Toeplitz Matrix Method and Volterra-Hammerstien Integral Equation with a Generalized Singular Kernel

A. M. Al-Bugami

Abstract


In this work, the existence of a unique solution of Volterra-Hammerstein integral equation of the second kind (V-HIESK) is discussed. The Volterra integral term (VIT) is considered in time with a continuous kernel, while the Fredholm integral term (FIT) is considered in position with a generalized singular kernel. Using a numerical technique, V-HIESK is reduced to a nonlinear system of Fredholm integral equations (SFIEs). Using Toeplitz matrix method we have  a nonlinear algebraic system of equations. Also, many important theorems related to the existence and uniqueness of the produced algebraic system are derived. Finally, some numerical examples when the kernel takes the logarithmic, Carleman, and Cauchy forms, are considered.

Keywords


Singular integral equation; Nonlinear Volterra-Fredholm integral equation; Toeplitz matrix; Cauchy kernel; Carleman kernel

Full Text:

PDF

References


[1] Abdou, M. A., & Salama, F. A. (2004). Volterra-Fredholm integral equation of the first kind and spectral relationships. Appl. Math. Comput., 153, 141-153.

[2] EL-Borai, M. M., Abdou, M. A., & EL-Kojok, M. M. (2006). On a discussion of nonlinear integral equation of type volterra-fredholm. J. KSIAM, 10(2), 59-83.

[3] Maleknejad, K., & Sohrabi, S. (2008). Legendre polynomial solution of nonlinear volterra–fredholm integral equations. IUST IJES, 19(5-2), 49-52

[4] Ezzati, R. & Najufalizadeh, S. (2011). Numerical solution of nonlinear Volterra-Fredholm integral equation by using Chebyshev polynomials. Mathematical Sciences, 5(1), 1-12.

[5] Ahmed, S. S. (2011). Numerical solution for Volterra-Fredholm integral equation of the second kind by using least squares technique. Iraqi Journal of Science, 52(4), 504-512.

[6] J. Ahmadi Shali, A. A. Joderi Akbarfam, G. Ebadi, (2012). Approximate Solution of Nonlinear Volterra–Fredholm integral equation. International Journal of Nonlinear Science, 14(4), 425-433.

[7] Abdou, M. A., Mohamed, K. I., & Ismail, A. S. (2003). On the numerical solutions of fredholm–volterra integral equation. Appl. Math. Comput., 146, 713-728

[8] Abdou,M. El-Borai,A. M. & Kojak, M. M. (2009). Toeplitz matrix method and nonlinear integral equation of hammerstein type. J. Comp. Appl. Math., 223, 765-776.

[9] Atkinson, K. E. (1997). The numerical solution of integral equation of the second kind. Cambridge.

[10] Delves, L. M., & Mohamed, J. L. (1985). Computational methods for integral equations.




DOI: http://dx.doi.org/10.3968/j.pam.1925252820130602.2593

DOI (PDF): http://dx.doi.org/10.3968/g5264

Refbacks

  • There are currently no refbacks.


Copyright (c)



Share us to:   

Reminder

If you have already registered in Journal A and plan to submit article(s) to Journal B, please click the "CATEGORIES", or "JOURNALS A-Z" on the right side of the "HOME".


We only use the follwoing mailboxes to deal with issues about paper acceptance, payment and submission of electronic versions of our journals to databases:
[email protected]
[email protected]

 Articles published in Progress in Applied Mathematics are licensed under Creative Commons Attribution 4.0 (CC-BY).

 ROGRESS IN APPLIED MATHEMATICS Editorial Office

Address: 1055 Rue Lucien-L'Allier, Unit #772, Montreal, QC H3G 3C4, Canada.

Telephone: 1-514-558 6138
Http://www.cscanada.net
Http://www.cscanada.org
E-mail:[email protected] [email protected] [email protected]

Copyright © 2010 Canadian Research & Development Center of Sciences and Cultures