scholarly journals On Newton-Kantorovich Method for Solving the Nonlinear Operator Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
Hameed Husam Hameed ◽  
Z. K. Eshkuvatov ◽  
Anvarjon Ahmedov ◽  
N. M. A. Nik Long
Keyword(s):  
Approximate Solution ◽  
Integral Equations ◽  
Operator Equation ◽  
Unknown Function ◽  
Existence And Uniqueness ◽  
Nonlinear Operator ◽  
Volterra Integral Equations ◽  
Nonlinear Operator Equation ◽  
Kantorovich Method ◽  
Majorant Function

We develop the Newton-Kantorovich method to solve the system of2×2nonlinear Volterra integral equations where the unknown function is in logarithmic form. A new majorant function is introduced which leads to the increment of the convergence interval. The existence and uniqueness of approximate solution are proved and a numerical example is provided to show the validation of the method.

scholarly journals SOLUTION OF A NONLINEAR OPERATOR EQUATION OF THE FIRST KIND WITH A CONTINUOUS POSITIVE LINEAR OPERATOR

2021 ◽  
pp. 118-123
Author(s):  
I.A. Usenov ◽  
R.K. Usenova ◽  
A. Nurkalieva
Keyword(s):  
Hilbert Space ◽  
Exact Solution ◽  
Approximate Solution ◽  
Linear Operator ◽  
Operator Equation ◽  
Nonlinear Operator ◽  
Regularization Parameter ◽  
Positive Linear Operator ◽  
Nonlinear Operator Equation ◽  
The Right

In the space H, a nonlinear operator equation of the first kind is studied, when the linear, nonlinear operator and the right-hand side of the equation are given approximately. Based on the method of Lavrent'ev M.M. an approximate solution of the equation in Hilbert space is constructed. The dependence of the regularization parameter on errors was selected. The rate of convergence of the approximate solution to the exact solution of the original equation is obtained.

scholarly journals An iterative method for solution of nonlinear operator equation

2016 ◽  
Vol 13 (1) ◽  
pp. 6-10
Author(s):  
Nguyễn Bường
Keyword(s):  
Hilbert Space ◽  
Iterative Method ◽  
Integral Equations ◽  
Operator Equation ◽  
Iterative Process ◽  
Nonlinear Operator ◽  
Nonlinear Operator Equation ◽  
Nonlinear Integral Equations

In the note, for finding a solution of nonlinear operator equation of Hammerstein’s type an iterative process in infinite-dimentional Hilbert space is shown, where a new iteration is constructed basing on two last steps. An example in the theory of nonlinear integral equations is given for illustration.

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

2014 ◽  
Vol 6 (2) ◽  
Author(s):  
Z.K. Eshkuvatov ◽  
A. Akhmedov ◽  
N.M.A. Nik Long ◽  
O. Shafiq
Keyword(s):  
Approximate Solution ◽  
Integral Operator ◽  
Integral Equations ◽  
Iteration Method ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
Kantorovich Method ◽  
Nonlinear Integral Equations ◽  
System Of Integral Equations ◽  
Nonlinear Integral Operator

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.

scholarly journals Fuzzy Volterra integral equations with in-finite delay

2009 ◽  
Vol 40 (1) ◽  
pp. 19-29 ◽  
Author(s):  
P. Prakash ◽  
V. Kalaiselvi
Keyword(s):  
Integral Equations ◽  
Existence And Uniqueness ◽  
Infinite Delay ◽  
Volterra Integral Equations ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Uniqueness Of Solutions ◽  
Finite Delay

In this paper, we study the existence and uniqueness of solutions for a class of fuzzy Volterra integral equations with infinite delay by using the method of successive approximations.

Oscillation analysis of numerical solution of Nonlinear Operator Equation

2021 ◽  
Vol 7 (5) ◽  
pp. 2111-2126
Author(s):  
Yang Zhou ◽  
Cuimei Li
Keyword(s):  
Numerical Solution ◽  
Operator Equation ◽  
Vibration Analysis ◽  
Nonlinear Operator ◽  
Vibration Spectrum ◽  
Nonlinear Operator Equation ◽  
Analysis Method ◽  
Analysis Model ◽  
Oscillation Analysis ◽  
Analysis Equation

There is a problem of low accuracy in the analysis of the vibration of the numerical solution of the nonlinear operator equation. In this work, the vibration analysis equation is constructed by the step-by-step search method, and the vibration quadrant of the equation is divided by the dichotomy method. The vibration spectrum is determined by the iteration method, and the vibration analysis model of the numerical solution of the nonlinear operator equation is constructed. The vibration analysis of the numerical solution of the nonlinear operator equation is completed based on the solution of the model and the numerical calculation and display of the step-by-step Fourier. The experimental results show that the proposed method has higher accuracy than the traditional vibration analysis method, which meets the requirements of the vibration analysis of the numerical solution of nonlinear operator equation.

scholarly journals On the Fredholm integral equation associated with pairs of dual integral equations

1969 ◽  
Vol 16 (3) ◽  
pp. 185-194 ◽  
Author(s):  
V. Hutson
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Bessel Function ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Unknown Function ◽  
Sufficient Conditions ◽  
Neumann Series ◽  
Dual Integral Equations ◽  
Image Position

Consider the Fredholm equation of the second kindwhereand Jv is the Bessel function of the first kind. Here ka(t) and h(x) are given, the unknown function is f(x), and the solution is required for large values of the real parameter a. Under reasonable conditions the solution of (1.1) is given by its Neumann series (a set of sufficient conditions on ka(t) for the convergence of this series is given in Section 4, Lemma 2). However, in many applications the convergence of the series becomes too slow as a→∞ for any useful results to be obtained from it, and it may even happen that f(x)→∞ as a→∞. It is the aim of the present investigation to consider this case, and to show how under fairly general conditions on ka(t) an approximate solution may be obtained for large a, the approximation being valid in the norm of L2(0, 1). The exact conditions on ka(t) and the main result are given in Section 4. Roughly, it is required that 1 -ka(at) should behave like tp(p>0) as t→0. For example, ka(at) might be exp ⌈-(t/ap)⌉.

scholarly journals Periodic solutions of Volterra integral equations

1988 ◽  
Vol 11 (4) ◽  
pp. 781-792 ◽  
Author(s):  
M. N. Islam
Keyword(s):  
Periodic Solutions ◽  
Integral Equations ◽  
Existence And Uniqueness ◽  
Contraction Mapping ◽  
Contraction Mapping Principle ◽  
Volterra Integral Equations ◽  
System Of Equations ◽  
Convolution Kernel ◽  
Basic Tool ◽  
Resolvent Function

Consider the system of equationsx(t)=f(t)+∫−∞tk(t,s)x(s)ds,           (1)andx(t)=f(t)+∫−∞tk(t,s)g(s,x(s))ds.       (2)Existence of continuous periodic solutions of (1) is shown using the resolvent function of the kernelk. Some important properties of the resolvent function including its uniqueness are obtained in the process. In obtaining periodic solutions of (1) it is necessary that the resolvent ofkis integrable in some sense. For a scalar convolution kernelksome explicit conditions are derived to determine whether or not the resolvent ofkis integrable. Finally, the existence and uniqueness of continuous periodic solutions of (1) and (2) are btained using the contraction mapping principle as the basic tool.

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