Some classes of nonlinear synthesis problems for radiation systems: theory and methods of solution

The Numerical Solution of Integral Equations on the Half-Line

Numerial Methods for Volterra Integral Equations with Singular Kernels

In this paper, first, we use the Rich type iterative method for numerical solution of nonlinear fuzzy Fredholm integral equation. The main structure of this method is based on the Richardson iterative method and trapezoidal quadrature formula for Numerical solution of nonlinear fuzzy integral equations. Convergence, error analysis and numerical stability of the proposed method are also discussed.

This paper discussed some existence theorems for nonlinear functional integral equations in the space L^1 of Lebesgue integrable functions,by using the Darbo fixed point theorem associated with the Hausdorff measure of noncompactness. Also, as an application, we discuss the existence of solutions for some nonlinear integral equations with fractional order.

A new approach for numerical solution of two-dimensional nonlinear Fredholm integral equations in the most general kind of kernel, based on Bernstein polynomials and its convergence analysis

When considering systems of differential equations with very general boundary conditions, exact solution methods encounter great difficulties, which become insurmountable in the study of nonlinear problems. In this case it is necessary to apply to certain numerical methods. It is important to note that the use of numerical methods often allows you to abandon the simplified interpretation of the mathematical model of the process. The problems of numerical solution of nonlinear Volterra integral equations of the first kind with a differentiable kernel, which degenerates at the initial point of the diagonal, are studied in the paper. This equation is reduced to the Volterra integral equation of the third kind and a numerical method is developed on the basis of that regularized equation. The convergence of the numerical solution to the exact solution of the Volterra integral equation of the first kind is proved, an estimate of the permissible error and a recursive formula of the computational process are obtained. Keywords: nonlinear integral equation, system of nonlinear algebraic equations, error vectors, the Volterra equation, small parameter, numerical methods.

An iterative solution of a particular non-linear Fredholm integral equation is considered. The method is based on the reduction of the equation for the residual at any stage of the iteration to a linear integral equation for the required perturbation, using Newton' s method. This equation is then solved by using a low order expansion in terms of Chebyshev polynomials. The numerical integrations required in the basic iterative process are carried out by the powerful Clenshaw-Curtis quadrature prescription. The non-linear integral equation considered arises from the unstable two-point boundary value problem for a tubular chemical reactor. The capacity of the present method in dealing with such numerical difficulties is illustrated in representative calculations. Comparisons are made with conventional solutions of the parent differential equation and also with attempts at direct iterative solution of the integral equation by standard Neumann-Liouville series and finally with the direct algebraic approach.

Experiments have been carried out using on-line control of a digital computer to obtain solutions for certain non-linear integral equations. The computer program did not anticipate any particular method of solution; each user constructed his own method during the solution process. Easy cases, those for which the straight-forward iteration converges, were solved with half as many iterations. Difficult cases that were both globally and locally divergent were solved through careful control by the problem solver. Photographs displaying the solution process for two sample cases are included.

Analysis of the Euler and trapezoidal discretization methods for the numerical solution of nonlinear functional Volterra integral equations of Urysohn type

Wavelet Galerkin method for numerical solution of nonlinear integral equation

The problems of Engineering and Science can easily represent by developing their mathematical models in the terms of integral equations. Various analytical and numerical methods are available that can be used for solving integral equations of different kinds. In this paper, authors have considered recently developed integral transform “Rishi Transform” for obtaining the exact solution of non-linear Volterra integral equation of first kind (NLVIEFK). Four numerical problems have considered for demonstrating the complete procedure of determining the exact solution. Results of these problems depict that Rishi transform is very effective integral transform and it provides the exact solution of NLVIEFK without doing complicated calculation work.

The purpose of this note is to apply a generalized Kantorovich majorization principle to existence and uniqueness results for Hölder solutions of nonlinear singular integral equations. In contrast to the classical Kantorovich priziciple, we do notrequire differentiability, but only a local Lipschitz condition.

Numerical solution of nonlinear two-dimensional integral equations using rationalized Haar functions

Location, separation and approximation of solutions of nonlinear Hammerstein-type integral equations

The synthesis problem of the waveguide array according to the prescribed amplitude radiation pattern is considered. The mutual coupling of the separate elements of array is taken into account in the process of solution of a direct problem (analysis problem). Variational statement of the synthesis problem and application of the aperture orthogonal polynomials (AOP) method, are considered as alternative approaches to solve the synthesis problem. The functional representing a mean-square deviation of the prescribed and received amplitude radiation patterns is minimized, as well as the expansion of sought amplitude-phase distribution of current, in the series by some kind of AOP that implies the finding the expansion coefficients, are used for finding the optimal amplitude distributions of current in array. The modeling results are presented.