About this Journal Submit a Manuscript Table of Contents 
ISRN Computational Mathematics
Volume 2012 (2012), Article ID 139514, 5 pages
http://dx.doi.org/10.5402/2012/139514
Research Article

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

Abdelwahab Zerarka, Abdesselam Soukeur, and Hanane Saidi

Laboratory of Applied Mathematics, Mohamed Khider University of Biskra, BP145, 07000 Biskra, Algeria

Received 6 September 2011; Accepted 20 October 2011

Academic Editors: V. Rai and K. Wang

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.

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 𝑙𝑘=1𝛾𝑘𝑑(𝑥)𝑘𝑓(𝑥)𝑑𝑥𝑘,=𝛼(𝑥)𝑓(𝑥)+𝛽(𝑥)+𝜆𝑊(𝑥,𝑧),𝑎≤𝑥≤𝑇(1) where 𝑊(𝑥,𝑧) is given by 𝑊(𝑥,𝑧)≡𝑧𝑎(𝑥−𝑠)−𝜈𝐾(𝑥,𝑠)𝑓(𝑠)𝑑𝑠,(2) and we set 𝑊(𝑥,𝑧)=𝑊(𝑥,𝑥),forVolterratype,𝑊(𝑥,𝑇),forFredholmtype,(3) with prescribed boundary condition 𝑓(𝑎)=𝑓𝑎.

𝜆 and 𝜈 are parameters with 0<𝜈<1. 𝛼(𝑥),𝛽(𝑥), and 𝛾𝑘(𝑥)(𝑘=0,…,𝑚) are given functions, and 𝐾(𝑥,𝑠) is a regular kernel of the integrodifferential equation. It is assumed that the function involved in (1) is sufficiently regular. The above equations, for 𝜈=0, become regular integrodifferential equations.

In order to avoid the singularity, (1) is integrated by part and with little effort; we obtain 1𝑊(𝑥,𝑧)=(𝜈−1)(𝑥−𝑧)1−𝜈𝐾(𝑥,𝑧)𝑓(𝑧)−(𝑥−𝑎)1−𝜈−1𝐾(𝑥,𝑎)𝑓(𝑎)(𝜈−1)𝑧𝑎(𝑥−𝑠)1−𝜈𝑑[]𝐾(𝑥,𝑠)𝑓(𝑠)𝑑𝑠𝑑𝑠.(4) Now the relation (1) becomes 𝑙𝑘=1𝛾𝑘𝑑(𝑥)𝑘𝑓(𝑥)𝑑𝑥𝑘=𝛼(𝑥)𝑓(𝑥)+𝛽𝜈+𝜆(𝑥,𝑧)(1−𝜈)𝑧𝑎(𝑥−𝑠)1−𝜈𝑑[]𝐾(𝑥,𝑠)𝑓(𝑠)𝑑𝑠𝑑𝑠,(5) where 𝛽𝜈(𝑥,𝑧)=𝛽(𝑥)+(𝜆/(𝜈−1))((𝑥−𝑧)1−𝜈𝐾(𝑥,𝑧)𝑓(𝑧)−(𝑥−𝑎)1−𝜈𝐾(𝑥,𝑎)𝑓(𝑎)), and 𝑧=𝑥 and 𝑧=𝑇 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 by𝑑𝑘𝑓𝑥𝑚𝑑𝑥𝑘=𝑁𝑗=0𝐷𝑘𝑚𝑗𝑓𝑥𝑗,(6)𝑥𝑚𝑎𝑥𝑚−𝑠1−𝜈𝑑𝐾𝑥𝑚𝑓,𝑠(𝑠)𝑑𝑠𝑑𝑠=𝑁𝑗=0𝐼𝑚𝑗𝑓𝑥𝑗,forVolterratype,(7)𝑇𝑎𝑥𝑚−𝑠1−𝜈𝑑𝐾𝑥𝑚,𝑠𝑓(𝑠)𝑑𝑠𝑑𝑠=𝑁𝑗=0𝐽𝑚𝑗𝑓𝑥𝑗,forFredholmtype,(8) where {𝑥𝑘}𝑁𝑘=0 are interpolating points, taken as the points of Tchebychev of the form 𝑥𝑗=(1/2)𝑇[1−cos((2𝑗+1)/(2𝑁+2)𝜋)], 0≤𝑗≤𝑁. 𝐷𝑖𝑗 and 𝐼𝑚𝑗 are the weighting coefficients linked with Lagrange interpolated polynomials 𝑃𝑁,𝑗(𝑥) and are explicitly given by 𝐷𝑘𝑚𝑗=𝑑𝑘𝑃𝑁,𝑗𝑥𝑚𝑑𝑥𝑘,(9)𝐼𝑚𝑗=𝑥𝑚0𝑥𝑚−𝑠1−𝜈𝑑𝐾𝑥𝑚𝑃,𝑠𝑁,𝑗(𝑠)𝐽𝑑𝑠𝑑𝑠,𝑚𝑗=𝑇0𝑥𝑚−𝑠1−𝜈𝑑𝐾𝑥𝑚𝑃,𝑠𝑁,𝑗(𝑠)𝑑𝑠𝑑𝑠.(10)

With little effort the original problems (1) can be converted to an algebraic system and take, respectively, the following compact forms: 𝛼𝑥𝑚𝑓𝑥𝑚+𝛽𝜈𝑥𝑚,𝑥𝑚=𝑁𝑗=0𝑈𝐼𝑚𝑗𝑓𝑥𝑗,𝑚=0,…,𝑁,(11) for Volterra type, and 𝛼𝑥𝑚𝑓𝑥𝑚+𝛽𝜈𝑥𝑚=,𝑇𝑁𝑗=0𝑈𝐽𝑚𝑗𝑓𝑥𝑗,𝑚=0,…,𝑁,(12) for Fredholm type, where 𝑈𝐼𝑚𝑗=𝑙𝑘=0𝛾𝑘𝑥𝑚𝐷𝑘𝑚𝑗−𝜆𝐼(1−𝜈)𝑚𝑗,(13)𝑈𝐽𝑚𝑗=𝑙𝑘=0𝛾𝑘𝑥𝑚𝐷𝑘𝑚𝑗−𝜆𝐽(1−𝜈)𝑚𝑗.(14)

The relations (12) and (13) are real systems of 𝑀×𝑀 equations in the 𝑀×𝑀 unknown real coefficients 𝑈𝐼𝑚𝑗 and 𝑈𝐽𝑚𝑗, respectively, where 𝑀=𝑁+1. In order to avoid unnecessary calculation and to optimize the computational aspect, it is therefore more convenient to get the desired coefficients 𝐷𝑖𝑖, 𝐼𝑖𝑖, and 𝐽𝑖𝑖 in the following forms: 𝐷𝑘𝑖𝑖=−𝑁𝑗=0,𝑗≠𝑖𝐷𝑘𝑖𝑗,for𝑖=0,…,𝑁,𝑘=1,…,𝑙,(15)𝐼𝑖𝑖=𝑥𝑖𝑎𝑥𝑖−𝑠1−𝜈𝑥𝑑𝐾𝑖,𝑠𝑑𝑠𝑑𝑠−𝑁𝑗=0,𝑗≠𝑖𝐼𝑖𝑗,for𝑖=0,…,𝑁,(16)𝐽𝑖𝑖=𝑇𝑎𝑥𝑖−𝑠1−𝜈𝑥𝑑𝐾𝑖,𝑠𝑑𝑠𝑑𝑠−𝑁𝑗=0,𝑗≠𝑖𝐽𝑖𝑗,for𝑖=0,…,𝑁.(17)

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 𝑓(𝑥)=𝑁𝑗=0𝑓𝑥𝑗𝑃𝑁,𝑗(𝑥).(18)

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 𝛼𝑝𝑓𝑝+𝛽𝜈,𝑝𝑝=𝑁𝑗=0𝑈𝐼𝑝𝑗𝑓𝑗,𝑝=0,…,𝑁,(19)𝛼𝑝𝑓𝑝+𝛽𝜈,𝑝𝑇=𝑁𝑗=0𝑈𝐽𝑝𝑗𝑓𝑗,𝑝=0,…,𝑁,(20) where 𝑓𝑗 and 𝛼𝑖 stand for the values of 𝑓(𝑥𝑖) and 𝛼(𝑥𝑖), respectively. 𝛽𝜈,𝑝𝑝 and 𝛽𝜈,𝑝𝑇 denote the values of 𝛽𝜈(𝑥𝑝,𝑥𝑝) and 𝛽𝜈(𝑥𝑝,𝑇), respectively.

Evaluating (20) and (21) at 𝑥𝑝, the linear systems of algebraic equations follow, respectively, 𝐼𝐴−𝑈1𝐹=𝐵1,(21)𝐼𝐴−𝑈2𝐹=𝐵2,(22) where the vectors 𝐹 and 𝐵1(𝐵2) have components 𝑓𝑝 and 𝛽𝜈,𝑝𝑝(𝛽𝜈,𝑝𝑇), respectively, 𝑈1(𝑈2) is a matrix with the elements 𝑈1,𝑖𝑗=𝑈𝐼𝑖𝑗(𝑈2,𝑖𝑗=𝑈𝐽𝑖𝑗), and the components of 𝑈𝐼 and 𝑈𝐼 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 𝑂((𝑁+1)2) operations.

The accuracy of solutions can be checked by using the error tolerance 𝐸(𝑥𝑖)=|𝑓(𝑥𝑖)−𝑓exact(𝑥𝑖)|=10−𝑗, where “𝑗” 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 𝑑𝑓(𝑥)𝑑𝑥=𝑓(𝑥)+𝛽(𝑥)−𝑥0(𝑥−𝑠)−1/2𝑓(𝑠)𝑑𝑠,0≤𝑥≤1,𝑓(0)=0,(23) where √𝛽(𝑥)=(3/2)√𝑥+𝑥𝑥+(3𝜋/8)𝑥2, from the governing equation (1) and (24) subject to 𝐾(𝑥,𝑠)=1, we deduce the following: 𝑙=1,𝛾1(𝑥)=1,𝛼(𝑥)=1,𝜆=−1,𝜈=1/2,𝑎=0,𝑇=1, and 𝑓0=0. The exact solution is 𝑓(𝑥)=𝑥3/2.

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, TOLV=10−5.

139514.fig.001
Figure 1: Solution 𝑓(𝑥) of (23). Solid curve: exact solution: 𝑓(𝑥)=𝑥3/2, dash curve: present work.
3.2. Example  2

We consider the numerical solution of the following Volterra’s integrodifferential equation [40]: 𝑑2𝑓(𝑥)𝑑𝑥2=𝑓(𝑥)+𝛽(𝑥)+𝑥0𝑥𝑠𝑓(𝑠)𝑑𝑠,0≤𝑥≤1,𝑓(0)=1,(24) where 𝛽(𝑥)=−𝑥(𝑥+1)+𝑥(1−𝑥)exp(𝑥)+2−𝑥5/4, and from this equation we deduce the following: 𝑙=2,𝛾1(𝑥)=0,𝛾2(𝑥)=1,𝛼(𝑥)=1,𝜆=1,𝜈=0,𝑎=0,𝑇=1,𝑓0=1, and the kernel 𝐾(𝑥,𝑠)=𝑥𝑠.

The above equation has the exact solution 𝑓(𝑥)=𝑥2+exp(𝑥).

The result is displayed in Figure 2, where the error tolerance, TOLV=10−6. With this tolerance and for the number of the interpolation points 𝑁=10, the results under consideration are very satisfactory.

139514.fig.002
Figure 2: Solution 𝑓(𝑥) of (24). Solid line: exact solution: 𝑓(𝑥)=𝑥2+exp(𝑥), dash curve: present work.
3.3. Example  3

Consider the Fredholm integrodifferential equation [41]: 𝑑2𝑓(𝑥)𝑑𝑥2+𝑥𝑑𝑓(𝑥)𝑑𝑥=𝑥𝑓(𝑥)+𝛽(𝑥)+1−1𝑒−𝑠1sin(𝑥)𝑓(𝑠)𝑑𝑠,−1≤𝑥≤1,𝑓(−1)=𝑒,(25) where 𝛽(𝑥)=exp(𝑥)−2sin(𝑥), and from this equation we deduce the following: 𝑙=2,𝛾1(𝑥)=𝑥, 𝛾2(𝑥)=1, 𝛼(𝑥)=𝑥, 𝜆=1, 𝜈=0, 𝑎=−1, 𝑇=1, 𝑓−1=1/𝑒, and the kernel 𝐾(𝑥,𝑠)=𝑒−𝑠sin(𝑥). The exact solution has the form 𝑓(𝑥)=exp(𝑥).

In Figure 3, the number of interpolation points is taken equal to 𝑁=10 to solve this problem with TOLV=10−8. We see from this Figure that the method provides accurate results in the full range, where the error tolerance, TOLV=10−6.

139514.fig.003
Figure 3: Solution 𝑓(𝑥) of (25). Solid curve: exact solution: 𝑓(𝑥)=exp(𝑥), dash curve: present work.
3.4. Example  4

We apply the mentioned technique to solve the fifth-order Fredholm integrodifferential equation [42]: 𝑑5𝑓(𝑥)𝑑𝑥5−𝑥2𝑑3𝑓(𝑥)𝑑𝑥3−𝑑𝑓(𝑥)𝑑𝑥=𝑥𝑓(𝑥)+𝛽(𝑥)+1−1𝑓(𝑠)𝑑𝑠,−1≤𝑥≤1,𝑓(−1)=−sin(1),(26) where 𝛽(𝑥)=𝑥2cos(𝑥)−𝑥sin(𝑥), and from this equation we deduce the following: 𝑙=5,𝛾1(𝑥)=−1,𝛾2(𝑥)=𝛾4(𝑥)=0,𝛾3(𝑥)=−𝑥2,𝛾5(𝑥)=1,𝛼(𝑥)=𝑥,𝜆=1,𝜈=0,𝑎=−1,𝑇=1,𝑓−1=−sin(1), and the kernel 𝐾(𝑥,𝑠)=1. The exact solution has the form 𝑓(𝑥)=sin(𝑥). Figure 4 presents computer simulations of this example, that is generated using the number of interpolating points 𝑁=8. 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 TOLV=10−6.

139514.fig.004
Figure 4: Solution 𝑓(𝑥) of the equation (26). Solid curve: exact solution: 𝑓(𝑥)=sin(𝑥), 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 𝑓(𝑥𝑖) 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 TOLV=10−6.

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 · View at 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 · View at 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 · View at Google Scholar
  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 · View at 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 · View at 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 · View at 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 · View at 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 · View at 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 · View at 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 · View at 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 · View at 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 · View at 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 · View at 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 · View at 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 · View at 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 · View at 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 · View at 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 · View at 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 · View at 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 · View at 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 · View at 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 · View at Google Scholar