Approximate solutionn of a nonllinear system of integral equations using modified Newton-Kantorovich method


  • Z.K. Eshkuvatov
  • A. Akhmedov
  • N.M.A. Nik Long
  • O. Shafiq



Newton-Kantorovich method, Nonnlinear operator | Volterra integral equation, Trapezoiddal rule, Convergence,


Modified Newton-Kantorovich method is developed to obtain an approximate solution for a system of nonlinear integral equations. The system of nonlinear integral equations is reduced to find the roots of nonlinear integral operator. This nonlinear integral operator is solved by the modified Newton-Kantorovich method with initial conditions and this procedure is continued by iteration method to find the unknown functions. The existence and uniqueness of the solutions of the system are also proven.


Alefeld, G., On the convergence of Halley’s method, Amer. Math. Monthly 88(7) (1981) 530-536.

Argyros, I.K., On the approximation of some nonlinear equations, Aequationes Math.32 (1987) 87–95

Baker, C.T.H., A perspective on the numerical treatment of Volterra equations, J. of Comp. and App. Math 2000, vol. 125, pp. 217 -249.

Boikov, I.V., and Tynda, A.V, Approximate solution of nonlinear integral equations of the theory of developing systems, Vol. 39, No .9, 2003,

pp. 1277-1288.

Glushkov, V.M., Ivanov, V.V., and Yanenko, V.M., Modeling of Developing Systems, Moscow, 1983.

Kantorovich, L.V., The method of successive approximations for functional analysis, Acta. Math. 71 (1939) 63-97.

Kantorovich, L.V., On Newton’s method for functional equations, Dokl Akad. Nauk SSSR 59 (1948) 1237-1240 (in Russian).

Kantorovich, L.V., Akilov, G.P., Functional Analysis, 2nd ed., Pergamon Press, Oxford, 1982.

Smirnov, V.I., Course of Higher Mathematics, Moscow, 1957, vol. 4.