Error for N = 15. It shows the decreasing of the absolute error Error by collocation method based MQ-RBFs. The convergence behavior of the MQ-RBFs method in terms of the L2-error norm versus reciprocal of number of collocation points N is shown in Fig. 3. The Fig. 3 shows that the L2-error norm decreases by increasing the number of collocation points N for MQ-RBFs. 5.2. Problem 2 Consider the nonlinear Volterra–Fredholm–Hammerstein integral equation with the exact solution y(x) = cos (x) [64,65]. 2
yðxÞ ¼ 1 þ sin ðxÞ þ
Z
1
Kðx; tÞy2 ðtÞdt;
ð17Þ
0
Table 2 Absolute error between the exact and numerical solution with c = 1.8 for various value of N and 0 6 x 6 1; L2-norm error, L1-norm error, RMS error, condition number of s = 2,1 at various values of N for problem 1. x
N=5
N=8
N = 10
N = 15
0.0 0.2 0.4 0.6 0.8 1.0
2.6866e 003 7.4963e 004 7.3269e 004 1.9736e 004 1.3495e 004 2.7461e 003
6.1015e 005 1.5098e 005 7.0374e 006 3.4187e 006 2.0792e 006 1.1342e 004
6.8393e 007 1.2665e 007 7.7416e 008 6.6926e 009 2.7495e 008 6.9435e 008
1.4562e 010 2.0810e 011 4.9287e 012 5.6513e 012 2.1146e 012 2.0294e 011
L2 L1 RMS K2(A) K1(A)
1.2650e 003 1.0908e 003 6.3250e 004 5.0348e + 005 8.0345e + 005
8.8171e 005 8.6438e 005 3.3325e 005 2.1713e + 010 3.9677e + 010
7.1095e 008 3.1291e 008 2.3698e 008 2.4771e + 012 4.8599e + 012
1.4373e 011 5.5922e 012 3.8412e 012 6.9386e + 018 1.5087e + 019
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Fig. 1. Absolute error between MQ-RBFs solution and exact solution for c = 1.8 and N = 15 of problem 1.
Fig. 2. Error plot versus shape parameter c for N = 15 of problem 1.
Fig. 3. Convergence in space for MQ-RBFs with c = 1.8 of problem 1, N is number of collocation points.
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K1(A)
L1 error
c
K1(A)
L1 error
0.7 0.8 0.9 1.0 1.1 1.2 1.3
1.4135e + 015 5.7373e + 015 1.4723e + 016 2.8363e + 016 4.6954e + 016 8.0433e + 016 1.5891e + 017
1.7023e 009 3.0839e 009 3.3808e 009 2.1572e 009 9.8427e 010 4.1733e 010 1.6781e 010
1.4 1.5 1.6 1.7 1.8 1.9 2.0
3.1764e + 017 6.2056e + 017 1.3559e + 018 3.0431e + 018 6.9386e + 018 1.5929e + 019 3.6542e + 019
7.6028e 011 3.8676e 011 1.9269e 011 1.0133e 011 5.5922e 012 3.1229e + 001 3.3107e + 001
where
Kðx; tÞ ¼
3 sinðx tÞ; 0 6 t 6 x; 0;
ð18Þ
x 6 t 6 1:
We applied our method and solved Eq. (17) for different values of N. Table 4 shows the Error, L2-error, L1-error and RMSerror norms, K2(A) and K1(A) in some points of the interval [0, 1] obtained for N = 5, 10, 15, 18. Then in Fig. 4 the absolute error (Error) for N = 18 is represented. The convergence behavior of the MQ-RBFs method in terms of the L2-error norm versus reciprocal of number of collocation points N is shown in Fig. 5. 5.3. Problem 3 Consider the nonlinear Volterra–Fredholm–Hammerstein integral equation of the form [35,37–39,66].
yðxÞ ¼ gðxÞ þ
Z
x
ðx tÞy2 ðtÞdt þ
0
Z
1
ðx þ tÞyðtÞdt;
ð19Þ
0
Table 4 Absolute error between the exact and numerical solution with c = 2 for various value of N and 0 6 x 6 1; L2-norm error, L1-norm error, RMS error, condition number of s = 2,1 at various values of N for problem 2. x
N=5
N = 10
N = 15
N = 18
0.0 0.2 0.4 0.6 0.8 1.0
9.7774e 005 1.4953e 005 3.0243e 005 2.7392e 005 1.1666e 005 6.1572e 005
2.3698e 009 2.5583e 010 4.3742e 011 1.9526e 010 5.3780e 010 2.4312e 009
2.6890e 012 3.2997e 013 9.4818e 015 7.9988e 015 1.6686e 013 8.9004e 013
2.4514e 014 4.1423e 015 3.3900e 015 2.4645e 015 1.6125e 015 5.3448e 015
L2 L1 RMS K2(A) K1(A)
1.9697e 006 1.8373e 006 9.8485e 007 9.2332e + 005 1.4219e + 006
3.3018e 012 2.0912e 012 1.1006e 012 1.3164e + 013 2.4592e + 013
8.3896e 016 3.3923e 016 2.2422e 016 3.6542e + 019 7.4388e + 019
4.5392e 018 1.7346e 018 1.1009e 018 7.3870e + 023 1.5798e + 024
Fig. 4. Absolute error between MQ-RBFs solution and exact solution for c = 2 and N = 18 of problem 2.
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Fig. 5. Convergence in space for MQ-RBFs with c = 2 of problem 2, N is number of collocation points.
where
gðxÞ ¼
1 6 1 4 5 5 x þ x x2 þ x ; 30 3 3 4
ð20Þ
which has the exact solution y(x) = x2 2. We applied the RBFs approach and solved Eq. (19). Table 5 shows the Error, L2error, L1-error and RMS-error norms, K2(A) and K1(A) in some points of the interval [0,1] obtained for N = 10,14,16,20. Fig. 6 represents the Error for N = 14. The convergence behavior of the MQ-RBFs method in terms of the L2-error norm versus reciprocal of number of collocation points N is shown in Fig. 7. 5.4. Problem 4 Consider the nonlinear Volterra–Fredholm–Hammerstein integral equation given in [38,39,66] by
yðxÞ ¼ 2 cosðxÞ 2 þ 3
Z
x
sinðx tÞy2 ðtÞdt þ
0
6 7 6 cosð1Þ
Z
1
ð1 tÞ cos2 ðxÞðt þ yðtÞÞdt;
ð21Þ
0
which has the exact solution y(x) = cos(x). We solve Eq. (21) for different values of N. Table 6 shows the Error, L2-error, L1-error and RMS-error norms, K2(A) and K1(A) norms in some points of the interval [0, 1] obtained for N = 5, 10, 14, 19. Then in Fig. 8 the Error for N = 14 is represented. The convergence behavior of the MQ-RBFs method in terms of the L2-error norm versus reciprocal of number of collocation points N is shown in Fig. 9. 5.5. Problem 5 Consider the nonlinear Volterra–Hammerstein integral equation given in [52] by
yðxÞ ¼
1 4 5 2 3 x þ x þ þ 10 6 8
Z
x
0
1 2 y ðtÞdt; 2x
ð22Þ
Table 5 Absolute error between the exact and numerical solution with c = 2 for various value of N and 0 6 x 6 1; L2-norm error, L1-norm error, RMS error, condition number of s = 2,1 at various values of N for problem 3. x
N = 10
N = 14
N = 16
N = 20
0.0 0.2 0.4 0.6 0.8 1.0
1.9247e 008 4.1585e 009 2.5313e 009 1.9238e 009 1.8319e 009 5.0722e 009
3.1852e 011 5.5758e 012 3.0841e 012 1.5462e 012 6.3014e 013 3.1567e 013
1.1713e 013 3.5184e 015 2.9987e 014 3.6562e 014 1.3020e 014 1.1518e 013
1.4281e 015 2.2529e 016 1.1229e 016 7.5228e 017 6.8774e 017 2.0931e 016
L2 L1 RMS K2(A) K1(A)
8.4109e 012 3.7303e 012 2.6598e 012 3.3021e + 013 4.1241e + 013
6.0600e 015 2.8521e 015 1.6807e 015 1.8008e + 018 3.6587e + 018
4.5610e 017 2.2962e 017 1.1776e 017 1.0935e + 021 2.2940e + 021
1.2417e 019 4.6900e 020 2.8486e 020 5.2299e + 026 1.1443e + 027
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Fig. 6. Absolute error between MQ-RBFs solution and exact solution for c = 2 and N = 14 of problem 3.
Fig. 7. Convergence in space for MQ-RBFs with c = 2 of problem 3, N is number of collocation points.
Table 6 Absolute error between the exact and numerical solution with c = 2 for various value of N and 0 6 x 6 1; L2-norm error, L1-norm error, RMS error, condition number of s = 2,1 at various values of N for problem 4. x
N=5
N = 10
N = 14
N = 19
0.0 0.2 0.4 0.6 0.8 1.0
1.0572e 004 7.7160e 006 3.8609e 005 1.4363e 005 3.0035e 005 4.1888e 005
2.3692e 009 2.5393e 010 4.7045e 011 1.9227e 010 5.3574e 010 2.4327e 009
4.8595e 012 1.5440e 012 1.5124e 012 1.4149e 012 1.2960e 014 3.9911e 013
6.2514e 015 1.6328e 017 6.4699e 016 4.4481e 016 5.2436e 018 1.0257e 015
L2 L1 RMS K2(A) K1(A)
3.1710e 005 1.7367e 005 1.5855e 005 9.2332e + 005 1.4219e + 006
3.1991e 012 1.9265e 012 1.0663e 012 1.3164e + 013 2.4592e + 013
2.9194e 015 1.1718e 015 8.0970e 016 1.8008e + 018 3.6587e + 018
9.2477e 019 3.4760e 019 2.1797e 019 1.9249e + 025 4.1728e + 025
which has the exact solution yðxÞ ¼ x2 þ 12. We solve Eq. (22) for different values of N. Table 7 shows the Error, L2-error, L1error and RMS-error norms, K2(A) and K1(A) in some points of the interval [0, 1] obtained for N = 6, 9, 11, 14. Then in Fig. 10 the Error for N = 14 is represented. The convergence behavior of the MQ-RBFs method in terms of the L2-error norm versus reciprocal of number of collocation points N is shown in Fig. 11.
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Fig. 8. Absolute error between MQ-RBFs solution and exact solution for c = 2 and N = 14 of problem 4.
Fig. 9. Convergence in space for MQ-RBFs with c = 2 of problem 4, N is number of collocation points.
Table 7 Absolute error between the exact and numerical solution with c = 2 for various value of N and 0 6 x 6 1; L2-norm error, L1-norm error, RMS error, condition number of s = 2,1 at various values of N for problem 5. x
N=6
N=9
N = 11
N = 14
0.0 0.2 0.4 0.6 0.8 1.0
5.2922e 004 7.8748e 005 2.3950e 005 5.0644e 006 7.0737e 005 1.9844e 004
1.4841e 006 7.9423e 008 2.1415e 007 6.6108e 008 8.4048e 008 2.0520e 007
3.2574e 008 2.0604e 009 7.9366e 010 2.8076e 009 2.5851e 009 3.0117e 009
5.1507e 011 6.4333e 012 3.8578e 012 2.4023e 012 1.6677e 012 1.0692e 012
L2 L1 RMS K2(A) K1(A)
1.8018e 004 1.6275e 004 8.0580e 005 6.0029e + 007 9.2776e + 007
5.0168e 007 4.5471e 007 1.7737e 007 1.8699e + 011 3.6261e + 011
1.0971e 008 9.9696e 009 3.4694e 009 3.3020e + 013 4.1242e + 013
1.7275e 011 1.5740e 011 4.7912e 012 1.8007e + 018 3.6586e + 018
K. Parand, J.A. Rad / Applied Mathematics and Computation 218 (2012) 5292–5309
5303
Fig. 10. Absolute error between MQ-RBFs solution and exact solution for c = 2 and N = 14 of problem 5.
Fig. 11. Convergence in space for MQ-RBFs with c = 2 of problem 5, N is number of collocation points.
5.6. Problem 6 Consider the nonlinear Volterra–Hammerstein integral equation given in [52] by
yðxÞ ¼ x þ
1 ðxþ1Þ e 1 þ xþ1
Z 0
x
1 t=x yðtÞ e e dt; x
ð23Þ
which has the exact solution y(x) = x. We solve Eq. (23) for different values of N. Table 8 shows the Error, L2-error, L1-error and RMS-error norms, K2(A) and K1(A) in some points of the interval [0, 1] obtained for N = 5, 10, 12, 14. Then in Fig. 12 the Error for N = 14 is represented. The convergence behavior of the MQ-RBFs method in terms of the L2-error norm versus reciprocal of number of collocation points N is shown in Fig. 13. 5.7. Problem 7 In this example the collocation approximation based on RBFs is used to solve the integral equation reformulation of the nonlinear two-point boundary value problem (BVP)
(
y00 ðxÞ eyðxÞ ¼ 0; yð0Þ ¼ yð1Þ ¼ 0;
x 2 ½0; 1;
ð24Þ
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Table 8 Absolute error between the exact and numerical solution with c = 2.5 for various value of N and 0 6 x 6 1; L2-norm error, L1-norm error, RMS error, condition number of s = 2,1 at various values of N for problem 6. x
N=5
N = 10
N = 12
N = 14
0.0 0.2 0.4 0.6 0.8 1.0
4.3370e 005 5.4298e 006 2.2459e 005 2.0180e 005 1.0315e 005 5.7406e 005
1.0675e 007 4.1211e 008 1.8549e 008 1.6673e 008 2.4316e 008 7.5796e 008
1.2962e 010 6.9576e 012 3.0518e 011 2.4874e 011 2.5851e 009 3.0117e 009
1.1881e 012 4.0754e 013 2.9210e 013 2.1907e 013 1.6937e 013 4.6278e 013
L2 L1 RMS K2(A) K1(A)
1.1918e 005 1.0692e 005 5.9588e 006 3.74209e + 006 5.29853e + 006
2.9624e 008 2.6094e 008 9.8747e 009 4.2650e + 015 7.0544e + 015
3.6033e 011 3.1652e 011 1.0864e 011 2.9787e + 017 5.1076e + 017
3.3073e 013 2.8994e 013 9.1728e 014 1.5237e + 020 2.7419e + 020
Fig. 12. Absolute error between MQ-RBFs solution and exact solution for c = 2.5 and N = 14 of problem 6.
Fig. 13. Convergence in space for MQ-RBFs with c = 2.5 of problem 6, N is number of collocation points.
which is of great interest in hydrodynamics [33,67]. This problem has the exact solution given in [33] as
yðxÞ ¼ ln
2 cos2
h2 h 2
! x 12
;
ð25Þ
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here, h is the root of the equation
h ¼ 4 cos
h : 4
ð26Þ
Eq. (24) can be reformulated as the Hammerstein integral equation
yðxÞ ¼
Z
1
Kðx; tÞeyðtÞ dt;
ð27Þ
0
where
Kðx; tÞ ¼
tð1 xÞ; 0 6 t 6 x;
ð28Þ
xð1 tÞ; x 6 t 6 1:
We applied the new method proposed in the current paper to approximate the solution of Eq. (24) for different values of N. Table 9 shows the Error, L2-error, L1-error and RMS-error norms, K2(A) and K1(A) in some points of the interval [0, 1] obtained for N = 7, 10, 13, 15. Then in Fig. 14 the Error for N = 15 is represented. The convergence behavior of the MQ-RBFs method in terms of the L2-error norm versus reciprocal of number of collocation points N is shown in Fig. 15. 5.8. Problem 8 In this example we consider the mathematical model for an adiabatic tubular chemical reactor discussed in [68,69], which in the case of steady state solutions, can be stated as the ordinary differential equation
y00 ðxÞ ky0 ðxÞ þ Fðk; l; b; yðxÞÞ ¼ 0;
x 2 ½0; 1;
ð29Þ
Table 9 Absolute error between the exact and numerical solution with c = 2 for various value of N and 0 6 x 6 1; L2-norm error, L1-norm error, RMS error, condition number of s = 2,1 at various values of N for problem 7. x
N=7
N = 10
N = 13
N = 15
0.0 0.2 0.4 0.6 0.8 1.0
1.4876e 005 4.8786e 006 4.2779e 006 3.9278e 006 3.8000e 006 1.0022e 005
1.3448e 007 2.9107e 008 1.2368e 008 8.9588e 009 1.0984e 008 2.6040e 008
4.8773e 010 7.8393e 011 3.9109e 011 3.2126e 011 4.3877e 011 1.9084e 010
1.9743e 011 2.4293e 012 7.1551e 014 5.7228e 014 1.2811e 012 7.0232e 012
L2 L1 RMS K2(A) K1(A)
5.9703e 009 2.9327e 009 2.4374e 009 5.5040e + 008 9.8990e + 008
1.6664e 011 7.6147e 012 5.5546e 012 1.3164e + 013 2.4592e + 013
3.1465e 014 1.2283e 014 9.0833e 015 4.3266e + 016 7.8419e + 016
8.9736e 016 3.2744e 016 2.3983e 016 3.6542e + 019 7.4388e + 019
Fig. 14. Absolute error between MQ-RBFs solution and exact solution for c = 2 and N = 15 of problem 7.
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Fig. 15. Convergence in space for MQ-RBFs with c = 2 of problem 7, N is number of collocation points.
with boundary conditions
y0 ð0Þ ¼ kyð0Þ;
y0 ð1Þ ¼ 0;
ð30Þ
where
Fðk; l; b; yðxÞÞ ¼ klðb yðxÞÞeyðxÞ :
ð31Þ
The problem can be converted into a Hammerstein integral equation of the form [69]
yðxÞ ¼
Z
1
Kðx; tÞGðt; yðtÞÞdt;
ð32Þ
0
Table 10 Comparison between solution of Adomian [70] and Marzban et al. [39] and present solution with c = 1.8 for various value of N and 0 6 x 6 1 for problem 8. x
N = 10
N = 13
N = 15
Adomian [70]
Marzban et al. [39]
0.0 0.2 0.4 0.6 0.8 1.0
0.0060480837 0.0181927441 0.0304249181 0.0426694912 0.0543707757 0.0614628310
0.0060481548 0.0181928964 0.0304246478 0.0426691388 0.0543716843 0.0614588885
0.0060483431 0.0181929325 0.0304246703 0.0426691182 0.0543716557 0.0614587507
0.006048 0.018192 0.030424 0.042669 0.054371 0.061458
0.0060483739 0.0181929364 0.0304246702 0.0426691183 0.0543716533 0.0614587374
kResk2 K2(A) K1(A)
4.6241e 013 2.4771e + 012 4.8599e + 012
1.7420e 015 7.2428e + 014 1.4945e + 015
2.3401e 017 6.9386e + 018 1.5087e + 019
-
-
Fig. 16. Convergence in space for MQ-RBFs with c = 1.8 of problem 8, N is number of collocation points.
K. Parand, J.A. Rad / Applied Mathematics and Computation 218 (2012) 5292–5309
5307
where
Kðx; tÞ ¼
1; 0 6 t 6 x; kðxtÞ ; x6t61 e
ð33Þ
and
Gðt; yðtÞÞ ¼ lðb yðtÞÞeyðtÞ :
ð34Þ
The existence and uniqueness of the solution for this Hammerstein integral equation with respect to the values of parameters k, l, b are given in [69]. In [70], the Adomian’s method is used to solve the integral equation Eq. (32) for the particular values of the parameters k = 10, l = 0.02 and b = 3. Also, Marzban et al. [39] applied a composite collocation method for solving the integral equation Eq. (32) for the particular values of the parameters k = 10, l = 0.02 and b = 3. We applied the new method proposed in the current paper to approximate the solution of Eq. (32) for different values of N. Table 10 shows the comparison between solution of Adomian [70] and Marzban et al. [39] and presents the solution with c = 1.8 for various values of N and 0 6 x 6 1 obtained for N = 10, 13, 15. The convergence behavior of the MQ-RBFs method in terms of the kResk2 versus reciprocal of number of collocation points N is shown in Fig. 16. 6. Appropriate interval for shape parameter DRBFs uses differential operator which possesses a sharp noise. It means that if we make a small change to the shape parameter c, it makes a sharp noise will emerge in the solution of the problem [71,72]. For the limiting value of c, the condition number of the RBFs system becomes so large that the system leads to ill-conditioning. The condition number also grows with N for fixed values of the shape parameter c [20,73]. Generally for a fixed number of collocation points N, smaller values of c produce better approximations, but the matrix A will be more ill-conditioned [13,74]. When the number of mesh points is further increased after a critical value, the accuracy of numerical results is decreased [9,12]. The reason may be due to the fact that, when the number of mesh points increases, the condition number of the matrix becomes very large and the matrix tends to be ill-conditioned. For example, co-relation between the condition number of the matrix A and the different values of the shape parameter c is shown in Table 3 corresponding to the problem 1 for MQ-RBFs. The critical values of the shape parameter c in this case are 0.9 and 1.8 and the condition numbers of the matrix A corresponding to these values are given by 1.4723e + 016 and 6.9386e + 018, respectively. The interval of stability in this case is (0.9, 1.8). It is clear from the Table 3, that if the values of the shape parameter c are the outputs of the interval (0.9, 1.8), then the solution will down and hence the method will become unstable. This phenomenon is shown in Fig. 2, where L1-error norm is plotted against different values of the shape parameter c. 7. Conclusions and remarks In this study, we have applied an approximation technique to solve Volterra–Fredholm–Hammerstein integral equations. The method is based on the collocation method and multiquadric radial basis function. We used an irregular mesh-grid in the RBF collocation instead of finite difference method which enhanced our options. Additionally, through the comparison with exact solutions we show that the RBFs methods have good reliability and efficiency. Also high convergence rates and good accuracy are obtained with the proposed method using relatively low numbers of data points. Acknowledgements The authors are very grateful to the reviewers and the subject editor for carefully reading the paper and for their comments and suggestions which have greatly improved the paper. The research of first author (K. Parand) was supported by a grant from Shahid Beheshti University. References [1] G.E. Fasshauer, Meshfree Approximation Methods with MATLAB, Word Scientific Publishing, 2007. [2] E.J. Kansa, Multiquadrics-A scattered data approximation scheme with applications to computational fluid-dynamics-I surface approximations and partial derivative estimates, Comput. Math. Appl. 19 (1990) 127–145. [3] M. Sharan, E.J. Kansa, S. Gupta, Application of the multiquadric method for numerical solution of elliptic partial differential equations, Appl. Math. Comput. 84 (1997) 275–302. [4] M. Zerroukat, H. Power, C.S. Chen, A numerical method for heat transfer problems using collocation and radial basis functions, Int. J. Numer. Meth. Eng. 42 (1998) 1263–1278. [5] N. Mai-Duy, T. Tran-Cong, Numerical solution of differential equations using multiquadric radial basis function networks, Neural Netw 14 (2001) 185– 199. [6] M. Tatari, M. Dehghan, A method for solving partial differential equations via radial basis functions: application to the heat equation, Eng. Anal. Bound. Elem. 34 (2010) 206–212. [7] M. Dehghan, A. Shokri, Numerical solution of the nonlinear Klein–Gordon equation using radial basis functions, J. Comput. Appl. Math. 230 (2009) 400– 410. [8] A. Alipanah, M. Dehghan, Numerical solution of the nonlinear Fredholm integral equations by positive definite functions, Appl. Math. Comput. 190 (2007) 1754–1761.
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Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Numerical solution of nonlinear Volterra–Fredholm–Hammerstein integral equations via collocation method based on radial basis functions K. Parand, J.A. Rad ⇑ Department of Computer Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Evin, Tehran 19839, Iran
a r t i c l e
i n f o
Keywords: Volterra–Fredholm–Hammerstein Collocation method Radial basis functions Legendre polynomials Nonlinear integral equations
a b s t r a c t A numerical technique based on the spectral method is presented for the solution of nonlinear Volterra–Fredholm–Hammerstein integral equations. This method is a combination of collocation method and radial basis functions (RBFs) with the differentiation process (DRBF), using zeros of the shifted Legendre polynomial as the collocation points. Different applications of RBFs are used for this purpose. The integral involved in the formulation of the problems are approximated based on Legendre–Gauss–Lobatto integration rule. The results of numerical experiments are compared with the analytical solution in illustrative examples to confirm the accuracy and efficiency of the presented scheme. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction In recent decades, the so-called meshless methods have been extensively used to find approximate solutions of various types of linear and nonlinear equations [1] such as differential equations (DEs) and integral equations (IEs). Unlike the other methods which were used to mesh the domain of the problem, meshless methods don’t require a structured grid and only make use of a scattered set of collocation points regardless of the connectivity information between the collocation points. For the last years, the radial basis functions (RBFs) method was known as a powerful tool for the scattered data interpolation problem. The main advantage of numerical methods which use radial basis functions is the meshless characteristic of these methods. The use of radial basis functions as a meshless method for the numerical solution of ordinary differential equations (ODEs) and partial differential equations (PDEs) is based on the collocation method. One of the domain-type meshless methods is given in [2] in 1990, which directly collocates radial basis functions, particularly the multiquadric (MQ), to find an approximate solution of linear and nonlinear DEs. Kansa’s method has recently received a great deal of attention from researchers [3–8]. Recently, Kansa’s method was extended to solve various ordinary and partial differential equations including the nonlinear Klein–Gordon equation [7], regularized long wave (RLW) equation [9], high order ordinary differential equations [10], the case of heat transfer equations [11], Hirota–Satsuma coupled KdV equations [12], second-order parabolic equation with nonlocal boundary conditions [13], second-order hyperbolic telegraph equation [14] and so on [15–18]. All of the radial basis functions have global support, and in fact many of them, such as multiquadrics (MQ), do not even have isolated zeros [7,9,19]. The RBFs can be compactly and globally supported, infinitely differentiable, and contain a free parameter c, called the shape parameter [9,19,20]. The interested reader is referred to the recent books and paper by Buhmann [19,21] and Wendland [22] for more basic details about RBFs, compactly and globally supported and convergence rate of the radial basis functions.
⇑ Corresponding author. E-mail addresses: [email protected] (K. Parand), [email protected] (J.A. Rad). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.11.013
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Despite many research works which are done to finding algorithms for selecting the optimum values of c [23–25], the optimal choice of shape parameter is an open problem which is still under intensive investigation. For example, Carlson and Foley [24] found that the shape parameter is problem dependent. They [24] observed that for rapidly varying functions, a small value of c should be used, but a large value should be used if the function has a large curvature. Tarwater [25] found that by increasing c, the Root-Mean-Square (RMS) of error dropped to a minimum and then increased sharply afterwards. In general, as c increases, the system of equations to be solved becomes ill-conditioned. Cheng et al. [23] showed that when c is very large then the RBFs system error is of exponential convergence. But there is a certain limit for the value c after which the solution breaks down. In general, as the value of the shape parameter c increases, the matrix of the system to be solved becomes highly ill-conditioned and hence the condition number can be used in determining the critical value of the shape parameter for an accurate solution [23]. For some new work on optimal choice of shape parameter, we refer the interested reader to the recent work of [26]. There are two basic approaches for obtaining basis functions from RBFs, namely direct approach (DRBF) based on a differential process [27] and indirect approach (IRBF) based on an integration process [5,10,28]. Both approaches were tested on the solution of second order DEs and the indirect approach was found to be superior to the direct approach [5]. Some of the infinitely smooth RBFs choices are listed in Table 1. The RBFs can be of various types, for example: polynomials of any chosen degree such as linear, cubic, etc., thin plate spline (TPS), multiquadrics (MQ), inverse multiquadrics (IMQ), Gaussian forms (GA), hyperbolic secant (sech) form etc. In the cases of inverse quadratic, inverse multiquadric (IMQ), hyperbolic secant (sech) and Gaussian (GA), the coefficient matrix of RBFs interpolating is positive definite and, for multiquadric (MQ), it has one positive eigenvalue and the remaining ones are all negative [29]. In this paper, we use the multiquadrics direct radial basis function for finding the solution of Volterra–Feredholm–Hemmerstein equations. The MQ was first developed by Hardy [30] in 1971 as a multidimensional scattered interpolation method in modeling of the earth gravitational field. It was not recognized by most of the academic researchers until Franke [31] published a review paper on the evaluation of two-dimensional interpolation methods, whereas MQ was ranked as the best based on its accuracy, visual aspect, sensitivity to parameters, execution time, storage requirements, and ease of implementation. There is a considerable approximation method that discusses approximating the solution of linear and nonlinear mixed Hammerstein integral equations [32–39]. For Fredholm–Hammerstein integral equations, the classical method of successive approximations was introduced in [32]. A variation of the Nystrom method was presented in [40], and a collocation-type method was developed in [33]. Marzban et al. [39] solved the nonlinear Volterra-Fredholm–Hammerstein integral equation by using the Hybrid of block-pulse functions. In [34], Brunner applied a collocation-type method to nonlinear Volterra–Hammerstein integral equations and integro-differential equations and discussed its connection with the iterated collocation method. Han [41] introduced and discussed the asymptotic error expansion of a collocation-type method for Volterra–Hammerstein integral equations. The existence of the solutions to nonlinear Hammerstein integral equations was discussed in [42]. Ezquerro and Hernández applied new fourth-order iterative methods to obtain solution of Hammerstein integral equations [43]. In [44] a numerical procedure based on Sinc basis functions was presented for solving a class of nonlinear integral equations of the Hammerstein-type. For more references about numerical methods for Volterra–Hammerstein integral equations see [75–79]. Maleknejad et al. [45] provided a cubic semiorthogonal compactly supported B-spline wavelets as a basis functions for solution of nonlinear Fredholm–Hammerstein integral equations of the second kind. More recently, Ghoreishi and Hadizadeh [46] gave an extension of the algebraic formulation of the Tau method for the numerical solution of the nonlinear Volterra– Hammerstein integral equations. Also, Maleknejad el al. applied the various numerical methods for solving Hammerstein integral equations [47–50]. For more references about Volterra–Hammerstein integral equations see [51–54]. Several authors considered the nonlinear mixed Volterra–Fredholm integral equation of the form
yðxÞ ¼ f ðxÞ þ k1
Z
x
K 1 ðx; tÞG1 ðt; yðtÞÞdt þ k2
0
Z
1
K 2 ðx; tÞG2 ðt; yðtÞÞdt;
0 6 x 6 1;
ð1Þ
0
where k1 and k2 are constants and f(x) and the kernels K1(x, t) and K2(x, t) are given functions assumed to have nth derivatives on the interval 0 6 x 6 1 and 0 6 t 6 1. Yalcinbas [35] applied Taylor series and Bildik and Inc [36] used the modified decomposition method to find the solution of Eq. (1) for the case G1(t, y(t)) = yp(t) and G2(t, y(t)) = yq(t), where p and q are nonnegative integers. For the case G1(t, y(t)) = F1(y(t)) and G2(t, y(t)) = F2(y(t)), where F1(y(t)) and F2(y(t)) are given continuous functions which are nonlinear with respect to y(t), Yousefi and Razzaghi [37] applied Legendre wavelets to obtain the
Table 1 Some well-known radial basis functions (r = kx xik2 = ri), c > 0. Name of functions
Inverse multiquadrics (IMQ)
Definition pffiffiffiffiffiffiffiffiffiffiffiffiffi r þ c2 pffiffiffiffiffiffiffiffiffiffiffiffiffi 1= r þ c2
Gaussian (GA) Hyperbolic secant (sech)
exp(cr) pffiffiffi sech c r
Multiquadrics (MQ)
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solution, and for the general case, where G1(t, y(t)) and G2(t, y(t)) are given continuous functions which are nonlinear with respect to t and y(t), Ordokhani [38] applied the rationalized Haar functions to get the solution. Also in this case, Marzban et al. [39] solved Eq. (1) by using the Hybrid of block-pulse functions. In this paper we propose a meshless collocation method based on the MQ-RBFs to solve the nonlinear Volterra–Fredholm–Hammerstein integral equations of the form of Eq. (1). For convenience the solution we use RBFs with collocation nodes fxj gNj¼1 which are the zeros of the shifted Legendre polynomial LN(x), 0 6 x 6 1. The shifted Legendre polynomials Li(x) are defined on the interval [0, 1] and satisfy the following formulae [55]:
L0 ðxÞ ¼ 1;
L1 ðxÞ ¼ 2x 1;
2i þ 1 i Liþ1 ðxÞ ¼ ð2x 1ÞLi ðxÞ Li1 ðxÞ; iþ1 iþ1
i ¼ 1; 2; 3; . . . :
ð2Þ
This paper is arranged as follows: in Section 2, we describe the properties of radial basis functions. In Section 3, we introduce the Legendre–Gauss–Lobatto nodes and weights. In Section 4 we implement the problem with the proposed method and in Section 5, we report our numerical finding and demonstrate the accuracy of the proposed methods. In Section 6 we apply a numerical algorithm [9,12] for finding the optimum shape parameter. The conclusions are discussed in the final Section. 2. Radial basis function 2.1. Definition of radial basis function Let Rþ ¼ fx 2 R; x P 0g be the non-negative half-line and let / : Rþ ! R be a continuous function with /(0) P 0. A radial basis functions on Rd is a function of the form
/ðkX X i kÞ; where X; X i 2 Rd and k.k denotes the Euclidean distance between X and Xis. If one chooses N points fX i gNi¼1 in Rd then by custom
sðXÞ ¼
N X
ki /ðkX X i kÞ;
ki 2 R
i¼1
is called a radial basis functions as well [56,57]. The standard radial basis functions are categorized into two major classes [12]: Class 1. Infinitely smooth RBFs [12,58]: These basis functions are infinitely differentiable and heavily depend on the shape parameter c e.g. Hardy multiquadric (MQ), Gaussian (GA), inverse multiquadric (IMQ), and inverse quadric (IQ) (See Table 1). Class 2. Infinitely smooth (except at centers) RBFs [12,58]: The basis functions of this category are not infinitely differentiable. These basis functions are shape parameter free and have comparatively less accuracy than the basis functions discussed in the Class 1. For example, thin plate spline, etc. [12]. 2.2. RBF interpolation The one dimensional function y(x) to be interpolated or approximated can be represented by an RBFs as:
yðxÞ yN ðxÞ ¼
N X
ki /i ðxÞ ¼ UT ðxÞK;
ð3Þ
i¼1
where
/i ðxÞ ¼ uðkx xi kÞ; UT ðxÞ ¼ ½/1 ðxÞ; /2 ðxÞ; . . . ; /N ðxÞ; K ¼ ½k 1 ; k 2 ; . . . ; k N T ; fki gNi¼1
x is the input and are the set of coefficients to be determined [8]. By choosing N interpolate nodes approximate the function y(x).
yj ¼
N X
ki /i ðxj Þ;
j ¼ 1; 2; . . . ; N:
i¼1
To brief discussion on coefficient matrix, we define:
AK ¼ Y;
ð4Þ fxi gNi¼1 ,
we can
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where
y ¼ ½y1 ; y2 ; . . . ; yN T ; A ¼ ½UT ðx1 Þ; UT ðx2 Þ; . . . ; UT ðxN ÞT ; 3 2 /1 ðx1 Þ /2 ðx1 Þ . . . /N ðx1 Þ 6 / ðx2 Þ / ðx2 Þ . . . / ðx2 Þ 7 2 N 7 6 1 7: ¼6 . .. .. 7 6 .. . 5 4 . . . .
ð5Þ
/1 ðxN Þ /2 ðxN Þ . . . /N ðxN Þ Note that /i(xj) = u(kxi xjk) therefore we have /i(xj) = /j(xi) consequently A = AT. All the infinitely smooth RBFs choices are listed in Table 1 will give coefficient matrices A in Eq. (5) which are symmetric and nonsingular [29], i.e. there is a unique interpolant of the form Eq. (3) no matter how the distinct data points are scattered in any number of space dimensions. In the cases of inverse quadratic, inverse multiquadric (IMQ), hyperbolic secant (sech) and Gaussian (GA) the matrix A is positive definite and, for multiquadric (MQ), it has one positive eigenvalue and the remaining ones are all negative [29]. We have the following theorem about the convergence of RBFs interpolation: Theorem 1 (See [59,60]). Assume fxi gN i¼1 are N nodes in X which is convex, let
h ¼ max min kx xi k2 ; x2X
16i6N
R ^ gÞ < cð1 þ jgjÞð2lþdÞ for any y(x) satisfies ðy ^ gÞdg < 1 we have ^ðgÞÞ2 =/ð when /ð
ða Þ la yN yðaÞ 6 ch ; 1
where /(x) is RBFs and the constant c depends on the RBFs, d is space dimension, l and a are nonnegative integer. It can be seen that not only RBFs itself but also its any order derivative has a good convergence. Proof. A complete proof is given by authors [59,60]. The results of this section can be summarized in the following algorithm. Algorithm. The algorithm works in the following manner: 1. 2. 3. 4. 5.
Choose N center points fxj gNj¼1 from the domain set [a, b]. Approximate y(x) as yN(x) = UT(x)K. Substitute yN(x) into the main problem and create residual function Res(x). Substitute collocation points fxj gNj¼1 into the Res(x) and create the N equations. Solve the N equations with N unknown coefficients of members of K and find the numerical solution.
3. Legendre–Gauss–Lobatto nodes and weights Let HN [1, 1] denote the space of algebraic polynomials of degree 6N
hP i ; Pj i ¼
2 dij : 2j þ 1
ð6Þ
Here, h.,.i represents the usual L2[1, 1] inner product and {Pi}iP0 are the well-known Legendre polynomials of order i which are orthogonal with respect to the weight function w(x) = 1 on the interval [1, 1], and satisfy the following formulae:
P0 ðxÞ ¼ 1; P1 ðxÞ ¼ x; 2i þ 1 i xPi ðxÞ P i1 ðxÞ; Piþ1 ðxÞ ¼ iþ1 iþ1
i ¼ 1; 2; 3; . . . :
Next, we let fxj gNj¼0 as:
_ j Þ ¼ 0; ð1 x2j ÞPðx 1 ¼ x0 < x1 < x2 < . . . < xN ¼ 1;
ð7Þ
_ where PðxÞ is derivative of P(x). No explicit formula for the nodes fxj gN1 j¼1 is known. However, they are computed numerically using the existing subroutines [61,62]. Now, we assume f 2 H2N1 ½1; 1, we have
Z
1
1
f ðxÞdx ’
N X j¼0
xj f ðxj Þ ¼ IG ðf Þ;
ð8Þ
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where xj are the Legendre–Gauss–Lobatto weights given in [55]
xj ¼
2 1 : NðN þ 1Þ ðPN ðxj ÞÞ2
ð9Þ
4. Solution of nonlinear Volterra–Fredholm–Hammerstein integral equations via DRBF Consider the nonlinear Volterra–Fredholm–Hammerstein integral equations given in Eq. (1). In order to use the radial basis functions, we first approximate y(x) as
yðxÞ ’ yN ðxÞ ¼
N X
ki /i ðxÞ ¼ UT ðxÞK;
ð10Þ
i¼1
where UT(x) and K are defined in Eq. (4). Then, from substituting Eq. (10) into Eq. (1), we have
UT ðxÞK ¼ f ðxÞ þ k1
Z
x
K 1 ðx; tÞG1 ðt; UT ðtÞKÞdt þ k2
0
Z
1
K 2 ðx; tÞG2 ðt; UT ðtÞKÞdt:
ð11Þ
0
We now collocate Eq. (11) at points fxi gNi¼1 as
UT ðxi ÞK ¼ f ðxi Þ þ k1
Z
xi
K 1 ðxi ; tÞG1 ðt; UT ðtÞKÞdt þ k2
Z
1
K 2 ðxi ; tÞG2 ðt; UT ðtÞKÞdt:
ð12Þ
0
0
In order to use the Legendre–Gauss–Lobatto integration formula [55] for Eq. (12), we transfer the t-intervals [0, xi] and [0, 1] into the g1 and g2-intervals [1, 1], respectively, by means of the transformations g1 ¼ x2i t 1 and g2 = 2t 1. Let
H1 ðxi ; tÞ ¼ K 1 ðxi ; tÞG1 ðt; UT ðtÞKÞ
ð13Þ
H2 ðxi ; tÞ ¼ K 2 ðxi ; tÞG2 ðt; UT ðtÞKÞ:
ð14Þ
and
Eq. (12) may then be restated as
UT ðxi ÞK ¼ f ðxi Þ þ k1
xi 2
Z
Z x k2 1 1 i H1 xi ; ðg1 þ 1Þ dg1 þ H2 ðxi ; ðg2 þ 1ÞÞdg2 ; 2 2 2 1 1 1
by using the Legendre–Gauss–Lobatto integration formula [55] we have the residual function Res(x)
Resðxi Þ ¼ UT ðxi ÞK þ f ðxi Þ þ k1
r1 r2 x k X xi X 1 2 i w1j H1 xi ; ðg1j þ 1Þ þ w2j H2 xi ; ðg2j þ 1Þ ; 2 2 j¼0 2 2 j¼0
i ¼ 1 . . . N:
ð15Þ
Eq. (15) generates an N set of nonlinear equations which can be solved by mathematical software for the unknowns K. 5. Numerical results In order to illustrate the performance of the RBFs method in solving Hammerstein integral equations and justify the accuracy and efficiency of the method presented in this paper, we consider the following examples. In all examples, we use the multiquadrics (MQ) RBFs. To study the convergence behavior of the MQ-RBFs method, we applied the following laws: (1) Absolute error (Error) between the exact solution and the present solution defined by
Error ¼ yexact ðxÞ yimplicit ðxÞ ; N
x 2 ½0; 1;
(2) The L2 error norm of the solution which is defined by
" # N 2 1=2 X exact implicit implict exact L2 ¼ y ðxÞ yN ðxÞ ¼ y ðxj Þ yN ðxj Þ ; 2
j¼1
fxj gNj¼1
where are interpolate nodes which are the zeros of the shifted Legendre polynomial LN(x), 0 6 x 6 1. (3) The L1 error norm of the solution which is defined by
L1 ¼ yexact ðxÞ yimplicit ðxÞ ¼ maxj yexact ðxj Þ yimplicit ðxj Þ ; N N 1
where 1 6 j 6 N.
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(4) The root mean square (RMS) is defined by
RMS ¼
" # N 2 1=2 X 1 yexact ðxj Þ yimplict ðx Þ : j N N 1 j¼1
(5) The condition number Ks(A) of the coefficient matrix A is given by
K s ðAÞ ¼ kAks kA1 ks ;
s ¼ 2; 1:
The condition number Ks(A) has the following properties: The condition number shows that a small perturbation in initial data may produce a large amount of perturbation in the solution [13]. The condition number of an interpolation matrix gives information on the numerical stability of the interpolation process. The condition number depends more on the separation distance than on the number N of centers [22]. The condition number grows with N for fixed values of the shape parameter c. In practices, the shape parameter must be adjusted with the number of interpolating points in order to produce an interpolation matrix that is well-conditioned enough to be inverted in finite precision arithmetic [20]. The condition number of the system matrix can be used in determining the optimal value of the shape parameter c for an accurate solution [63]. When the number of mesh points increases, the condition number of the matrix becomes very large and the matrix tends to be ill-conditioned [13]. The numerical implementation is carried out in microsoft.maple.13, with hardware configuration: desktop 64-bit Intel Core 2 Duo CPU, 4 GB of RAM, 64-bit Operating System (Windows Vista). 5.1. Problem 1 Consider the following Hammerstein integral equation with the exact solution y(x) = ex [35–37].
1 1 yðxÞ ¼ ex e3x þ þ 3 3
Z
x
y3 ðtÞdt:
ð16Þ
0
We solve Eq. (16) for different values of N. Table 2 shows the absolute error (Error) between the exact and MQ-RBFs solutions, L2-error and L1-error, RMS-error norms, K2(A) and K1(A) in some points of the interval [0, 1] obtained for N = 5, 8, 10, 15. This table indicates that as N increases, the Error decreases more rapidly. From Table 2, it can be observed that the accuracy increases with the increase of number of collocation points. Accuracy of the present approximation is examined in the L2-error and L1-error, RMS-error norms. The results of RMS-error are comparatively better than the L2-error and L1error norms. Fig. 1 represented the Error for N = 15. It shows the decreasing of the absolute error Error by collocation method based MQ-RBFs. The convergence behavior of the MQ-RBFs method in terms of the L2-error norm versus reciprocal of number of collocation points N is shown in Fig. 3. The Fig. 3 shows that the L2-error norm decreases by increasing the number of collocation points N for MQ-RBFs. 5.2. Problem 2 Consider the nonlinear Volterra–Fredholm–Hammerstein integral equation with the exact solution y(x) = cos (x) [64,65]. 2
yðxÞ ¼ 1 þ sin ðxÞ þ
Z
1
Kðx; tÞy2 ðtÞdt;
ð17Þ
0
Table 2 Absolute error between the exact and numerical solution with c = 1.8 for various value of N and 0 6 x 6 1; L2-norm error, L1-norm error, RMS error, condition number of s = 2,1 at various values of N for problem 1. x
N=5
N=8
N = 10
N = 15
0.0 0.2 0.4 0.6 0.8 1.0
2.6866e 003 7.4963e 004 7.3269e 004 1.9736e 004 1.3495e 004 2.7461e 003
6.1015e 005 1.5098e 005 7.0374e 006 3.4187e 006 2.0792e 006 1.1342e 004
6.8393e 007 1.2665e 007 7.7416e 008 6.6926e 009 2.7495e 008 6.9435e 008
1.4562e 010 2.0810e 011 4.9287e 012 5.6513e 012 2.1146e 012 2.0294e 011
L2 L1 RMS K2(A) K1(A)
1.2650e 003 1.0908e 003 6.3250e 004 5.0348e + 005 8.0345e + 005
8.8171e 005 8.6438e 005 3.3325e 005 2.1713e + 010 3.9677e + 010
7.1095e 008 3.1291e 008 2.3698e 008 2.4771e + 012 4.8599e + 012
1.4373e 011 5.5922e 012 3.8412e 012 6.9386e + 018 1.5087e + 019
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Fig. 1. Absolute error between MQ-RBFs solution and exact solution for c = 1.8 and N = 15 of problem 1.
Fig. 2. Error plot versus shape parameter c for N = 15 of problem 1.
Fig. 3. Convergence in space for MQ-RBFs with c = 1.8 of problem 1, N is number of collocation points.
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K1(A)
L1 error
c
K1(A)
L1 error
0.7 0.8 0.9 1.0 1.1 1.2 1.3
1.4135e + 015 5.7373e + 015 1.4723e + 016 2.8363e + 016 4.6954e + 016 8.0433e + 016 1.5891e + 017
1.7023e 009 3.0839e 009 3.3808e 009 2.1572e 009 9.8427e 010 4.1733e 010 1.6781e 010
1.4 1.5 1.6 1.7 1.8 1.9 2.0
3.1764e + 017 6.2056e + 017 1.3559e + 018 3.0431e + 018 6.9386e + 018 1.5929e + 019 3.6542e + 019
7.6028e 011 3.8676e 011 1.9269e 011 1.0133e 011 5.5922e 012 3.1229e + 001 3.3107e + 001
where
Kðx; tÞ ¼
3 sinðx tÞ; 0 6 t 6 x; 0;
ð18Þ
x 6 t 6 1:
We applied our method and solved Eq. (17) for different values of N. Table 4 shows the Error, L2-error, L1-error and RMSerror norms, K2(A) and K1(A) in some points of the interval [0, 1] obtained for N = 5, 10, 15, 18. Then in Fig. 4 the absolute error (Error) for N = 18 is represented. The convergence behavior of the MQ-RBFs method in terms of the L2-error norm versus reciprocal of number of collocation points N is shown in Fig. 5. 5.3. Problem 3 Consider the nonlinear Volterra–Fredholm–Hammerstein integral equation of the form [35,37–39,66].
yðxÞ ¼ gðxÞ þ
Z
x
ðx tÞy2 ðtÞdt þ
0
Z
1
ðx þ tÞyðtÞdt;
ð19Þ
0
Table 4 Absolute error between the exact and numerical solution with c = 2 for various value of N and 0 6 x 6 1; L2-norm error, L1-norm error, RMS error, condition number of s = 2,1 at various values of N for problem 2. x
N=5
N = 10
N = 15
N = 18
0.0 0.2 0.4 0.6 0.8 1.0
9.7774e 005 1.4953e 005 3.0243e 005 2.7392e 005 1.1666e 005 6.1572e 005
2.3698e 009 2.5583e 010 4.3742e 011 1.9526e 010 5.3780e 010 2.4312e 009
2.6890e 012 3.2997e 013 9.4818e 015 7.9988e 015 1.6686e 013 8.9004e 013
2.4514e 014 4.1423e 015 3.3900e 015 2.4645e 015 1.6125e 015 5.3448e 015
L2 L1 RMS K2(A) K1(A)
1.9697e 006 1.8373e 006 9.8485e 007 9.2332e + 005 1.4219e + 006
3.3018e 012 2.0912e 012 1.1006e 012 1.3164e + 013 2.4592e + 013
8.3896e 016 3.3923e 016 2.2422e 016 3.6542e + 019 7.4388e + 019
4.5392e 018 1.7346e 018 1.1009e 018 7.3870e + 023 1.5798e + 024
Fig. 4. Absolute error between MQ-RBFs solution and exact solution for c = 2 and N = 18 of problem 2.
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Fig. 5. Convergence in space for MQ-RBFs with c = 2 of problem 2, N is number of collocation points.
where
gðxÞ ¼
1 6 1 4 5 5 x þ x x2 þ x ; 30 3 3 4
ð20Þ
which has the exact solution y(x) = x2 2. We applied the RBFs approach and solved Eq. (19). Table 5 shows the Error, L2error, L1-error and RMS-error norms, K2(A) and K1(A) in some points of the interval [0,1] obtained for N = 10,14,16,20. Fig. 6 represents the Error for N = 14. The convergence behavior of the MQ-RBFs method in terms of the L2-error norm versus reciprocal of number of collocation points N is shown in Fig. 7. 5.4. Problem 4 Consider the nonlinear Volterra–Fredholm–Hammerstein integral equation given in [38,39,66] by
yðxÞ ¼ 2 cosðxÞ 2 þ 3
Z
x
sinðx tÞy2 ðtÞdt þ
0
6 7 6 cosð1Þ
Z
1
ð1 tÞ cos2 ðxÞðt þ yðtÞÞdt;
ð21Þ
0
which has the exact solution y(x) = cos(x). We solve Eq. (21) for different values of N. Table 6 shows the Error, L2-error, L1-error and RMS-error norms, K2(A) and K1(A) norms in some points of the interval [0, 1] obtained for N = 5, 10, 14, 19. Then in Fig. 8 the Error for N = 14 is represented. The convergence behavior of the MQ-RBFs method in terms of the L2-error norm versus reciprocal of number of collocation points N is shown in Fig. 9. 5.5. Problem 5 Consider the nonlinear Volterra–Hammerstein integral equation given in [52] by
yðxÞ ¼
1 4 5 2 3 x þ x þ þ 10 6 8
Z
x
0
1 2 y ðtÞdt; 2x
ð22Þ
Table 5 Absolute error between the exact and numerical solution with c = 2 for various value of N and 0 6 x 6 1; L2-norm error, L1-norm error, RMS error, condition number of s = 2,1 at various values of N for problem 3. x
N = 10
N = 14
N = 16
N = 20
0.0 0.2 0.4 0.6 0.8 1.0
1.9247e 008 4.1585e 009 2.5313e 009 1.9238e 009 1.8319e 009 5.0722e 009
3.1852e 011 5.5758e 012 3.0841e 012 1.5462e 012 6.3014e 013 3.1567e 013
1.1713e 013 3.5184e 015 2.9987e 014 3.6562e 014 1.3020e 014 1.1518e 013
1.4281e 015 2.2529e 016 1.1229e 016 7.5228e 017 6.8774e 017 2.0931e 016
L2 L1 RMS K2(A) K1(A)
8.4109e 012 3.7303e 012 2.6598e 012 3.3021e + 013 4.1241e + 013
6.0600e 015 2.8521e 015 1.6807e 015 1.8008e + 018 3.6587e + 018
4.5610e 017 2.2962e 017 1.1776e 017 1.0935e + 021 2.2940e + 021
1.2417e 019 4.6900e 020 2.8486e 020 5.2299e + 026 1.1443e + 027
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Fig. 6. Absolute error between MQ-RBFs solution and exact solution for c = 2 and N = 14 of problem 3.
Fig. 7. Convergence in space for MQ-RBFs with c = 2 of problem 3, N is number of collocation points.
Table 6 Absolute error between the exact and numerical solution with c = 2 for various value of N and 0 6 x 6 1; L2-norm error, L1-norm error, RMS error, condition number of s = 2,1 at various values of N for problem 4. x
N=5
N = 10
N = 14
N = 19
0.0 0.2 0.4 0.6 0.8 1.0
1.0572e 004 7.7160e 006 3.8609e 005 1.4363e 005 3.0035e 005 4.1888e 005
2.3692e 009 2.5393e 010 4.7045e 011 1.9227e 010 5.3574e 010 2.4327e 009
4.8595e 012 1.5440e 012 1.5124e 012 1.4149e 012 1.2960e 014 3.9911e 013
6.2514e 015 1.6328e 017 6.4699e 016 4.4481e 016 5.2436e 018 1.0257e 015
L2 L1 RMS K2(A) K1(A)
3.1710e 005 1.7367e 005 1.5855e 005 9.2332e + 005 1.4219e + 006
3.1991e 012 1.9265e 012 1.0663e 012 1.3164e + 013 2.4592e + 013
2.9194e 015 1.1718e 015 8.0970e 016 1.8008e + 018 3.6587e + 018
9.2477e 019 3.4760e 019 2.1797e 019 1.9249e + 025 4.1728e + 025
which has the exact solution yðxÞ ¼ x2 þ 12. We solve Eq. (22) for different values of N. Table 7 shows the Error, L2-error, L1error and RMS-error norms, K2(A) and K1(A) in some points of the interval [0, 1] obtained for N = 6, 9, 11, 14. Then in Fig. 10 the Error for N = 14 is represented. The convergence behavior of the MQ-RBFs method in terms of the L2-error norm versus reciprocal of number of collocation points N is shown in Fig. 11.
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Fig. 8. Absolute error between MQ-RBFs solution and exact solution for c = 2 and N = 14 of problem 4.
Fig. 9. Convergence in space for MQ-RBFs with c = 2 of problem 4, N is number of collocation points.
Table 7 Absolute error between the exact and numerical solution with c = 2 for various value of N and 0 6 x 6 1; L2-norm error, L1-norm error, RMS error, condition number of s = 2,1 at various values of N for problem 5. x
N=6
N=9
N = 11
N = 14
0.0 0.2 0.4 0.6 0.8 1.0
5.2922e 004 7.8748e 005 2.3950e 005 5.0644e 006 7.0737e 005 1.9844e 004
1.4841e 006 7.9423e 008 2.1415e 007 6.6108e 008 8.4048e 008 2.0520e 007
3.2574e 008 2.0604e 009 7.9366e 010 2.8076e 009 2.5851e 009 3.0117e 009
5.1507e 011 6.4333e 012 3.8578e 012 2.4023e 012 1.6677e 012 1.0692e 012
L2 L1 RMS K2(A) K1(A)
1.8018e 004 1.6275e 004 8.0580e 005 6.0029e + 007 9.2776e + 007
5.0168e 007 4.5471e 007 1.7737e 007 1.8699e + 011 3.6261e + 011
1.0971e 008 9.9696e 009 3.4694e 009 3.3020e + 013 4.1242e + 013
1.7275e 011 1.5740e 011 4.7912e 012 1.8007e + 018 3.6586e + 018
K. Parand, J.A. Rad / Applied Mathematics and Computation 218 (2012) 5292–5309
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Fig. 10. Absolute error between MQ-RBFs solution and exact solution for c = 2 and N = 14 of problem 5.
Fig. 11. Convergence in space for MQ-RBFs with c = 2 of problem 5, N is number of collocation points.
5.6. Problem 6 Consider the nonlinear Volterra–Hammerstein integral equation given in [52] by
yðxÞ ¼ x þ
1 ðxþ1Þ e 1 þ xþ1
Z 0
x
1 t=x yðtÞ e e dt; x
ð23Þ
which has the exact solution y(x) = x. We solve Eq. (23) for different values of N. Table 8 shows the Error, L2-error, L1-error and RMS-error norms, K2(A) and K1(A) in some points of the interval [0, 1] obtained for N = 5, 10, 12, 14. Then in Fig. 12 the Error for N = 14 is represented. The convergence behavior of the MQ-RBFs method in terms of the L2-error norm versus reciprocal of number of collocation points N is shown in Fig. 13. 5.7. Problem 7 In this example the collocation approximation based on RBFs is used to solve the integral equation reformulation of the nonlinear two-point boundary value problem (BVP)
(
y00 ðxÞ eyðxÞ ¼ 0; yð0Þ ¼ yð1Þ ¼ 0;
x 2 ½0; 1;
ð24Þ
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Table 8 Absolute error between the exact and numerical solution with c = 2.5 for various value of N and 0 6 x 6 1; L2-norm error, L1-norm error, RMS error, condition number of s = 2,1 at various values of N for problem 6. x
N=5
N = 10
N = 12
N = 14
0.0 0.2 0.4 0.6 0.8 1.0
4.3370e 005 5.4298e 006 2.2459e 005 2.0180e 005 1.0315e 005 5.7406e 005
1.0675e 007 4.1211e 008 1.8549e 008 1.6673e 008 2.4316e 008 7.5796e 008
1.2962e 010 6.9576e 012 3.0518e 011 2.4874e 011 2.5851e 009 3.0117e 009
1.1881e 012 4.0754e 013 2.9210e 013 2.1907e 013 1.6937e 013 4.6278e 013
L2 L1 RMS K2(A) K1(A)
1.1918e 005 1.0692e 005 5.9588e 006 3.74209e + 006 5.29853e + 006
2.9624e 008 2.6094e 008 9.8747e 009 4.2650e + 015 7.0544e + 015
3.6033e 011 3.1652e 011 1.0864e 011 2.9787e + 017 5.1076e + 017
3.3073e 013 2.8994e 013 9.1728e 014 1.5237e + 020 2.7419e + 020
Fig. 12. Absolute error between MQ-RBFs solution and exact solution for c = 2.5 and N = 14 of problem 6.
Fig. 13. Convergence in space for MQ-RBFs with c = 2.5 of problem 6, N is number of collocation points.
which is of great interest in hydrodynamics [33,67]. This problem has the exact solution given in [33] as
yðxÞ ¼ ln
2 cos2
h2 h 2
! x 12
;
ð25Þ
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here, h is the root of the equation
h ¼ 4 cos
h : 4
ð26Þ
Eq. (24) can be reformulated as the Hammerstein integral equation
yðxÞ ¼
Z
1
Kðx; tÞeyðtÞ dt;
ð27Þ
0
where
Kðx; tÞ ¼
tð1 xÞ; 0 6 t 6 x;
ð28Þ
xð1 tÞ; x 6 t 6 1:
We applied the new method proposed in the current paper to approximate the solution of Eq. (24) for different values of N. Table 9 shows the Error, L2-error, L1-error and RMS-error norms, K2(A) and K1(A) in some points of the interval [0, 1] obtained for N = 7, 10, 13, 15. Then in Fig. 14 the Error for N = 15 is represented. The convergence behavior of the MQ-RBFs method in terms of the L2-error norm versus reciprocal of number of collocation points N is shown in Fig. 15. 5.8. Problem 8 In this example we consider the mathematical model for an adiabatic tubular chemical reactor discussed in [68,69], which in the case of steady state solutions, can be stated as the ordinary differential equation
y00 ðxÞ ky0 ðxÞ þ Fðk; l; b; yðxÞÞ ¼ 0;
x 2 ½0; 1;
ð29Þ
Table 9 Absolute error between the exact and numerical solution with c = 2 for various value of N and 0 6 x 6 1; L2-norm error, L1-norm error, RMS error, condition number of s = 2,1 at various values of N for problem 7. x
N=7
N = 10
N = 13
N = 15
0.0 0.2 0.4 0.6 0.8 1.0
1.4876e 005 4.8786e 006 4.2779e 006 3.9278e 006 3.8000e 006 1.0022e 005
1.3448e 007 2.9107e 008 1.2368e 008 8.9588e 009 1.0984e 008 2.6040e 008
4.8773e 010 7.8393e 011 3.9109e 011 3.2126e 011 4.3877e 011 1.9084e 010
1.9743e 011 2.4293e 012 7.1551e 014 5.7228e 014 1.2811e 012 7.0232e 012
L2 L1 RMS K2(A) K1(A)
5.9703e 009 2.9327e 009 2.4374e 009 5.5040e + 008 9.8990e + 008
1.6664e 011 7.6147e 012 5.5546e 012 1.3164e + 013 2.4592e + 013
3.1465e 014 1.2283e 014 9.0833e 015 4.3266e + 016 7.8419e + 016
8.9736e 016 3.2744e 016 2.3983e 016 3.6542e + 019 7.4388e + 019
Fig. 14. Absolute error between MQ-RBFs solution and exact solution for c = 2 and N = 15 of problem 7.
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Fig. 15. Convergence in space for MQ-RBFs with c = 2 of problem 7, N is number of collocation points.
with boundary conditions
y0 ð0Þ ¼ kyð0Þ;
y0 ð1Þ ¼ 0;
ð30Þ
where
Fðk; l; b; yðxÞÞ ¼ klðb yðxÞÞeyðxÞ :
ð31Þ
The problem can be converted into a Hammerstein integral equation of the form [69]
yðxÞ ¼
Z
1
Kðx; tÞGðt; yðtÞÞdt;
ð32Þ
0
Table 10 Comparison between solution of Adomian [70] and Marzban et al. [39] and present solution with c = 1.8 for various value of N and 0 6 x 6 1 for problem 8. x
N = 10
N = 13
N = 15
Adomian [70]
Marzban et al. [39]
0.0 0.2 0.4 0.6 0.8 1.0
0.0060480837 0.0181927441 0.0304249181 0.0426694912 0.0543707757 0.0614628310
0.0060481548 0.0181928964 0.0304246478 0.0426691388 0.0543716843 0.0614588885
0.0060483431 0.0181929325 0.0304246703 0.0426691182 0.0543716557 0.0614587507
0.006048 0.018192 0.030424 0.042669 0.054371 0.061458
0.0060483739 0.0181929364 0.0304246702 0.0426691183 0.0543716533 0.0614587374
kResk2 K2(A) K1(A)
4.6241e 013 2.4771e + 012 4.8599e + 012
1.7420e 015 7.2428e + 014 1.4945e + 015
2.3401e 017 6.9386e + 018 1.5087e + 019
-
-
Fig. 16. Convergence in space for MQ-RBFs with c = 1.8 of problem 8, N is number of collocation points.
K. Parand, J.A. Rad / Applied Mathematics and Computation 218 (2012) 5292–5309
5307
where
Kðx; tÞ ¼
1; 0 6 t 6 x; kðxtÞ ; x6t61 e
ð33Þ
and
Gðt; yðtÞÞ ¼ lðb yðtÞÞeyðtÞ :
ð34Þ
The existence and uniqueness of the solution for this Hammerstein integral equation with respect to the values of parameters k, l, b are given in [69]. In [70], the Adomian’s method is used to solve the integral equation Eq. (32) for the particular values of the parameters k = 10, l = 0.02 and b = 3. Also, Marzban et al. [39] applied a composite collocation method for solving the integral equation Eq. (32) for the particular values of the parameters k = 10, l = 0.02 and b = 3. We applied the new method proposed in the current paper to approximate the solution of Eq. (32) for different values of N. Table 10 shows the comparison between solution of Adomian [70] and Marzban et al. [39] and presents the solution with c = 1.8 for various values of N and 0 6 x 6 1 obtained for N = 10, 13, 15. The convergence behavior of the MQ-RBFs method in terms of the kResk2 versus reciprocal of number of collocation points N is shown in Fig. 16. 6. Appropriate interval for shape parameter DRBFs uses differential operator which possesses a sharp noise. It means that if we make a small change to the shape parameter c, it makes a sharp noise will emerge in the solution of the problem [71,72]. For the limiting value of c, the condition number of the RBFs system becomes so large that the system leads to ill-conditioning. The condition number also grows with N for fixed values of the shape parameter c [20,73]. Generally for a fixed number of collocation points N, smaller values of c produce better approximations, but the matrix A will be more ill-conditioned [13,74]. When the number of mesh points is further increased after a critical value, the accuracy of numerical results is decreased [9,12]. The reason may be due to the fact that, when the number of mesh points increases, the condition number of the matrix becomes very large and the matrix tends to be ill-conditioned. For example, co-relation between the condition number of the matrix A and the different values of the shape parameter c is shown in Table 3 corresponding to the problem 1 for MQ-RBFs. The critical values of the shape parameter c in this case are 0.9 and 1.8 and the condition numbers of the matrix A corresponding to these values are given by 1.4723e + 016 and 6.9386e + 018, respectively. The interval of stability in this case is (0.9, 1.8). It is clear from the Table 3, that if the values of the shape parameter c are the outputs of the interval (0.9, 1.8), then the solution will down and hence the method will become unstable. This phenomenon is shown in Fig. 2, where L1-error norm is plotted against different values of the shape parameter c. 7. Conclusions and remarks In this study, we have applied an approximation technique to solve Volterra–Fredholm–Hammerstein integral equations. The method is based on the collocation method and multiquadric radial basis function. We used an irregular mesh-grid in the RBF collocation instead of finite difference method which enhanced our options. Additionally, through the comparison with exact solutions we show that the RBFs methods have good reliability and efficiency. Also high convergence rates and good accuracy are obtained with the proposed method using relatively low numbers of data points. Acknowledgements The authors are very grateful to the reviewers and the subject editor for carefully reading the paper and for their comments and suggestions which have greatly improved the paper. The research of first author (K. Parand) was supported by a grant from Shahid Beheshti University. References [1] G.E. Fasshauer, Meshfree Approximation Methods with MATLAB, Word Scientific Publishing, 2007. [2] E.J. Kansa, Multiquadrics-A scattered data approximation scheme with applications to computational fluid-dynamics-I surface approximations and partial derivative estimates, Comput. Math. Appl. 19 (1990) 127–145. [3] M. Sharan, E.J. Kansa, S. Gupta, Application of the multiquadric method for numerical solution of elliptic partial differential equations, Appl. Math. Comput. 84 (1997) 275–302. [4] M. Zerroukat, H. Power, C.S. Chen, A numerical method for heat transfer problems using collocation and radial basis functions, Int. J. Numer. Meth. Eng. 42 (1998) 1263–1278. [5] N. Mai-Duy, T. Tran-Cong, Numerical solution of differential equations using multiquadric radial basis function networks, Neural Netw 14 (2001) 185– 199. [6] M. Tatari, M. Dehghan, A method for solving partial differential equations via radial basis functions: application to the heat equation, Eng. Anal. Bound. Elem. 34 (2010) 206–212. [7] M. Dehghan, A. Shokri, Numerical solution of the nonlinear Klein–Gordon equation using radial basis functions, J. Comput. Appl. Math. 230 (2009) 400– 410. [8] A. Alipanah, M. Dehghan, Numerical solution of the nonlinear Fredholm integral equations by positive definite functions, Appl. Math. Comput. 190 (2007) 1754–1761.
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