Numerical solution of nonlinear fuzzy Volterra integral equations of the second kind for changing sign kernels

Saberirad, Fateme; Karbassi, S. M.; Heydari, Mohammad Hossein

doi:10.1007/s00500-018-3668-x

Cited by 9 publications

(12 citation statements)

References 17 publications

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“…Several numerical techniques have been used successfully for fuzzy integral equations [1,2,4,9,13,15,16,17]. In many cases it is not proved that the approximate solution is a fuzzy function.…”

Section: Methodsmentioning

confidence: 99%

“…In [4] also a convergence rate of O(h) is obtained. Numerical solution of fuzzy Volterra integral equations is considered in [13,15,16,17], but in many cases it is not proved that the approximate solution is a fuzzy function (in some cases it may be trivial, but in other cases it is not true in general). The convergence rates have usually not been considered.…”

Section: Introductionmentioning

confidence: 99%

“…The convergence rates have usually not been considered. Fuzzy Volterra integral equations with changing sign kernels have, to our knowledge, been considered only in [15], but there, only a trivial special case where the sign can only change on horizontal lines is considered, the smoothness of the solution is not proved, and the convergence results are only valid under additional assumptions not mentioned in the paper. There are also a lot of papers which only describe some numerical method for solving fuzzy integral equations and give some numerical examples, but do not provide any analysis at all.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Collocation Method for Fuzzy Volterra Integral Equations of the Second Kind

Alijani

Kangro

2020

Mathematical Modelling and Analysis

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In this paper we consider fuzzy Volterra integral equation of the second kind whose kernel may change sign. We give conditions for smoothness of the upper and lower functions of the solution. For numerical solution we propose the collocation method with two different basis function sets: triangular and rectangular basis. The smoothness results allow us to obtain the convergence rates of the methods. The proposed methods are illustrated by numerical examples, which confirm the theoretical convergence estimates.

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Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Collocation Method for Fuzzy Volterra Integral Equations of the Second Kind

Alijani

Kangro

2020

Mathematical Modelling and Analysis

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“…Although some numerical approaches in last decade were used to solve FVIE2 such as trapezoidal quadrature rule method [15] solved several types of FVIE2, in [16] a numerical method based on quadrature rules is suggested to solve nonlinear FVIE2, Iterative numerical method [17] is introduced to solve some examples involved nonlinear FVIE2 and so on. From the previous analysis, it would be important to analyze the RKM6 which was not used to solve FVIE2.…”

Section: Introductionmentioning

confidence: 99%

Numerical Treatment for Solving Fuzzy Volterra Integral Equation by Sixth Order Runge-Kutta Method

Esa¹,

Ibraheem²,

Jameel³

2021

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There has recently been considerable focus on finding reliable and more effective numerical methods for solving different mathematical problems with integral equations. The Runge-Kutta methods in numerical analysis are a family of iterative methods, both implicit and explicit, with different orders of accuracy, used in temporal and modification for the numerical solutions of integral equations. Fuzzy Integral equations (known as FIEs) make extensive use of many scientific analysis and engineering applications. They appear because of the incomplete information from their mathematical models and their parameters under fuzzy domain. In this paper, the sixth order Runge-Kutta is used to solve second-kind fuzzy Volterra integral equations numerically. The proposed method is reformulated and updated for solving fuzzy second-kind Volterra integral equations in general form by using properties and descriptions of fuzzy set theory. Furthermore a Volterra fuzzy integral equation, based on the parametric form of a fuzzy numbers, transforms into two integral equations of the second kind in the crisp case under fuzzy properties. We apply our modified method using the specific example with a linear fuzzy integral Volterra equation to illustrate the strengths and accurateness of this process. A comparison of evaluated numerical results with the exact solution for each fuzzy level set is displayed in the form of table and figures. Such results indicate that the proposed approach is remarkably feasible and easy to use.

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“…With time, this concept found interesting applications in the modeling of many real-world problems and it motivated many researchers to develop the area of fuzzy integral equations where the values of functions involved are fuzzy numbers. The topics related to fuzzy integral equations have received particular attention from the research community during the last few decades [4,10,17,18].…”

Section: Introductionmentioning

confidence: 99%

Collocation Method for Fuzzy Volterra Integral Equation of the Second Kind with Weakly Singular Kernels

Alijani

Kangro

2021

Preprint

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In this paper we consider fuzzy Volterra integral equation of the second kind with weakly singular kernel. We propose piecewise spline collocation methods with a graded mesh. By increasing the number of collocation points we show that the numerical solution exists and converges to the exact solution. We obtain exact convergence vlues depending on smoothness of the solution and on the grading parameter of the mesh.We give sufficient conditions for the fuzziness of the approximate solution. The proposed method is illustrated by numerical examples which confirm the theoretical convergence estimates.

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Numerical solution of nonlinear fuzzy Volterra integral equations of the second kind for changing sign kernels

Cited by 9 publications

References 17 publications

Collocation Method for Fuzzy Volterra Integral Equations of the Second Kind

Collocation Method for Fuzzy Volterra Integral Equations of the Second Kind

Numerical Treatment for Solving Fuzzy Volterra Integral Equation by Sixth Order Runge-Kutta Method

Collocation Method for Fuzzy Volterra Integral Equation of the Second Kind with Weakly Singular Kernels

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