Iterative numerical method for fuzzy Volterra linear integral equations in two dimensions

Bica, Alexandru Mihai; Ziari, Shokrollah

doi:10.1007/s00500-016-2085-2

Cited by 19 publications

(4 citation statements)

References 37 publications

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“…As we mentioned before, equation ( 11) is approximate form of equation (3). Thus, equation ( 18) is approximate form of equation (17).…”

Section: Existence Of a Fuzzy Approximate Solutionmentioning

confidence: 90%

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F¹-transform in Fuzzy Fredholm Integral Equations

Pham¹,

Perfilieva²

2021

Atlantis Studies in Uncertainty Modelling

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The goal of this study is to solve fuzzy Fredholm integral equations of the second kind. In order to solve these equations with respect to fuzzy valued functions, we propose a very powerful and relatively simple technique called fuzzy transform. This approach allows the transformation of a fuzzy Fredholm integral equation to a system of algebraic equations. A solution to this algebraic system gives the appropriate parameters of the inverse F1-transform. Hence, we can estimate the approximate solution to the original problem. The existence and uniqueness of the exact solution and approximate solution are also discussed.

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“…As we mentioned before, equation ( 11) is approximate form of equation (3). Thus, equation ( 18) is approximate form of equation (17).…”

Section: Existence Of a Fuzzy Approximate Solutionmentioning

confidence: 90%

“…The presence of fuzziness makes these equations more complicated then their classical versions. We can solve the equations by iterative computation [3,12] or we can approximate the solution by simple functions [9,10]. In this paper, we use the second method.…”

Section: Introductionmentioning

confidence: 99%

F¹-transform in Fuzzy Fredholm Integral Equations

Pham¹,

Perfilieva²

2021

Atlantis Studies in Uncertainty Modelling

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“…In many applications, certain problem parameters are typically defined by a fuzzy number rather than a crisp number, and it is therefore important to establish mathematical models and numerical procedures for the proper handling of fuzzy integral equations. Numerical approaches to fuzzy integral equations have inspired many research works in the last decade due to their use in scientific phenomena [7][8][9][10][11]. The existence and uniqueness of a second kind fuzzy Volterra equation solution was introduced in [12].…”

Section: Introductionmentioning

confidence: 99%

Solution and Analysis of the Fuzzy Volterra Integral Equations via Homotopy Analysis Method

Jameel¹,

Anakira²,

Alomari³

et al. 2021

Computer Modeling in Engineering &Amp; Sciences

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Homotopy Analysis Method (HAM) is semi-analytic method to solve the linear and nonlinear mathematical models which can be used to obtain the approximate solution. The HAM includes an auxiliary parameter, which is an efficient way to examine and analyze the accuracy of linear and nonlinear problems. The main aim of this work is to explore the approximate solutions of fuzzy Volterra integral equations (both linear and nonlinear) with a separable kernel via HAM. This method provides a reliable way to ensure the convergence of the approximation series. A new general form of HAM is presented and analyzed in the fuzzy domain. A qualitative convergence analysis based on the graphical method of a fuzzy HAM is discussed. The solutions sought by the proposed method show that the HAM is easy to implement and computationally quite attractive. Some solutions of fuzzy second kind Volterra integral equations are solved as numerical examples to show the potential of the method. The results also show that HAM provides an easy way to control and modify the convergence area in order to obtain accurate solutions.

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“…Also, two-dimensional fuzzy integral equations have been noticed by a lot of researchers because of their broad applications in engineering sciences. Some of the most important papers in this area are trapezoidal quadrature rule and iterative method [10][11][12], triangular functions [13], quadrature iterative [14], Bernstein polynomials [15], collocation fuzzy wavelet like operator [16], homotopy analysis method (HAM) [17], open fuzzy cubature rule [18], kernel iterative method [19], modified homotopy pertubation [20], block-pulse functions [21], optimal fuzzy quadrature formula [22], and finally, iterative method and fuzzy bivariate block-pulse functions [23]. Also, some researchers have solved one-dimensional fuzzy Fredholm integral equations by using fuzzy interpolation via iterative method such as: iterative interpolation method [9], Lagrange interpolation based on the extension principle [5], and spline interpolation [7].…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Two-Dimensional Linear Fuzzy Fredholm Integral Equations by the Fuzzy Lagrange Interpolation

Nouriani

Ezzati

2018

Advances in Fuzzy Systems

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In this study, at first, we propose a new approach based on the two-dimensional fuzzy Lagrange interpolation and iterative method to approximate the solution of two-dimensional linear fuzzy Fredholm integral equation (2DLFFIE). Then, we prove convergence analysis and numerical stability analysis for the proposed numerical algorithm by two theorems. Finally, by some examples, we show the efficiency of the proposed method.

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Iterative numerical method for fuzzy Volterra linear integral equations in two dimensions

Cited by 19 publications

References 37 publications

F¹-transform in Fuzzy Fredholm Integral Equations

F¹-transform in Fuzzy Fredholm Integral Equations

Solution and Analysis of the Fuzzy Volterra Integral Equations via Homotopy Analysis Method

Numerical Solution of Two-Dimensional Linear Fuzzy Fredholm Integral Equations by the Fuzzy Lagrange Interpolation

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Iterative numerical method for fuzzy Volterra linear integral equations in two dimensions

Cited by 19 publications

References 37 publications

F1-transform in Fuzzy Fredholm Integral Equations

F1-transform in Fuzzy Fredholm Integral Equations

Solution and Analysis of the Fuzzy Volterra Integral Equations via Homotopy Analysis Method

Numerical Solution of Two-Dimensional Linear Fuzzy Fredholm Integral Equations by the Fuzzy Lagrange Interpolation

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F¹-transform in Fuzzy Fredholm Integral Equations

F¹-transform in Fuzzy Fredholm Integral Equations