Numerical and Analytic Method for Solving Proposal Fuzzy Nonlinear Volterra Integral Equation by Using Homotopy Analysis Method

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.

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

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.

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

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.

The Numerical Solution of Integral Equations on the Half-Line

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.

Solving linear and nonlinear Abel fuzzy integral equations by homotopy analysis method

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.

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.

In this paper, an iterative numerical method has been developed to solve nonlinear fuzzy Volterra integral equations based on three-point quadrature formula. The error estimation of the method is obtained based on Lipschitz condition and in order to confirm the yielded theoretical results, we perform the iterative method on some numerical examples.

<div>This paper mainly focuses on the recent advances in the some approximated methods for solving fuzzy Volterra-Fredholm integral equations, namely, Adomian decomposition method, variational iteration method and homotopy analysis method. We converted fuzzy Volterra-Fredholm integral equation to a system of Volterra-Fredholm integral equation in crisp case. The approximated methods using to find the approximate solutions of this system and hence obtain an approximation for the fuzzy solution of the fuzzy Volterra-Fredholm integral equation. To assess the accuracy of each method, algorithms with Mathematica 6 according is used. Also, some numerical examples are included to demonstrate the validity and applicability</div><div>of the proposed techniques.</div>This paper mainly focuses on the recent advances in the some approximated methods for solvingfuzzy Volterra-Fredholm integral equations, namely, Adomian decomposition method, variational iterationmethod and homotopy analysis method. We converted fuzzy Volterra-Fredholm integral equation to asystem of Volterra-Fredholm integral equation in crisp case. The approximated methods using to find theapproximate solutions of this system and hence obtain an approximation for the fuzzy solution of the fuzzyVolterra-Fredholm integral equation. To assess the accuracy of each method, algorithms with Mathematica 6according is used. Also, some numerical examples are included to demonstrate the validity and applicabilityof the proposed techniques.

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

Numerical solution for a generalized form of nonlinear cordial Volterra integral equations using quasilinearization and Legendre-collocation methods