volume | PIER Journals

Abstract

A new and effective direct method to determine the numerical solution of specific nonlinear Volterra-Fredholm integral and integro-differential equations is proposed. The method is based on vector forms of block-pulse functions (BPFs). By using BPFs and its operational matrix of integration, an integral or integro-differential equation can be transformed to a nonlinear system of algebraic equations. Some numerical examples are provided to illustrate accuracy and computational efficiency of the method. Finally, the error evaluation of this method is presented. The benefits of this method are low cost of setting up the equations without applying any projection method such as Galerkin, collocation, . . . . Also, the nonlinear system of algebraic equations is sparse.

1. Arnold, M. D., "An efficient solution for scattering by a perfectly conducting strip grating," Journal of Electromagnetic Waves and Applications, Vol. 20, No. 7, 891-900, 2006. Google Scholar

2. Zhao, J. X., "Numerical and analytical formulations of the extended MIE theory for solving the sphere scattering problem," Journal of Electromagnetic Waves and Applications, Vol. 20, No. 7, 967-983, 2006. Google Scholar

3. Rui, P.-L. and R. Chen, "Implicity restarted gmres fast Fourier transform method for electromagnetic scattering," Journal of Electromagnetic Waves and Applications, Vol. 21, No. 7, 973-976, 2007. Google Scholar

4. Wang, M. Y., J. Xu, J. Wu, Y. Yan, and H.-L. Li, "FDTD study on scattering of metallic column covered by doublenegative metamaterial ," Journal of Electromagnetic Waves and Applications, Vol. 21, No. 14, 1905-1914, 2007. Google Scholar

5. Liu, X.-F., B. Z. Wang, and S.-J. Lai, "Element-free Galerkin method in electromagnetic scattering field computation," Journal of Electromagnetic Waves and Applications, Vol. 21, No. 14, 1915-1923, 2007. Google Scholar

6. Kumar, P., T. Chakravarty, S. Bhooshan, S. K. Khah, and A. De, "Numerical computation of resonant frequency of gap coupled circular microstrip antennas," Journal of Electromagnetic Waves and Applications, Vol. 21, No. 10, 1303-1311, 2007. Google Scholar

7. Ding, W., L. Chen, and C. H. Liang, "Numerical study of Goos-Hanchen shift on the surface of anisotropic left-handed materials," Progress In Electromagnetics Research B, Vol. 2, 151-164, 2008. Google Scholar

8. Popov, A. V. and V. V. Kopeikin, "Electromagnetic pulse propagation over nonuniform earth surface: Numerical simulation," Progress In Electromagnetics Research B, Vol. 6, 37-64, 2008. Google Scholar

9. Suyama, T., Y. Okuno, A. Matsushima, and M. Ohtsu, "A numerical analysis of stop band characteristics by multilayered dielectric gratings with sinusoidal profile," Progress In Electromagnetics Research B, Vol. 2, 83-102, 2008. Google Scholar

10. Mokari, H. and P. Derakhshan-Barjoei, "Numerical analysis of homojunction Gallium arsenide avalanche photodiodes (GAAs-APDs)," Progress In Electromagnetics Research B, Vol. 7, 159-172, 2008. Google Scholar

11. Steinbauer, M., R. Kubasek, and K. Bartusek, "Numerical method of simulation of material influences in MR tomography," Progress In Electromagnetics Research Letters, Vol. 1, 205-210, 2008. Google Scholar

12. Hatamzadeh-Varmazyar, S., M. Naser-Moghadasi, E. Babolian, and Z. Masouri, "Numerical approach to survey the problem of electromagnetic scattering from resistive strips based on using a set of orthogonal basis functions," Progress In Electromagnetics Research, Vol. 81, 393-412, 2008. Google Scholar

13. Hatamzadeh-Varmazyar, S., M. Naser-Moghadasi, and Z. Masouri, "A moment method simulation of electromagnetic scattering from conducting bodies," Progress In Electromagnetics Research, Vol. 81, 99-119, 2008. Google Scholar

14. Hatamzadeh-Varmazyar, S., M. Naser-Moghadasi, E. Babolian, and Z. Masouri, "Calculating the radar cross section of the resistive targets using the Haar wavelets," Progress In Electromagnetics Research, Vol. 83, 55-80, 2008. Google Scholar

15. Hatamzadeh-Varmazyar, S. and M. Naser-Moghadasi, "New numerical method for determining the scattered electromagnetic fields from thin wires," Progress In Electromagnetics Research B, Vol. 3, 207-218, 2008. Google Scholar

16. Hatamzadeh-Varmazyar, S. and M. Naser-Moghadasi, "An integral equation modeling of electromagnetic scattering from the surfaces of arbitrary resistance distribution," Progress In Electromagnetics Research B, Vol. 3, 157-172, 2008. Google Scholar

17. Jiang, G.-X., H.-B. Zhu, and W. Cao, "Implicit solution of modified form of time-domain electric field integral equation," Journal of Electromagnetic Waves and Applications, Vol. 21, No. 5, 697-707, 2007. Google Scholar

18. Franceschini, G., A. Abubakar, T. M. Habashy, and A. Massa, "A comparative assessment among iterative linear solvers dealing with electromagnetic integral equations in 3D inhomogeneous anisotropic media," Journal of Electromagnetic Waves and Applications, Vol. 21, No. 7, 899-914, 2007. Google Scholar

19. Wu, C. and G.-X. Jiang, "Stabilization procedure for the timedomain integral equation," Journal of Electromagnetic Waves and Applications, Vol. 21, No. 11, 1507-1512, 2007. Google Scholar

20. Tong, M. S., "A stable integral equation solver for electromagnetic scattering by large scatterers with concave surface," Progress In Electromagnetics Research, Vol. 74, 113-130, 2007. Google Scholar

21. Hussein, K. F. A., "Fast computational algorithm for EFIE applied to arbitrarily-shaped conducting surfaces," Progress In Electromagnetics Research, Vol. 68, 339-357, 2007. Google Scholar

22. Maleknejad, K. and M. Hadizadeh, "The numerical analysis of Adomian's decomposition method for nonlinear Volterra integral and integro-differential equations," International Journal of Engineering Science, Iran University of Science & Technology, Vol. 8, No. 2a, 33-48, 1997. Google Scholar

23. Wazwaz, A. M., A First Course in Integral Equations, World Scientific, 1997.

24. Akyuz-Dascioglu, A., "A Chebyshev polynomial approach for linear Fredholm-Volterra integro-differential equations in the most general form ," Appl. Math. Comput., Vol. 181, 103-112, 2006. Google Scholar

25. Brunner, H., Collocation Method for Volterra Integral and Relation Functional Equations, Cambridge University Press, 2004.

26. Delves, L. M. and J. L. Mohamed, Computational Methods for Integral Equations, Cambridge University Press, 1985.

27. Maleknejad, K. and H. Derili, "The collocation method for Hammerstein equations by Daubechies wavelets," Appl. Math. Comput., Vol. 172, 846-864, 2006. Google Scholar

28. Maleknejad, K., M. Karami, and M. Rabbani, "Using the Petrov-Galerkin elements for solving Hammerstein integral equations," Appl. Math. Comput., Vol. 172, 831-845, 2006. Google Scholar

29. Maleknejad, K. and Y. Mahmoudi, "Taylor polynomial solution of high-order nonlinear Volterra-Fredholm integro-differential equations," Appl. Math. Comput., Vol. 145, 641-653, 2003. Google Scholar

30. Babolian, E. and Z. Masouri, "Direct method to solve Volterra integral equation of the first kind using operational matrix with block-pulse functions," Journal of Computational and Applied Mathematics, in press, 2007, doi: 10.1016/j.cam.2007.07.029. Google Scholar

31. Babolian, E. and A. Salimi Shamloo, "Numerical solution of Volterra integral and integro-differential equations of convolution type by using operational matrices of piecewise constant orthogonal functions ," Journal of Computational and Applied Mathematics, Vol. 214, 495-508, 2008. Google Scholar

32. Xu, L., Y.-C. Guo, and X.-W. Shi, "Dielectric half space model for the analysis of scattering from objects on ocean surface," Journal of Electromagnetic Waves and Applications, Vol. 21, No. 15, 2287-2296, 2007. Google Scholar

33. Zhong, X. J., T. Cui, Z. Li, Y.-B. Tao, and H. Lin, "Terahertzwave scattering by perfectly electrical conducting objects," Journal of Electromagnetic Waves and Applications, Vol. 21, No. 15, 2331-2340, 2007. Google Scholar

34. Li, Y.-L., J.-Y. Huang, and M.-J. Wang, "Scattering cross section for airborne and its application," Journal of Electromagnetic Waves and Applications, Vol. 21, No. 15, 2341-2349, 2007. Google Scholar

35. Wang, M. Y., J. Xu, J. Wu, Y. Yan, and H.-L. Li, "FDTD study on scattering of metallic column covered by doublenegative metamaterial ," Journal of Electromagnetic Waves and Applications, Vol. 21, No. 14, 1905-1914, 2007. Google Scholar

36. Liu, X.-F., B. Z. Wang, and S.-J. Lai, "Element-free Galerkin method in electromagnetic scattering field computation," Journal of Electromagnetic Waves and Applications, Vol. 21, No. 14, 1915-1923, 2007. Google Scholar

37. Du, P., B. Z. Wang, H. Li, and G. Zheng, "Scattering analysis of large-scale periodic structures using the sub-entire domain basis function method and characteristic function method," Journal of Electromagnetic Waves and Applications, Vol. 21, No. 14, 2085-2094, 2007. Google Scholar

38. Tuz, V. R., "Three-dimensional Gaussian beam scattering from a periodic sequence of bi-isotropic and material layers ," Progress In Electromagnetics Research B, Vol. 7, 53-73, 2008. Google Scholar

39. Sukharevsky, O. I. and V. A. Vasilets, "Scattering of reflector antenna with conic dielectric radome," Progress In Electromagnetics Research B, Vol. 4, 159-169, 2008. Google Scholar

40. Wang, M.-J., Z.-S. Wu, and Y. L. Li, "Investigation on the scattering characteristics of Gaussian beam from two dimensional dielectric rough surfaces based on the Kirchhoff approximation," Progress In Electromagnetics Research B, Vol. 4, 223-235, 2008. Google Scholar

41. Sun, X. and H. Ha, "Light scattering by large hexagonal column with multiple densely packed inclusions," Progress In Electromagnetics Research Letters, Vol. 3, 105-112, 2008. Google Scholar

42. Kokkorakis, G. C., "Scalar equations for scattering by rotationally symmetric radially inhomogeneous anisotropic sphere," Progress In Electromagnetics Research Letters, Vol. 3, 179-186, 2008. Google Scholar

Professor Esmail Babolian

Dr. Zahra Masouri

Dr. Saeed Hatamzadeh