Numerical solution of nonlinear mixed Volterra-Fredholm integral equations in complex plane via PQWs

The Numerical Solution of Integral Equations on the Half-Line

Numerial Methods for Volterra Integral Equations with Singular Kernels

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.

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.

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.

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.

In this paper, we propose a numerical scheme to solve a kind of nonlinear Fredholm integral equations of the second kind in the complex plane. The periodic quasi-wavelets (PQWs) constructed on [0; 2π] are utilised as a basis of iteration method. Using the Banach fixed point theorem, we obtain some results concerning the error analysis. Illustrative examples are included to demonstrate the validity and applicability of the technique.

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

In this paper for solving nonlinear system of mixed Volterra-Fredholm integral equations by using variational iteration method, we have used differentiation for converting problem to suitable form such that it can be useful for constructing a correction functional with general lagrange multiplier. The optimum of lagrange multiplier can be found by variational theorem and by choosing of restrict variations properly. By substituting of optimum lagrange multiplier in correction functional, we obtain convergent sequences of functions and by appropriate choosing initial approximation, we can get approximate of the exact solution of the problem with few iterations. Some applications of nonlinear mixed Volterra-Fredholm integral equations arise in mathematical modeling of the Spatio-temporal development of an epidemic. So nonlinear system of mixed Volterra-Fredholm integral equations is important and useful. The above method independent of small parameter in comparison with similar works such as perturbation method. Also this method does not require discretization or linearization. Accuracy of numerical results show that the method is very effective and it is better than Adomian decomposition method since it has faster convergence and it is more simple. Also this method has a closed form and avoids the round of errors for finding approximation of the exact solution. The looking forward the proposed method can be used for solving various kinds of nonlinear problems.

The random position of a particle in turbulent flow r(t;ω) is a vector random function of time t ≥ 0, ω ∈ Ω, where Ω is the supporting set of the underlying probability space (Ω,A,μ). If the statistics of a Lagrangian velocity field is known, the random position vector r(t;ω) is the solution of a nonlinear stochastic integral equation. Depending on the particular application, this equation may be homogeneous or inhomogeneous and with deterministic or random limits. A theorem is presented on the existence and uniqueness of the solution of this integral equation. The proof is based on Banach's fixed point theorem.

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.

Numerical solution of nonlinear weakly singular Volterra integral equations of the first kind: An hp-version collocation approach

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

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.

Numerical solution for three-dimensional nonlinear mixed Volterra–Fredholm integral equations via three-dimensional block-pulse functions