## On the solution of a mixed nonlinear integral equation.(English)Zbl 1209.65140

Summary: We consider a mixed nonlinear integral equation of the second kind in position and time. The existence of a unique solution of this equation is discussed and proved. A numerical method is used to obtain a system of Harmmerstein integral equations of the second kind in position. Then the modified Toeplitz matrix method, as a numerical method, is used to obtain a nonlinear algebraic system. Many important theorems related to the existence and uniqueness solution to the produced nonlinear algebraic system are derived. The rate of convergence of the total error is discussed. Finally, numerical examples when the kernel of position takes a logarithmic and Carleman forms, are presented and the error estimate, in each case, is calculated.

### MSC:

 65R20 Numerical methods for integral equations 45G05 Singular nonlinear integral equations 45G15 Systems of nonlinear integral equations
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### References:

 [1] Abdou, M. A., Fredholm-Volterra integral equation of the first kind and contact Problem, Appl. Math. Comput., 1259, 177-193 (2002) · Zbl 1028.45003 [2] Abdou, M. A., Integral equation of mixed type and integrals of orthogonal polynomials, J. Comput. Appl. Math., 138, 273-285 (2002) · Zbl 0994.45001 [3] Abdou, M. A., On the solution of linear and nonlinear integral equation, Appl. Math. Comput., 146, 857-871 (2003) · Zbl 1041.45006 [4] Abdou, M. A., On asymptotic method in contact problems of Fredholm integral equation of the second kind, Korean J. Comput. Appl. Math., 9, 261-275 (2002) · Zbl 0997.65140 [5] M.A. Abdou, K.M. Abu Anaja, Computational method for solving Volterra-Hammerstein integral equation in time and position, Int. J. Non. Sci., in press.; M.A. Abdou, K.M. Abu Anaja, Computational method for solving Volterra-Hammerstein integral equation in time and position, Int. J. Non. Sci., in press. [6] Abdou, M. A.; El-Borai, M. M.; El-Kojok, M. M., Toeplitz matrix method and nonlinear integral equation of Hammerstein type, J. Comput. Appl. Math., 223, 765-776 (2009) · Zbl 1156.65105 [7] Abdou, M. A.; El-Sayed, W. G.; Deebs, E. I., A solution of a nonlinear integral equation, Appl. Math. Comput., 160, 1-14 (2005) · Zbl 1068.45005 [8] M.A. Abdou, M.A. El-Sayed, D.A. Maturi, Spectral relationships of mixed integral equation using potential theory method (Part I), IJAMM, 1173, in press.; M.A. Abdou, M.A. El-Sayed, D.A. Maturi, Spectral relationships of mixed integral equation using potential theory method (Part I), IJAMM, 1173, in press. [9] K.E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge, 1977.; K.E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge, 1977. [10] Bannas, J., Integrable solution of Hammerstein and Uryshon integral equations, J. Integral Equat. Appl., 4, 89-94 (1992) · Zbl 0755.45005 [11] Brunner, H.; Makroglou, A.; Miller, R. K., On mixed collocation method for Volterra integral equations with periodic solution, Appl. Numer. Math., 24, 115-130 (1997) · Zbl 0878.65117 [12] T.A. Burton, Volterra Integral and Differential Equations, London, New York, 1985.; T.A. Burton, Volterra Integral and Differential Equations, London, New York, 1985. [13] Constanda, C., Integral equation of the first kind in plane elasticity, J. Quart. Appl. Math., L111, 4, 783-793 (1995) · Zbl 0923.45002 [14] El-Borai, M. M.; Abdou, M. A.; El-Kojok, M. M., On a discussion of nonlinear integral equation of type Volterra-Hammerstein, J. KSM Ser. B, 39, 1-15 (2008) · Zbl 1178.45007 [15] Emmanuele, G., Integrable solutions of Hammerstein equations, Appl. Anal., 50, 277-287 (1993) · Zbl 0795.45004 [16] Zhang, C.; He, Y., The extended one-leg methods for nonlinear neutral delay-integro-differential equations, Appl. Numer. Math., 59, 1409-1418 (2009) · Zbl 1163.65052 [17] Golbabai, A.; Keramati, B., Easy computational approach to solution of system of linear Fredholm integral equations, Chaos Solitons Fract., 38, 568-574 (2008) · Zbl 1146.65331 [18] Abou El-Seoud, A. A.M.; El-Kady, M. M.; El-Ameen, M. A., On approximation solution of Hammerstein integral equations in the space $$L_p (p$$⩾1), Appl. Math. Comput., 140, 91-144 (2003) · Zbl 1031.65147 [19] Diogo, T.; Lima, P., Superconvergence of collocation methods for a class of weakly singular Volterra integral equations, J. Comput. Appl. Math., 218, 307-316 (2008) · Zbl 1146.65084 [20] Linz, P., Analytical and Numerical Methods for Volterra Equations (1985), SIAM: SIAM Philadelphia · Zbl 0566.65094
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