On Some Volterra and Fredholm Problems via the Unified Integrodifferential Quadrature Method

Abstract

We present a new approach based on the formulation of the integrodifferential quadrature method (hereafter called IDQ) to handle Volterra's and Fredholm's equations. This approach is constructed and tested with some realistic numerical examples using the basic computational aspects.

1. Introduction

In the present work, we present a formulation of a new numerical approach which is based on the generalized integrodifferential quadrature method and applied to weakly singular Volterra and Fredholm integrodifferential equations in the linear case. This method studies the situation in which the unknown function is identified as the Lagrange polynomial and the interpolating points of the Tchebychev type are used. The accuracy and efficiency of solving integrodifferential equations are still an ongoing research in numerical analysis. Several different methods have also been modeled for the solutions of linear and nonlinear problem [1–27].

The central argument for the interest of the type of this problem comes naturally from its wide applications almost in any branches of science and engineering described by systems of ODEs and PDEs [28–38] which in some situations can be transformed into a set of integral equations. Much attention has been devoted to the investigation of new mathematical models and numerical approaches to evaluate the solutions of the integral equations.

New calculations are performed for the construction of the solution by a suitable choice of the interpolating points using the unified integrodifferential quadrature method. Based on this fact, this technique is a very promising and powerful methodology in successfully locating best solutions of the problem. Our main purpose is to develop a general numerical schema for the integrodifferential equations that is universally applicable.

The contents of this paper are organized as follows. In Section 2, a formulation adapted to the problem of construction of the solution of the one-dimensional weakly singular integrodifferential equation by using a summary limited to some aspects of the unified integrodifferential quadrature method is qualitatively presented. Section 3 exposes some essential examples for the construction of the unknown solution. The comments and conclusion are given in Section 4.

2. Unified Integrodifferential Quadrature Method

The description of generalized integrodifferential quadrature method is summarized as follows: the starting point of the weakly singular integrodifferential equation is written as where is given by and we set with prescribed boundary condition .

and are parameters with . , and are given functions, and is a regular kernel of the integrodifferential equation. It is assumed that the function involved in (1) is sufficiently regular. The above equations, for , become regular integrodifferential equations.

In order to avoid the singularity, (1) is integrated by part and with little effort; we obtain Now the relation (1) becomes where , and and for Volterra type and Fredholm type, respectively.

At this stage, we introduce the quadrature aspect of solutions: the derivatives and integral in expression (6) are approximated by where are interpolating points, taken as the points of Tchebychev of the form , . and are the weighting coefficients linked with Lagrange interpolated polynomials and are explicitly given by

With little effort the original problems (1) can be converted to an algebraic system and take, respectively, the following compact forms: for Volterra type, and for Fredholm type, where

The relations (12) and (13) are real systems of equations in the unknown real coefficients and , respectively, where . In order to avoid unnecessary calculation and to optimize the computational aspect, it is therefore more convenient to get the desired coefficients , , and in the following forms:

Now the expressions (14), (15), (16), (17), and (18) provide the charmingly practical formulae for the weighting coefficients. Once the function values at all grid points are obtained, it is then easy to determine the function values in the overall domain in terms of polynomial approximation, such that

The practical part of this study is examined in the following section.

3. Worked Examples

We shall consider the solution of some crack situations in order to show the application and the effectiveness of the method described in Section 2. Four examples are considered for this purpose.

We write (1) in the form where and stand for the values of and , respectively. and denote the values of and , respectively.

Evaluating (20) and (21) at , the linear systems of algebraic equations follow, respectively, where the vectors and have components and , respectively, is a matrix with the elements , and the components of and are given by (14) and (15) together with (16), (17), and (18). The equations (20) and (21) are explicit prescriptions that give the solutions in operations.

The accuracy of solutions can be checked by using the error tolerance , where “” is a positive integer. We continue evaluating the solution until the above relation was satisfied.

In order to verify the efficiency of the method developed in the previous section, the following examples have been selected to provide a comparison with previously published work.

3.1. Example  1

In this example we use Volterra’s integrodifferential equation [39] given by where , from the governing equation (1) and (24) subject to , we deduce the following: , and . The exact solution is .

Illustrated in Figure 1 is the basic application of the problem (24). The numerical result displayed in Figure 1 is qualitatively in good agreement with the exact solution in which the error tolerance, .

Figure 1 

Solution of (23). Solid curve: exact solution: , dash curve: present work.

3.2. Example  2

We consider the numerical solution of the following Volterra’s integrodifferential equation [40]: where , and from this equation we deduce the following: , and the kernel .

The above equation has the exact solution .

The result is displayed in Figure 2, where the error tolerance, . With this tolerance and for the number of the interpolation points , the results under consideration are very satisfactory.

Figure 2 

Solution of (24). Solid line: exact solution: , dash curve: present work.

3.3. Example  3

Consider the Fredholm integrodifferential equation [41]: where , and from this equation we deduce the following: , , , , , , , , and the kernel . The exact solution has the form .

In Figure 3, the number of interpolation points is taken equal to to solve this problem with . We see from this Figure that the method provides accurate results in the full range, where the error tolerance, .

Figure 3 

Solution of (25). Solid curve: exact solution: , dash curve: present work.

3.4. Example  4

We apply the mentioned technique to solve the fifth-order Fredholm integrodifferential equation [42]: where , and from this equation we deduce the following: , and the kernel . The exact solution has the form . Figure 4 presents computer simulations of this example, that is generated using the number of interpolating points . The result is compared with the exact solution. From Figure 4, we see that the agreement between the numerical result of the present method and the exact result is very good, with an error tolerance .

Figure 4 

Solution of the equation (26). Solid curve: exact solution: , dash curve: present work.

4. Comments and Conclusions

This paper has introduced a new formulation that uses the unified integrodifferential quadrature method to handle some integrodifferential equations. The main advantage of Lagrange polynomial is that the weighting coefficients that have to be computed do not depend on the and it is independent on the manner the discrete points are ordered.

The preliminary results, obtained through the use of this method, show that the resulting solutions are quite good for all examples which have been selected here as a testbed. We can note that similar patterns of convergence are seen for all four examples with a common tolerance .

This method seems to be a powerful alternative and consequently gives very accurate solutions to the problems under consideration. It would be interesting to generalize this method to the nonlinear case. This matter deserves a further work.

Acknowledgments

The authors gratefully acknowledge helpful conversations with Professor W. Cramer. This work was sponsored in part by the M.E.R.S (Ministère de l’Enseignement et de la Recherche Scientifique): under contract no. D01420060012.

References

1. R. K. Miller and A. Feldstein, “Smoothness of solutions of Volterra integral equations with weakly singular kernels,” SIAM Journal on Mathematical Analysis, vol. 2, pp. 242–258, 1971. View at: Google Scholar
2. J. Norbury and A. M. Stuart, “Volterra integral equations and a new Gronwall inequality. I. The linear case,” Proceedings of the Royal Society of Edinburgh Section A, vol. 106, no. 3-4, pp. 361–373, 1987. View at: Google Scholar
3. J. Norbury and A. M. Stuart, “Volterra integral equations and a new Gronwall inequality. II. The nonlinear case,” Proceedings of the Royal Society of Edinburgh Section A, vol. 106, no. 3-4, pp. 375–384, 1987. View at: Google Scholar
4. A. Zerarka, S. Hassouni, H. Saidi, and Y. Boumedjane, “Energy spectra of the Schrödinger equation and the differential quadrature method,” Communications in Nonlinear Science and Numerical Simulation, vol. 10, no. 7, pp. 737–745, 2005. View at: Publisher Site | Google Scholar
5. H. Brunner and P. J. van der Houwen, The Numerical Solution of Volterra Equations, vol. 3 of CWI Monographs, North-Holland Publishing, Amsterdam, The Netherlands, 1986.
6. C. T. H. Baker, The Numerical Treatment of Integral Equations, Clarendon Press, Oxford, UK, 1977.
7. L. M. Delves and J. Walsh, Numerical Solution of Integral Equations, Clarendon Press, Oxford, UK, 1974.
8. J. A. Cochran, The Analysis of Linear Integral Equations, McGraw-Hill, New York, NY, USA, 1972.
9. K. E. Atkinson, A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1976.
10. P. H. M. Wolkenfelt, “The construction of reducible quadrature rules for Volterra integral and integro-differential equations,” IMA Journal of Numerical Analysis, vol. 2, no. 2, pp. 131–152, 1982. View at: Publisher Site | Google Scholar
11. H.-J. Dobner, “Bounds of high quality for first kind Volterra integral equations,” Reliable Computing, vol. 2, no. 1, pp. 35–45, 1996. View at: Google Scholar
12. D. Bugajewski, “On the Volterra integral equation in locally convex spaces,” Demonstratio Mathematica, vol. 25, no. 4, pp. 747–754, 1992. View at: Google Scholar
13. L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations, Cambridge University Press, Cambridge, UK, 1985. View at: Publisher Site
14. H. Brunner, A. Pedas, and G. Vainikko, “The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations,” Mathematics of Computation, vol. 68, no. 227, pp. 1079–1095, 1999. View at: Publisher Site | Google Scholar
15. J. P. Kauthen and H. Brunner, “Continuous collocation approximations to solutions of first kind Volterra equations,” Mathematics of Computation, vol. 66, no. 220, pp. 1441–1459, 1997. View at: Publisher Site | Google Scholar
16. Q. Hu, “Multilevel correction for discrete collocation solutions of Volterra integral equations with delay arguments,” Applied Numerical Mathematics, vol. 31, no. 2, pp. 159–171, 1999. View at: Publisher Site | Google Scholar
17. H. Brunner, Q. Hu, and Q. Lin, “Geometric meshes in collocation methods for Volterra integral equations with proportional delays,” IMA Journal of Numerical Analysis, vol. 21, no. 4, pp. 783–798, 2001. View at: Publisher Site | Google Scholar
18. V. Karlin, V. G. Maz'ya, A. B. Movchan, J. R. Willis, and R. Bullough, “Numerical solution of nonlinear hypersingular integral equations of the Peierls type in dislocation theory,” SIAM Journal on Applied Mathematics, vol. 60, no. 2, pp. 664–678, 2000. View at: Publisher Site | Google Scholar
19. M. H. Fahmy, M. A. Abdou, and M. A. Darwish, “Integral equations and potential-theoretic type integrals of orthogonal polynomials,” Journal of Computational and Applied Mathematics, vol. 106, no. 2, pp. 245–254, 1999. View at: Publisher Site | Google Scholar
20. T. Diogo and P. Lima, “Numerical solution of a nonuniquely solvable Volterra integral equation using extrapolation methods,” Journal of Computational and Applied Mathematics, vol. 140, no. 1-2, pp. 537–557, 2002. View at: Publisher Site | Google Scholar
21. F. Fagnani and L. Pandolfi, “A recursive algorithm for the approximate solution of Volterra integral equations of the first kind of convolution type,” Inverse Problems, vol. 19, no. 1, pp. 23–47, 2003. View at: Publisher Site | Google Scholar
22. P. Oja and D. Saveljeva, “Cubic spline collocation for Volterra integral equations,” Computing, vol. 69, no. 4, pp. 319–337, 2002. View at: Publisher Site | Google Scholar
23. I. Bock and J. Lovíšek, “On a reliable solution of a Volterra integral equation in a Hilbert space,” Applications of Mathematics, vol. 48, no. 6, pp. 469–486, 2003. View at: Publisher Site | Google Scholar
24. M. Federson, R. Bianconi, and L. Barbanti, “Linear Volterra integral equations,” Acta Mathematicae Applicatae Sinica, vol. 18, no. 4, pp. 553–560, 2002. View at: Publisher Site | Google Scholar
25. M. Federson, R. Bianconi, and L. Barbanti, “Linear Volterra integral equations as the limit of discrete systems,” Cadernos de Matemática, vol. 4, pp. 331–352, 2003. View at: Google Scholar
26. P. Linz, Analytical and Numerical Methods for Volterra Equations, vol. 7, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1985.
27. U. Kulisch and W. L. Miranker, A New Approach to Scientific Computation, Academic Press, New York, NY, USA, 1983.
28. R. Hammer, M. Hocks, U. Kulisch, and D. Ratz, Numerical Toolbox for Verified Computing I, Springer, Berlin, Germany, 1995.
29. D. Bugajewski, “On differential and integral equations in locally convex spaces,” Demonstratio Mathematica, vol. 28, no. 4, pp. 961–966, 1995. View at: Google Scholar
30. H.-J. Dobner, “Bounds for the solution of hyperbolic problems,” Computing, vol. 38, no. 3, pp. 209–218, 1987. View at: Publisher Site | Google Scholar
31. E. T. Copson, Partial Differential Equations, Cambridge University Press, Cambridge, UK, 1975.
32. D. Bugajewska, Topological properties of solution sets of some problems for differential equations, Ph.D. thesis, Poznan, 1999.
33. A. Constantin, “On the unicity of solutions for the differential equation x⁽ⁿ⁾ = f(t, x),” Rendiconti del Circolo Matematico di Palermo Serie II, vol. 42, no. 1, pp. 59–64, 1993. View at: Publisher Site | Google Scholar
34. D. Bugajewski and S. Szufla, “Kneser's theorem for weak solutions of the Darboux problem in Banach spaces,” Nonlinear Analysis, Theory, Methods & Applications, vol. 20, no. 2, pp. 169–173, 1993. View at: Publisher Site | Google Scholar
35. M. Maron and R. Lopez, Numerical Analysis, Wadsworth Publishing Company, Belmont, Calif, USA, 1991.
36. M. Hukuhara, “Théorèmes fondamentaux de la théorie des équations différentielles ordinaires dans l'espace vectoriel topologique,” Journal of the Faculty of Science, University of Tokyo, Section IA, vol. 8, pp. 111–138, 1959. View at: Google Scholar
37. R. K. Miller and A. Feldstein, “Smoothness of solutions of Volterra integral equations with weakly singular kernels,” SIAM Journal on Mathematical Analysis, vol. 2, pp. 242–258, 1971. View at: Google Scholar
38. G. Adomian, “Random Volterra integral equations,” Mathematical and Computer Modelling, vol. 22, no. 8, pp. 101–102, 1995. View at: Publisher Site | Google Scholar
39. H. Brunner, “Polynomial spline collocation methods for Volterra integrodifferential equations with weakly singular kernels,” IMA Journal of Numerical Analysis, vol. 6, no. 2, pp. 221–239, 1986. View at: Publisher Site | Google Scholar
40. E. Rawashdeh, D. McDowell, and L. Rakesh, “Polynomial spline collocation methods for second-order Volterra integrodifferential equations,” International Journal of Mathematics and Mathematical Sciences, vol. 2004, no. 56, pp. 3011–3022, 2004. View at: Publisher Site | Google Scholar
41. K. Al-Khaled and F. Allan, “Decomposition method for solving nonlinear integro-differential equations,” Journal of Applied Mathematics & Computing, vol. 19, no. 1-2, pp. 415–425, 2005. View at: Publisher Site | Google Scholar
42. A. Akyüz-Daşcıoğlu, “A Chebyshev polynomial approach for linear Fredholm-Volterra integro-differential equations in the most general form,” Applied Mathematics and Computation, vol. 181, no. 1, pp. 103–112, 2006. View at: Publisher Site | Google Scholar

Copyright

Copyright © 2012 Abdelwahab Zerarka et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.