Chebyshev Galerkin method for integro-differential equations of the second kind

Document Type : Research Article

Authors

• J. Biazar ¹
• F. Salehi ²

¹ Department of Applied Mathematics, Faculty of Mathematical Science, University of Guilan, Rasht, Iran.

² Department of Mathematics, Darab Branch, Islamic Azad University, Darab, Iran.

10.22067/ijnao.v6i1.37480

Abstract

‎In this paper‎, ‎we propose an efficient implementation of the Chebyshev Galerkin method for first order Volterra and Fredholm integro-differential equations of the second kind‎. ‎Some numerical examples are presented to show the accuracy of the method‎.

Keywords

• Volterra integro-differential equations
• Galerkin method
• Cheby- shev polynomials

20.1001.1.24236977.2016.6.1.2.4

---

References

1. Akyuz, A. Chebyshev Collocation method for solution of linear integro differential equations, M.Sc. Thesis, Dokuz Eylul University, Graduate School of Natural and Applied Sciences, 1997.

2. Atkinson, K. The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, 1997.

3. Avazzadeh, Z. and Heydari, M. Chebyshev polynomials for solving two dimensional linear and nonlinear integral equations of the second kind, Comp. Appl. Math. 31(1) (2012), 127-142.

4. Banks, H., Bortz, D. and Holt, S. Incorporation of variability into the modeling of viral delays in HIV infection dynamics, Math. Biosci 183 (2003) 63-91.

5. Barzkar, A., Oshagh, M., Assari, P. and Mehrpouya, M. Numerical So-lution of the Nonlinear Fredholm Integral Equation and the Fredholm Integro-differential Equation of Second Kind using Chebyshev Wavelets, World Appl. Sci. J. 18(12) (2012) 1774-1782.

6. Boland, J. The analytic solution of the differential equations describing heat ow in houses, Build. Environ. 37(2002), 1027-1035.

7. Brunner, H. The approximate solution of initial-value problems for general Volterra integro-differential equations, Comput. 40(40) (1988) 125-137.

8. Cerdik-Yaslan, H. and Akyuz-Dascioglu, A. Chebyshev polynomial solution of nonlinear Fredholm-Volterra integro-differential equations, J. Arts. Sci. 5 (2006) 89-101.

9. Chen, C. and Shih ,T. Finite Element Methods for Integro differential Equations, Singapore: World Scienti?c Publishing Co Ltd, 1997.

10. Dehghan, M. and Shakeri, F. Solution Of An Integro-Differential Equation Arising In Oscillating Magnetic Fields Using He's Homotopy Perturbation Method, Progr. In Electromagnetics Res., PIER 78 (2008) 361-376.

11. Delves, L. and Mohamed, J. Computional Methods for Integral Equations,Cambridge: Cambridge University Press, 1985.

12. El-gendi, S. Chebyshev solution of differential, integral and integro- differential equations, Comput. J. 12 (1969) 282-287.

13. Ezzati, R. and Najafalizadeh, S. Numerical solution of nonlinear Volterra-Fredholm integral equation by using Chebyshev polynomials, Mathematical Sciences 5(1) (2011) 1-12.

14. Fakhar-Izadi, F. and Dehghan, M. An efficient pseudo-spectral Legendre-Galerkin method for solving a nonlinear partial integro- differential equation arising in population dynamics, Math. Meth. Appl. Sci. 36(12) (2012) 485-1511.

15. Fung, B.Y.C. Mechanical Properties of Living Tissues, Springer-Verlag, New York, Springer-Verlag, New York, 1993.

16. Ghasemi, M., Tavassoli kajani, M. and Babolian, E. Numerical solutions of the nonlinear integro-differential equations: Wavelet-Galerkin method and homotopy perturbation method, Appl. Math. Comp. 188(1) (2007)

450-455.

17. Golbabai, A. and Seifollahi, S. Radial basis function networks in the numerical solution of linear integro-differential equations, Appl. Math. Comput. 188 (2007) 427-432.

18. Guo, H. and Rui, H. Crank{Nicolson least-squares Galerkin procedures for parabolic integro-differential equations, Appl. Math. Comp. 180 (2) (2006) 622-634.

19. Holmaker, K. Global asymptotic stability for a tationary solution of a system of integro-differential equations describing the formation of liver zones, SIAM J. Math. Anal. 24(1) (1993) 116-128.

20. Huang, Y. and Li, X. Approximate solution of a class of linear integro-differential equations by Taylor expansion method, Int. J. Comput. Math 15(3) (2009) 1-12.

21. Isik, O., Sezer, M. and Gney, Z. Bernstein series solution of a class of linear integro-differential equations with weakly singular kernel, Appl. Math,Comput. 217 (2011) 7009-7020.

22. Jangveladze, T., Kiguradze, Z. and Neta, B. Galerkin finite element method for one nonlinear integro-differential model, Appl. Math. Comp.217(16) (2011) 6883-6892.

23. Khater, A., Shamardan, A., Callebaut, D. and Sakran, M. Numerical solutions of integral and integro-differential equations using Legendre polynomials, Numer. Alg. 46 (2007) 195-218.

24. Lewis, B.A., On the numerical solution of Fredholm integral equations of the first kind, J. Inst. Math. Appl. 16(2) (1975) 207-220.

25. Lin, T., Lin, Y., Rao, M. and Zhang, S. Petrov-Galerkin Methods for Linear Volterra Integro-Differential Equations, SIAM J. Numer. Anal. 38(3) (2001) 937-963.

26. Makroglou, A. Numerical solution of some second order integro-differential equations arising in ruin theory, in: Proceedings of the Third Conference in, in: Proceedings of the Third Conference in Actuarial Sci-

ence and Finance Held in Samos, Greece, p. pp. 2-5, September 2004.

27. Maleknejad, K. and Tavassoli Kajani, M. Solving linear integro-differential equation system by Galerkin methods with hybrid functions, Appl. Math. Comput. 159 (2004) 603-612.

28. Maleknejad, K., Sohrabi, S. and Rostami, Y. Numerical solution of non-linear Volterra integral equation of the second kind by using Chebyshev polynomials, Appl. Math. Cmput. 188 (2007) 123-128.

29. Mason, J.C. and Handscomb, D. Chebyshev Polynomials, Boca Raton: Chapman & Hall/CRC, 2003.

30. Pedas, A. and Tamme, E. Discrete Galerkin method for Fredholm integro-differential equations with weakly singular kernels, Journal of Computa-tional and Applied Mathematics 213(1) (2008) 111-126.

31. Rashed, M. Lagrange interpolation to compute the numerical solutions of differential, integral and integro-differential equation, Appl. Math. Com-put. 151 (2004) 869-878.

32. Saberi-Nadjafi, J. and Ghorbani, A. He's homotopy perturbation method: an effective tool for solving nonlinear integral and integro-differential

equations, Comput. Math. Appl. 58 (2009) 2379-2390.

33. Sergeev, V. On stability of viscoelastic plate equilibrium, Automat. Remote Control 68(9) (2007) 1544-1550.

34. Thieme, H.R. A model for the spatial spread of an epidemic, J. Math. Biol. 4 (1977) 337-351.

35. Tomasiello, S. A note on three numerical procedures to solve Volterra integro-differential equations in structural analysis, Comput. Math. Appl. 62(8) (2011) 3183-3193.

36. Tomasiello, S. Some remarks on a new DQ-based method for solving a class of Volterra integro-differential equations, Appl. Math. Comput. 219 (2012) 399-407.

37. Van der Houwen, P. and Sommeijer, B. Euler{Chebyshev methods for integro-differential equations, Appl. Numer. Math. 24 (1997) 203-218.

38. Volk, W. The iterated Galerkin method for linear integro-differential equations, J. Comp. Appl. Math. 21(1) (1988) 63-74.

39. Wang, S. and He, J. Variational iteration method for solving integro-differential equations, Physics letters A 367 (2007) 188-191.

40. Wazwaz, A. Linear And Nonlinear Integral Equations, Methods And Applications, Springer, 2011.

41. Wolkowicz, G., Xia, H. and Ruan, S. Competition in the chemostat: a distributed delay model and its global asymptotic behavior, SIAM J. Appl. Math. 57 (1997) 1281-1310.

42. Yuzba.s., S., Sahidin, N. and Sezer, M. Bessel polynomial solutions of high-order linear Volterra integro-differential equations, Comput. Math. Appl 62(4) (2011) 1940-1956.

43. Zarebnia, M. Sinc numerical solution for the Volterra integro-differential equation, Commun. Nonlinear Sci. Numer. Simulat 15(3) (2010) 700-706.