Solving Fuzzy Volterra Integrodifferential Equations of Fractional Order by Bernoulli Wavelet Method

Abstract

A matrix method called the Bernoulli wavelet method is presented for numerically solving the fuzzy fractional integrodifferential equations. Using the collocation points, this method transforms the fuzzy fractional integrodifferential equation to a matrix equation which corresponds to a system of nonlinear algebraic equations with unknown coefficients. To illustrate the method, it is applied to certain fuzzy fractional integrodifferential equations, and the results are compared.

Iterative fuzzy Bernstein polynomials method for nonlinear fuzzy Volterra integral equations

“…If we compare the results obtained in Example 2 for n = 10 with the results obtained in Mosleh and Otadi (2013) with n = 3, in the context of fuzzy Volterra linear integral equations, we see that the accuracy is the same, but our method was applied both for linear and nonlinear integral equations. The accuracy of our method is not so good as was obtained in Mastani Shabestari et al (2018) with Bernoulli wavelets, but in Mastani Shabestari et al (2018), all the considered numerical examples are linear. We can conclude from these that our method has not high speed of convergence as the projection methods, but it requires smaller computational cost and could be applied both to linear and nonlinear fuzzy integral equations avoiding possible difficulties that could appear in the direct integration of nonlinear kernels.…”

“…In the literature, such numerical methods are based on various techniques such as variational iteration (see Attari and Yazdani 2011), iterative methods (see Bede and Gal 2004;Behzadi et al 2012;Bica and Popescu 2017;Friedman et al 1999;Bica 2016 andPark et al 2000) and fuzzy differential transforms (see Salahshour and Allahviranloo 2013). Recently, the Bernoulli wavelet method was applied for finding the numerical solution of fuzzy fractional integrodifferential equations using the collocation points which transform the equation into a system of nonlinear algebraic equations with unknown coefficients (see Mastani Shabestari et al 2018). In the construction of numerical methods for fuzzy integral equations, Bernstein polynomials are involved in Ezzati and Ziari (2011) and Mosleh and Otadi (2011) for approximating the solution of linear fuzzy Fredholm integral equations, while in Mosleh and Otadi (2013), the Bernstein polynomials are used to linear fuzzy Volterra integral equations.…”

Solving Fuzzy System of Volterra Integro-differential Equations by using Adomian Decomposition Method

“…The triangular fuzzy number [2,20,28], which is defined as a fuzzy set in R and characterized by an ordered triple u = (a, b, c) ∈ R 3 with a ≤ b ≤ c such that u (r) = a + (b − a) r and ū (r) = c − (c − b) r, are the endpoints of r−level sets for all r ∈ [0, 1] is a popular fuzzy number:…”

“…Bani Issaa et al [19] applied numerical methods to slove the fuzzy integro-differential equations of the second kind. Shabestari et al [20] have been solved Fuzzy Volterra Integrodifferential equations of fractional order by Bernoulli Wavelet method. Das and Talukdar [21] solved fuzzy integro-differential equation by using fuzzy Laplace Transformation.…”

Ulam‐Hyers Stability and Ulam‐Hyers‐Rassias Stability for Fuzzy Integrodifferential Equation

“…Fuzzy differential and integrodifferential equation (FD-FIDE) are a natural way to model dynamical systems subject to uncertainties. In the past few years, the study of fuzzy integrodifferential equations is an area of mathematics that has recently received a lot of attention (see, e.g., [1][2][3][4][5][6][7][8][9][10][11]). Alikhani et al [3] studied the existence and uniqueness of global solutions for fuzzy initial value problems via integrodifferential operators of Volterra type.…”