A Study of Some Iterative Methods for Solving Fuzzy Volterra-Fredholm Integral Equations

<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.

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Modified Adomian Decomposition Method for Solving Fuzzy Volterra-Fredholm Integral Equation

Journal of the Indian Mathematical Society ◽

10.18311/jims/2018/16260 ◽

2018 ◽

Vol 85 (1-2) ◽

pp. 53 ◽

Cited By ~ 4

Author(s):

Ahmed A. Hamoud ◽

Kirtiwant P. Ghadle

Keyword(s):

Integral Equation ◽

Integral Equations ◽

Fredholm Integral Equation ◽

Decomposition Method ◽

Adomian Decomposition Method ◽

Fredholm Integral Equations ◽

Modified Adomian Decomposition Method ◽

Adomian Decomposition ◽

Uniqueness Of The Solution ◽

Fuzzy Solution

In this paper, a modied Adomian decomposition method has been applied to approximate the solution of the fuzzy Volterra-Fredholm integral equations of the first and second Kind. That, a fuzzy Volterra-Fredholm integral equation has been converted to a system of Volterra-Fredholm integral equations in crisp case. We use MADM to find the approximate solution of this system and hence obtain an approximation for the fuzzy solution of the Fuzzy Volterra-Fredholm integral equation. A nonlinear evolution model is investigated. Moreover, we will prove the existence, uniqueness of the solution and convergence of the proposed method. Also, some numerical examples are included to demonstrate the validity and applicability of the proposed technique.

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THE ADOMIAN DECOMPOSITION METHOD APPLIED TO THE FUZZY SYSTEM OF FREDHOLM INTEGRAL EQUATIONS OF THE SECOND KIND

International Journal of Uncertainty Fuzziness and Knowledge-Based Systems ◽

10.1142/s0218488506003868 ◽

2006 ◽

Vol 14 (01) ◽

pp. 101-110 ◽

Cited By ~ 29

Author(s):

S. ABBASBANDY ◽

T. ALLAHVIRANLOO

Keyword(s):

Integral Equation ◽

Integral Equations ◽

Fredholm Integral Equation ◽

Fuzzy System ◽

Decomposition Method ◽

Adomian Decomposition Method ◽

Fredholm Integral Equations ◽

Extension Principle ◽

Numerical Example ◽

Adomian Decomposition

In this work, the Adomian decomposition(AD) method is applied to the Fuzzy system of linear Fredholm integral equations of the second kind(FFIE). First the crisp Fredholm integral equation is solved by AD method and then the crisp solution is fuzzified by extension principle. The proposed algorithm is illustrated by solving a numerical example.

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Numerical Solution of Nonlinear Fredholm Integral Equations by Using NKM and ADM

Dhaka University Journal of Science ◽

10.3329/dujs.v65i1.54506 ◽

2017 ◽

Vol 65 (1) ◽

pp. 61-66

Author(s):

MM Hasan ◽

MA Matin

Keyword(s):

Integral Equation ◽

Integral Equations ◽

Fredholm Integral Equation ◽

Decomposition Method ◽

Adomian Decomposition Method ◽

Fredholm Integral Equations ◽

Quadrature Method ◽

Adomian Decomposition ◽

Non Linear ◽

Nonlinear Fredholm Integral Equations

In this paper, we present a numerical method to solve a non-linear Fredholm integral equations. We intend to approximate the solution of this equation by Newton-Kantorovich-quadrature method and Adomian Decomposition method compare both the methods accurately for solving the non-linear Fredholm integral equation. Dhaka Univ. J. Sci. 65(1): 61-66, 2017 (January)

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Homotopy Analysis Method to Solve Two-Dimensional Nonlinear Volterra-Fredholm Fuzzy Integral Equations

Fractal and Fractional ◽

10.3390/fractalfract4010009 ◽

2020 ◽

Vol 4 (1) ◽

pp. 9

Author(s):

Atanaska Georgieva ◽

Snezhana Hristova

Keyword(s):

Integral Equation ◽

Integral Equations ◽

Homotopy Analysis Method ◽

Approximate Solutions ◽

Fredholm Integral Equations ◽

Fuzzy Integral ◽

Two Dimensional ◽

Analysis Method ◽

Homotopy Analysis ◽

Fuzzy Solution

The main goal of the paper is to present an approximate method for solving of a two-dimensional nonlinear Volterra-Fredholm fuzzy integral equation (2D-NVFFIE). It is applied the homotopy analysis method (HAM). The studied equation is converted to a nonlinear system of Volterra-Fredholm integral equations in a crisp case. Approximate solutions of this system are obtained by the help with HAM and hence an approximation for the fuzzy solution of the nonlinear Volterra-Fredholm fuzzy integral equation is presented. The convergence of the proposed method is proved and the error estimate between the exact and the approximate solution is obtained. The validity and applicability of the proposed method is illustrated on a numerical example.

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Solution of the space-fractional telegraph equations by using HAM

Ciência e Natura ◽

10.5902/2179460x20789 ◽

2015 ◽

Vol 37 ◽

pp. 320

Author(s):

Mehdi Abedi-Varaki ◽

Shahram Rajabi ◽

Vahid Ghorbani ◽

Farzad Hosseinzadeh

Keyword(s):

Homotopy Analysis Method ◽

Decomposition Method ◽

Adomian Decomposition Method ◽

Approximate Solutions ◽

Analysis Method ◽

Space And Time ◽

Adomian Decomposition ◽

Homotopy Analysis ◽

Telegraph Equations

In this study by using the Homotopy Analysis Method (HAM) obtained approximate solutions for the space and time-fractional telegraph equations. In Caputo sense (Yildirim, 2010)these equations considered. Examples are solved and the obtained results show to be more accurate than Adomian Decomposition Method (ADM) and are more efficient and commodious.

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Analytical Solutions for Rumor Spreading Dynamical Model in a Social Network

Nonlinear Engineering ◽

10.1515/nleng-2014-0025 ◽

2015 ◽

Vol 4 (1) ◽

Cited By ~ 3

Author(s):

R. Fallahpour ◽

S. Chakouvari ◽

H. Askari

Keyword(s):

Decomposition Method ◽

Adomian Decomposition Method ◽

Variational Iteration Method ◽

Analysis Method ◽

Rumor Spreading ◽

Spreading Model ◽

Adomian Decomposition ◽

Homotopy Analysis ◽

The Subject ◽

Forgetting Mechanism

AbstractIn this paper, Laplace Adomian decomposition method is utilized for evaluating of spreading model of rumor. Firstly, a succinct review is constructed on the subject of using analytical methods such as Adomian decomposion method, Variational iteration method and Homotopy Analysis method for epidemic models and biomathematics. In continue a spreading model of rumor with consideration of forgetting mechanism is assumed and subsequently LADM is exerted for solving of it. By means of the aforementioned method, a general solution is achieved for this problem which can be readily employed for assessing of rumor model without exerting any computer program. In addition, obtained consequences for this problem are discussed for different cases and parameters. Furthermore, it is shown the method is so straightforward and fruitful for analyzing equations which have complicated terms same as rumor model. By employing numerical methods, it is revealed LADM is so powerful and accurate for eliciting solutions of this model. Eventually, it is concluded that this method is so appropriate for this problem and it can provide researchers a very powerful vehicle for scrutinizing rumor models in diverse kinds of social networks such as Facebook, YouTube, Flickr, LinkedIn and Tuitor.

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The solutions of three dimensional Fredholm integral equations using Adomian decomposition method

10.1063/1.4952533 ◽

2016 ◽

Cited By ~ 1

Author(s):

Mohammad Almousa

Keyword(s):

Integral Equations ◽

Decomposition Method ◽

Three Dimensional ◽

Adomian Decomposition Method ◽

Fredholm Integral Equations ◽

Adomian Decomposition

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Solving two-dimensional nonlinear fuzzy Volterra integral equations by homotopy analysis method

Demonstratio Mathematica ◽

10.1515/dema-2021-0005 ◽

2021 ◽

Vol 54 (1) ◽

pp. 11-24

Author(s):

Atanaska Georgieva

Keyword(s):

Integral Equation ◽

Approximate Solution ◽

Integral Equations ◽

Volterra Integral Equation ◽

Homotopy Analysis Method ◽

Volterra Integral Equations ◽

Two Dimensional ◽

Analysis Method ◽

Homotopy Analysis ◽

Fuzzy Solution

Abstract The purpose of the paper is to find an approximate solution of the two-dimensional nonlinear fuzzy Volterra integral equation, as homotopy analysis method (HAM) is applied. Studied equation is converted to a nonlinear system of Volterra integral equations in a crisp case. Using HAM we find approximate solution of this system and hence obtain an approximation for the fuzzy solution of the nonlinear fuzzy Volterra integral equation. The convergence of the proposed method is proved. An error estimate between the exact and the approximate solution is found. The validity and applicability of the HAM are illustrated by a numerical example.

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