An iterative numerical method to solve nonlinear fuzzy Volterra-Hammerstein integral equations

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

Numerial Methods for Volterra Integral Equations with Singular Kernels

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.

Preface. 1. Introduction and Preliminaries. 2. Existence Theory for Nonlinear Fredholm and Volterra Integrodifferential Equations. 3. Solution Sets of Abstract Volterra Equations. 4. Existence Theory for Nonlinear Fredholm and Volterra Integral Equations on Compact Intervals. 5. Existence Theory for Nonlinear Fredholm and Volterra Integral Equations on Half-Open Intervals. 6. Existence Theory for Nonlinear Nonresonant Operator and Integral Equations. 7. Existence Theory for Nonlinear Resonant Operator and Integral Equations. 8. Integral Inclusions. 9. Approximation of Solutions of Operator Equations on the Half Line. 10. Operator Equations in Banach Spaces Relative to the Weak Topology. 11. Stochastic Integral Equations. 12. Periodic Solutions for Operator Equations. Index.

The Numerical Solution of Integral Equations on the Half-Line

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.

This research focuses on addressing both linear and nonlinear fuzzy Volterra integral equations that feature piecewise continuous kernels. The problem is tackled using the method of successive approximations. The study discusses the existence and uniqueness of solutions for these fuzzy Volterra integral equations with piecewise kernels. Numerical results are obtained by applying the successive approximations method to examples for both linear and nonlinear scenarios. Error analysis graphs are plotted to illustrate the accuracy of the method. Furthermore, a comparative analysis is presented through graphs of approximate solutions for different fuzzy parameter values. To highlight the effectiveness and significance of the successive approximations method, a comparison is made with the traditional homotopy analysis technique. The results indicate that the successive approximation method outperforms the homotopy analysis method in terms of accuracy and effectiveness.

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.

<abstract><p>This paper is concerned with obtaining approximate solutions of fuzzy Fredholm integral equations using Picard iteration method and bivariate Bernstein polynomials. We first present the way to approximate the value of the multiple integral of any fuzzy-valued function based on the two dimensional Bernstein polynomials. Then, it is used to construct the numerical iterative method for finding the approximate solutions of two dimensional fuzzy integral equations. Also, the error analysis and numerical stability of the method are established for such fuzzy integral equations considered here in terms of supplementary Lipschitz condition. Finally, some numerical examples are considered to demonstrate the accuracy and the convergence of the method.</p></abstract>

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

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.

The purpose of this paper is to provide an efficient numerical technique for solving two-dimensional nonlinear fuzzy Fredholm integral equations. The fuzzy Gauss quadrature rule is applied to the above mentioned equation, which reduces the fuzzy integral equations to the nonlinear approximate equations that can be easily solved by a numerical iterative algorithm. The convergence of the presented numerical method is investigated under several mild conditions. Finally, some illustrative numerical experiments are provided to illustrate the efficiency and accuracy of the proposed method.

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.

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

In the present study, first, we introduce an iterative method based on quadrature formula for solving two-dimensional nonlinear fuzzy Fredholm integral equations (2DNFFIE). Then, we present error estimation and the numerical stability analysis for the proposed method. Finally, to show the efficiency of the proposed method, supporting examples are also provided.