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

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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

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