Projection solution methods for one nonlinear singular integral equation

The Numerical Solution of Integral Equations on the Half-Line

Numerial Methods for Volterra Integral Equations with Singular Kernels

Methods for solving the amplitude-phase problem for one-dimensional signals are proposed. The amplitude-phase problem is investigated in the case of continuous and discrete signals. In both cases, the amplitude-phase problem is modeled by nonlinear singular integral equations. The study of continuous signals leads to nonlinear singular integral equations defined on the numerical axis; discrete ones to nonlinear singular integral equations defined on a unit circle in the plane of a complex variable. The obtained singular integral equations relate to the exceptional case - the symbols of the Frechet derivatives of singular operators degenerate in the entire domain of their definition.The continuous operator method for solution of nonlinear equations is used for solution these nonlinear singular integral equations. The numerical schemes for solving corresponding singular integral equations are constructed. Solutions of model examples have shown effectiveness of the proposed method and numerical algorithms.

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.

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.

By using a non-linear 3-D elastic waves real-time expert system, the new theory of ‘real-time expert seismology’ is proposed, for the exploration of the on-shore and off-shore oil and gas reserves all over the world. This highly innovative and groundbreaking technology is working under real time logic for searching the on-shore and off-shore petroleum reserves developed on the continental crust and in deeper water ranging from 300 m to 2,500 m, or even deeper. Furthermore, for the determination of the properties of the reservoir materials, when oil reserves mixed with water in multiphase flows are moving through porous media, a new mathematical device is proposed. The above problem is very much important for petroleum reservoir engineering and the oil industry. Hence, such a problem is reduced to the solution of a non-linear singular integral equation, which is numerically evaluated by using the singular integral operators method (SIOM). Also, several properties are analysed and investigated for the porous medium equation, defined as a Helmholtz differential equation.

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

Many different problems in mathematics, physics, engineering can be expressed in the form of integral equations.Among these are diffraction problems, population growth, heat transfer, particle transport problems, electrical engineering, elasticity, control, elastic waves, diffusion problems, quantum mechanics, heat radiation, electrostatics and contact problems.Therefore, the solutions which are obtained by the mathematical methods play an important role in these fields.The most two basic types of integral equations are called Fredholm (FIEs) and Volterra (VIEs).In many instances, the ordinary and partial differential equations can be converted into Fredhom and Volterra integral equations that are solved more effectively.We aim through this research to present an improved Adomian decomposition method based on modified Bernstein polynomials (ADM-MBP) to solve nonlinear integral equations of the second kind.We introduced efficient method, constructed on modified Bernstein polynomials.The formulation is developed to solve nonlinear Fredholm and Volterra integral equations of second kind.This method is tested for some examples from nonlinear integral equations.Maple software was used to obtain the solutions of these examples.The results demonstrate reliability of the proposed method.Generally, the proposed method is very convenient to apply to find the solutions of Fredholm and Volterra integral equations of second kind.

We describe the numerical solution of a nonlinear Cauchy singular integral equation depending on an unknown parameter and referring to a free boundary value problem of free surface from a channel. We propose a Newton collocation method and discuss various computational aspects in order to obtain an effective algorithm. Numerical results are presented.

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.

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.

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

HE simplified Oswatitsch equation is a one-dimensional nonlinear singular integral equation, occurring in transonic aerodynamics.l It is an approximate version of Oswatitsch's integral equation, whose kernel has a dipole singularity. According to transonic small perturbation theory, the Oswatitsch equation is the basic equation governing the steady inviscid irrotational flow of a perfect gas past a thin symmetric profile at zero incidence, with subsonic freestream Mach number Mx rstrud 4 and Nixon5 deserve special mention. In the present work, the simplified Oswatitsch equation has been solved by two different numerical procedures, viz., the direct iteration scheme (DIS) proposed by Niyogi and Chakraborty6 and by the recent perturbed iterative scheme (PIS) put forward by Dey7 for solving a system of nonlinear algebraic equations. Further, this nonlinear integral equation was used as a test case for studying the global convergence behavior of PIS. From computational results, it has been found that for a parabolic arc profile there exists a range of values for the reduced thickness ratio r (which is a transonic similarity parameter), where both the procedures lead to the same shock-free supercritical solution, and that in this range PIS converges much faster than DIS. However, for higher values of r beyond this range, there exists another range where, contrary to expectations, DIS converges but PIS fails to converge, indicating that DIS has a wider range of convergence.

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.

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