Newtonian iterative scheme with simultaneous iterations of inverse derivative Original Research Article

A modification of the Continuous Analogy of Newton Method for the numerical solving of nonlinear problems is suggested. It permits one to replace the inversion of the derivative operator on every step of iterations by its inversion only in the initial approximation point. Then the extended system of the differential equations in Hilbert space, introduced in the work, permits the realization of the iterative process with the simultaneous calculation of the inverse derivative operator. The convergence theorem is proved for the method under almost the same conditions as for CANM. Numerical calculations for the model problem (Kirchhoff equation) have shown the effectiveness and adequately fast convergence of the iterative schemes based on the suggested method.