Existence Theory for Nonlinear Integral and Integrodifferential Equations

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.

Numerial Methods for Volterra Integral Equations with Singular Kernels

The Numerical Solution of Integral Equations on the Half-Line

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

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 work proposes a new defintion of the nonlinear functional Volterra integral equation ib 2D of the second kind with continuous kernel (NT-DFVIE). Furthermore, the work is concerned to study this new equation numerically. The existence of a unique solution of the equation is proved. In addition, the approximate solutions are obtained by two powerful methods Adomian Decomposition Method (ADM) and Block by block method (BBM). The given numerical examples showed the efficiency and accuracy of the introduced methods.

In this paper, we use quasilinearization technique, product integration rule, and collocation method to present a new numerical method to solve nonlinear fractional Volterra integro‐differential equations with logarithmic weakly singular kernel. After examining the behavior of the solution of the integro‐differential equation, we convert it into a nonlinear Volterra integral equation with a logarithmic weakly singular kernel. We implement the quasilinearization method on the resulting nonlinear equation and derive a sequence of linear integral equations with low‐smooth solutions that are convergent to the solution of the nonlinear equation. By utilizing a regularization technique, we enhance the smoothness of the solutions of the linear equations. Then, we apply collocation method along with the product integration rule, to transform the regularized linear equations into a linear algebraic system with a unique solution. We discuss the convergence of the presented method and estimate the error bounds for the proposed numerical approach. Finally, through several numerical examples, we practically test the effectiveness and accuracy of the proposed numerical method, where the obtained numerical results show their agreement with the discussed theoretical results.

Discrete collocation method for solving two-dimensional linear and nonlinear fuzzy Volterra integral equations

: This document addresses 5 substantial problems in the theory and numerical analysis of ill-posed problems and integral equations: (1) Collocation, and Galerkin methods for Volterra and Abel equations of the first kind; (2) Galerkin and collocation methods for nonlinear Abel-Volterra integral equations on the half-line and on a finite interval; (3) a new approach to classification and regularization of ill-posed operator equations, and quantification of ill-posedness; (4) operator external problems in the theory of compensation and representation of control systems; (5) constrained least-squares solutions of linear inclusions and singular control problems in Hilbert space. New notions of bivariational and singular variational derivatives for functionals are also studies. They will be applied to extend the von Mises calculus for statistical functionals and its applications to robustness and approximation theorems. Keyboards: Multivalued linear mappings; Kernel functions.

In this paper, we presented the convergence of the solution for the nonlinear fuzzy volterra integral equation with high computational and complexity to find the solution in analytical method, so we describable this solution by using Homotopy analysis method, by using Banach fixed point theory for existence and uniqueness. That with explained numerical examples. Finally using the MAPLE program to solve our problem.

Wavelet Galerkin method for numerical solution of nonlinear integral equation

This paper presents a high accuracy combination algorithm for solving the systems of nonlinear Volterra integral and integro-differential equations with weakly singular kernels of the second kind. Two quadrature algorithms for solving the systems are discussed, which possess high accuracy order and the asymptotic expansion of the errors. By means of combination algorithm, we may obtain a numerical solution with higher accuracy order than the original two quadrature algorithms. Moreover an a posteriori error estimation for the algorithm is derived. Both of the theory and the numerical examples show that the algorithm is effective and saves storage capacity and computational cost.

To controlled systems under deterministic and random trade-off state, it is presented by general Langevin gradient equation while dynamic variational rule of the systems is described. The nonlinear differential equation is equivalent to its nonlinear Volterra's integral equation of the second kind, and any order approximate solution of the equation is also obtained, by successive approximation method. Finally, the influence of random effect for system state is discussed.

In this article, the Bielecki metric on the space is used to analyze the different types of stability results of nonlinear fractional integral equation with delay in its corresponding fractional boundary value problems. Sufficient conditions are obtained to prove stability results for fractional nonlinear Volterra and Fredholm integral equations with delay, given by Ulam, Hyer, and Rassias. Further, those stability results are extended to the fractional integral equations where the domain of integration is an unbounded interval. Two numerical examples are provided to assert the obtained stability results.

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.