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

The Numerical Solution of Integral Equations on the Half-Line

In this paper an application of the homotopy analysis method for some type of nonlinear and linear integral equations of the second kind is presented. A special case of considered equation is the Volterra- Fredholm integral equation. In homotopy analysis method a series is created. It has shown that if the series is convergent, its sum is the solution of the considered equation. It has been also shown that under proper assumptions the considered equation possesses a unique solution and the series obtained in homotopy analysis method is convergent. The error of the approximate solution was estimated. This approximate solution is obtained when we limit to the partial sum of the series.Application of the method is illustrated with examples.

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, a novel technique, that is to read, the New Homotopy Perturbation Method (NHPM) is utilized for solving a linear and non-linear differential equations and integral equations. The two most important steps in the application of the new homotopy perturbation method are to invent a suitable homotopy equation and to choose a suitable initial conditions. Comparing between the effects of the method (NHPM), is given exact solution, and the method (HPM), is given approximate solution, in this paper, we make some instances are provided to prove the ability of the method (NHPM). Show that the method (NHPM) is valid and effective, easy and accurate in solving linear and nonlinear differential equations, compared with the Homotopy Perturbation Method (HPM).

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

In this work, we consider the second order nonlinear integro-differential Equation (IDEs) of the Volterra-Fredholm type. One of the popular methods for solving Volterra or Fredholm type IDEs is the method of quadrature while the problem of consideration is a linear problem. If IDEs are nonlinear or integral kernel is complicated, then quadrature rule is not most suitable; therefore, other types of methods are needed to develop. One of the suitable and effective method is homotopy analysis method (HAM) developed by Liao in 1992. To apply HAM, we firstly reduced the IDEs into nonlinear integral Equation (IEs) of Volterra-Fredholm type; then the standard HAM was applied. Gauss-Legendre quadrature formula was used for kernel integrations. Obtained system of algebraic equations was solved numerically. Moreover, numerical examples demonstrate the high accuracy of the proposed method. Comparisons with other methods are also provided. The results show that the proposed method is simple, effective and dominated other methods.

Simulation and evaluation of interval-valued fuzzy linear Fredholm integral equations with interval-valued fuzzy neural network

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.

New iterative method for solving linear and nonlinear hypersingular integral equations

Integral equations occur naturally in many field of mechanics and mathematical physics. In this paper, an iterative scheme based on the homotopy analysis method (HAM) has been used for solving one of the most important cases in nonlinear integral equations which is called urysohn form. For this scope, we applied some examples with known exact solutions and the numerical solutions obtained confirm that the method is very effective and simple. Also, show the advantage of this method, the results obtained by (HAM) are compared with NewtonKantorovich-quadrature method. Convergence is also observed.

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.

In this paper, the numerical solution of the linear and nonlinear fuzzy Volterraintegro-differential equations have been investigated using the analytical method namely homotopy analysis method (HAM) and then the proposed method is illustrated by solving two numerical examples. It was found that the HAM provides a simple way to adjust and control the convergence region of solution series by introducing a nonzero auxiliary parameter h.

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.