Variational iterative method: an appropriate numerical scheme for solving system of linear Volterra fuzzy integro-differential equations

Abstract

In this research article, we focus on the system of linear Volterra fuzzy integro-differential equations and we propose a numerical scheme using the variational iteration method (VIM) to get a successive approximation under uncertainty aspects. We have U j (t) = f (t) + t a k(t, x)u(x) dx, (1) where j refers to the jth order of the integro-differential equation and j = 1, 2, 3,. .. , n. k(t, x) are integral kernel and a function of t andx, which arise in mathematical biology, physics and more. The variational iteration technique gives the more accurate results at the very small cost of iterations leading to exact solutions quickly. The benefits of the proposal, an algorithmic form of the VIM, are also designed. To illustrate the potentiality of the scheme, two test problems are given and the approximate solutions are compared with the exact solution and also represented graphically.

Efficient approximate analytical methods for nonlinear fuzzy boundary value problem

“…In order for the (1) to be solved using VIM, we must fuzzify and defuzzify the VIM as defined in (2) [34]. According to VIM in [22] and for all 𝑟 ∈ [0,1] we rewrite (1) ) + 𝛿 ∫ 𝜆(𝑡; 𝜂) {𝑦 𝑖 (𝑛) (𝜂; 𝑟)} 𝑑𝜂.…”

“…This approach is helpful for directly solving linear and nonlinear problems with 𝑛 -th order boundary value problems (BVPs) without reducing them to a BVP system. It has been reported by many authors, such as [22], that VIM is more robust than other analytical approaches, like ADM and HPM. Compared to HPM and ADM, where computer algorithms are commonly used for nonlinear terms, VIM is used explicitly without any nonlinear terms requirement or restrictive assumptions [23].…”

The solution of nonlinear parabolic equation using variational iteration method

“…Tidak seperti metode semi analitik lainnnya, kontruksi metode iterasi variasi tidak melibatkan linierisasi atau gangguan kecil. Metode ini juga telah sukses menyelesaikan berbagai tipe persamaan diferensial, baik elementer maupun parsial, seperti: persamaan Duffing dan pendulum [15], persaman Lotka-Volterra satu spesies [16], persamaan parabolik linear [17], sistem persamaan diferensial integral [18], problem Lane-Emden dan Emden-Fowler [19], dan sistem persamaan integral diferensial fuzzy Volterra linier [20].…”

Implementation of variational iteration method for various types of linear and nonlinear partial differential equations

“…Narayanamoorthy and Mathankumar [35] have claimed that VIM is more robust than other analytical methodologies like DTM and HPM. In contrast to HPM and DTM, which frequently use computer methods for nonlinear terms, VIM is used explicitly with no nonlinear term needs or restricted assumptions.…”