versão On-line ISSN 1807-0302
Comput. Appl. Math. vol.31 no.1 São Carlos 2012
R. Katani; S. Shahmorad*
In this paper, we propose an efficient numerical method for solving systems of linear and nonlinear integral equations of the first and second kinds, which avoids the need for special starting values. The method has also the advantages of simplicity of application and at least six order of convergence. A convergence analysis is given and accuracy of the method is clarified by numerical examples.
Mathematical subject classification: 65R20.
Key words: Urysohn type Volterra integral equations, Romberg quadrature rule, block by block method.
A number of problems in physics, engineering, biology, applied mathematics and many area of analysis, as well as other branches of science are described by system of integral equations.
Numerical solution of integral equations have attracted attention of many researchers. Considered methods are including: the methods that include applications of spline functions ; Runge-Kutta method ; Chebyshev polynomials method ; expansion method ; variational method ; HPM (Homotopy Perturbation Method), ADM (Adomian Decomposition Method)  and RBFN (Radial Basis Function Network) method .
In this paper, we propose a block by block method with Romberg quadrature rule which is a development of the presented method in .
The concept of a block by block method for integral equations seems to be described for the first time by Young . A similar technique for differential equations was given by Milne . A block method is essentially an extrapolation procedure and has the advantage of being self starting. As we shall see, it produces a block of values at a time and it is effective for the long intervals. The block by block method with Romberg quadrature rule has the following extra advantages:
1. For the given step size h, the order of convergence is at least h6 while it is h4 by using Simpson rule .
2. By increasing number of blocks, the order of convergence increases such that it would be at least h8 and h10 respectively for 8 and 16 blocks.
3. At the first step of Romberg rule one can use Simpson rule instead of trapezoidal rule, then the order of convergence for 4 blocks will be atleast h8.
The rest of the paper is organized as follows. In Section 2, we will present the method in a simple case (system of two equations) and in Section 3, we describe the general case. In Section 4, we prove a convergence result. Finally, we illustrate the performance of the method by comparing the numerical results of the HPM, RBFN method and the block by block method in Section 5 (see Table 1).
2 Description of the method
Consider a system of Volterra integral equations (VIEs) of the form
f(x) = [f1(x), f2(x), ..., fn(x)]T,g(x) = [g1(x), g2(x), ..., gn(x)]T,
We assume that the system (2.1) is uniquely solvable. Necessary and sufficient conditions for the existence and uniqueness of solution for (2.1) can be found in , therefore we assume
(i) g(x) is continuous (i.e. each component is continuous),
(ii) K(x, s, u) is a continuous function for
0 < s < x < X and 0 < ||u|| < ∞,
(iii) the kernel satisfies the Lipschtiz condition
||K(x, s, y) - K(x, s, z)|| < L||y - z||
where the norm is defined as ||f|| = |fi(t)|.
For simplicity, let n = 2 and the number of blocks to be 4. For 8, 16, ... blocks the process will be similar. Also, let 0 = x0 < x1 < ... < xN = X be a partition of [0, X] with xi = x0 + ih and h = X/N, (note that N must be multiple of the number of blocks). Then by putting x = x4m+p in (2.1), we have
for m = 0, 1, 2, ..., N/4 - 1 and p = 1, 2, 3, 4. Let Fi,j ≈ fi(xj) for i = 1, 2, ..., n and j = 0, 1, 2, ..., N.
If Fi,0, Fi,1, ..., Fi,4m (i = 1, 2) are known, then the first integral in (2.2) and (2.3) can be approximated by standard quadrature rules. The second integral is estimated by Romberg quadrature rule at the points x4m, x4m+1, x4m+2, x4m+3 and x4m+4, thus a system of eight simultaneous equations is obtained that is solved for a block of eight values of F. We use the trapezoidal rule for ki(x4m+p, s, f(s))ds (i = 1, 2) with xu = uh, xv = vh(u, v = 0, ..., N) and define
By using the Romberg rule we define
then the second integral in (2.2) and (2.3) can be written as
for p = 1, 2, 3, 4. If (j = 1, 2, 3) are not integers, then (i = 1, 2, ..., n) will be unknown in (2.4), but they can be approximated byLagrange interpolation at the points x4m, x4m+1, x4m+2, x4m+3 and x4m+4. Thus we obtain
When we use this method with very small step size, we must approximate the first integrals in a large interval, so we use the Romberg rule with three steps. If m is even, then
and we obtain from (2.2) and (2.3)
and we obtain
Therefore for p = 1, 2, 3, 4, (2.6) (or (2.7)) form a system of equations with the unknowns F4m+1, F4m+2, F4m+3 and F4m+4 which will be linear and nonlinear respectively for linear and nonlinear integral equations. For the linearcase, it is solved via a direct method but for the nonlinear case, the system may be solved by using an iterative method or by using a suitable software package such as Maple.
3 The general process
Consider the system of VIEs
where f, g and K are n-tuples vectors.
0 = x0 < x1 < ... < xN = X
be a partition of [0, X] with the step size h. Similar to previous section, if m be even, then we will have
for m = 0, 1, ..., N/4 - 1 and p = 1, ..., 4.
Otherwise (m be odd), we obtain
for m = 0, 1, ..., N/4 - 1. Consequently, at each step we get a system of 4n equations with unknowns F1,4m+1, ..., F1,4m+4, F2,4m+1,...,F2,4m+4, ..., Fn,4m+1, ..., Fn,4m+4.
4 Convergence analysis
Theorem 4.1. The approximation method given by the system (3.2), is convergent and its order of convergence is at least 6.
Proof. For simplicity, we prove the theorem for n = 2, the general case is proved similarly. Define εi,j := |Fi,j - fi(xj)|, then
By using (2.5), adding and diminishing the terms
and using the Lipschitz condition for k1, we obtain
where c' and c1, ..., c4 are constants and R is the error of the numerical integration.
Without diminishing universality, we assume that
where c = c1 + c2 + c3 + c4. Hence from Gronwall inequality , we have
It follow that if n approach to infinity then ε1,4m+1 will close to the 0 and for the functions k and f with at least sixth order derivatives, we have R = O(h6) and so ε1,4m+1 = O(h6) and the proof is completed.
5 Numerical results
We consider the following examples to illustrate the theoretical results ofTheorem 4.1 and compare numerical results of the method with the results of HPM and RBFN method.
Example 1 (). Consider the nonlinear system
with the exact solutions f1(x) = sin(x) and f2(x) = cos(x).
Example 2. Consider the linear system
with the exact solutions f1(x) = ex and f2(x) = x.
Example 3. Consider a triple system of VIEs as
with the exact solutions f1(x) = ex, f2(x) = x and f3(x) = x2.
Example 4 (). As a final example, we consider a non-linear system of the first kind VIEs of the form
with the exact solutions f1(x) = x2 and f2(x) = x.
The results in Tables 1-4 show the absolute errors for the examples 1-4. All results computed by programming in Maple 11. In Table 1, we show the superiority of the block by block method by comparing its results (for h = 0.05) with the results of RBFN-MshA (a modified version of Shi's algorithm)  and HPM , where the results of RBFN-MshA obtained with 6 hidden nodes and the results of HPM obtained with 4 iterations. Moreover,
1. The time of computation in the block by block method is less than that in the HPM whenever programming of both method is done using Maple package. Also, according to the structure of HPM, increasing number of iterations do not affect on the precision.
2. The values of the RBF widths affect significantly on the accuracy of results and determination of them is still a challenging problem whereas the block by block method dose not need any starting values.
3. At each step of the RBFN method, the weights are updated by using an optimization method, but the block by block method is independent of using any other method. Hence the RBFN method is more complicated than the block by block method.
In this paper, we have shown that the block by block method can achieve at least 6 order of convergence. Numerical results given in Tables 1-4 confirm this convergence order and show the high accuracy of the method. The idea can be applied to other types of integral equations and with other suitable quadrature rules.
Acknowledgements. The authors would like to thank so much the referee for carefully reading and his/her useful comments and suggestions on this paper.
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* Corresponding author.