Solving nonlinear fuzzy differential equations by using fuzzy variational iteration method

Allahviranloo, Tofigh; Abbasbandy, Saeid; Behzadi, Sh. Sadigh

doi:10.1007/s00500-013-1193-5

Cited by 19 publications

(7 citation statements)

References 44 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…VIM general structure for resolving problems TFPBVP is stated in [19]. 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.…”

Section: Description Of the Fuzzy Vimmentioning

confidence: 99%

Efficient approximate analytical methods for nonlinear fuzzy boundary value problem

Jameel

Saleh

Azmi

et al. 2022

IJECE

View full text Add to dashboard Cite

This paper aims to solve the nonlinear two-point fuzzy boundary value problem (TPFBVP) using approximate analytical methods. Most fuzzy boundary value problems cannot be solved exactly or analytically. Even if the analytical solutions exist, they may be challenging to evaluate. Therefore, approximate analytical methods may be necessary to consider the solution. Hence, there is a need to formulate new, efficient, more accurate techniques. This is the focus of this study: two approximate analytical methods-homotopy perturbation method (HPM) and the variational iteration method (VIM) is proposed. Fuzzy set theory properties are presented to formulate these methods from crisp domain to fuzzy domain to find approximate solutions of nonlinear TPFBVP. The presented algorithms can express the solution as a convergent series form. A numerical comparison of the mean errors is made between the HPM and VIM. The results show that these methods are reliable and robust. However, the comparison reveals that VIM convergence is quicker and offers a swifter approach over HPM. Hence, VIM is considered a more efficient approach for nonlinear TPFBVPs.

show abstract

Section: Description Of the Fuzzy Vimmentioning

confidence: 99%

Efficient approximate analytical methods for nonlinear fuzzy boundary value problem

Jameel

Saleh

Azmi

et al. 2022

IJECE

View full text Add to dashboard Cite

show abstract

“…The details of VIM to solve crisp differential equations have been introduced in [10][11][12]. The purpose of this section is to extend the fuzzy analysis of VIM to solve the system (1), from [14,15]: where the valued of the initial guessing � 0, ( ; ) that satisfy the initial conditions in Eq. (1).…”

Section: Description Of Fuzzy Vimmentioning

confidence: 99%

“…The VIM is an approximate method created by He and applied widely in many mathematics problems and engineering applications [10][11][12]. Moreover, in the last years, VIM have been used to find the approximate solution of a different order of FDEs involving linear and nonlinear problems [13][14][15]. In this work, by means of a linear FDE system, the unpredictable behaviours of CD4 + T-cells, CTLs [16] and the viral load in different patients are modelled.…”

Section: Introductionmentioning

confidence: 99%

Approximation treatment for Linear Fuzzy HIV Infection Model by Variational Iteration Method

Saleh¹,

Azmi²,

Jameel³

2021

View full text Add to dashboard Cite

There has recently been considerable focus on finding reliable and more effective approximate methods for solving biological mathematical models in the form of differential equations. One of the well-known approximate or semi-analytical methods for solving linear, nonlinear differential well as partial differential equations within various fields of mathematics is the Variational Iteration Method (VIM). This paper looks at the use of fuzzy differential equations in human immunodeficiency virus (HIV) infection modeling. The main advantage of the method lies in its flexibility and ability to solve nonlinear equations easily. VIM is introduced to provide approximate solutions for linear ordinary differential equation system including the fuzzy HIV infection model. The model explains the amount of undefined immune cells, and the immune system viral load intensity intrinsic that will trigger fuzziness in patients infected by HIV. CD4+T-cells and cytototoxic T-lymphocytes (CTLs) are known for the immune cells concerned. The dynamics of the immune cell level and viral burden are analyzed and compared across three classes of patients with low, moderate and high immune systems. A modification and formulation of the VIM in the fuzzy domain based on the use of the properties of fuzzy set theory are presented. A model was established in this regard, accompanied by plots that demonstrate the reliability and simplicity of the methods. The numerical results of the model indicate that this approach is effective and easily used in fuzzy domain.

show abstract

“…Jafari H. in 2012 in his article solved the n-th order linear and nonlinear fuzzy ordinary differential equations (FODE's) with fuzzy initial conditions by using variational iteration method (VIM), [3]. Subsequently, Allahvivallco T. et al in 2013 [4] solves fuzzy differential equations which are nonlinear by using the VIM and thereafter in 2014 Janeel A. F. used the VIM to approximate the analytical solution for an initial value problem involving fuzzy Duffing equation, [5]. Linear fuzzy ordinary differential equations of the n-th order were solved in 2013 at the same time by Xianbin Guo and Deauan Shang [6] and find the approximate solution in which the sign of the coefficient functions are maintained and investigated by the method of undetermined fuzzy coefficients.…”

Section: Introductionmentioning

confidence: 99%

Approximate Solution of Linear Fuzzy Initial Value Problems Using Modified Variaional Iteration Method

Sagban¹,

Fadhel²

2021

ANJS

View full text Add to dashboard Cite

The main objective of this paper is to solve fuzzy initial value problems, in which the fuzziness occurs in the initial conditions. The proposed approach, namely the modified variational iteration method, will be used to find the solution of fuzzy initial value problem approximately and to increase the rate of convergence of the variational iteration method. From the obtained results, as it is expected, the approximate results of the proposed method are more accurate than those results obtained without using the modified variational iteration method.

show abstract

Solving nonlinear fuzzy differential equations by using fuzzy variational iteration method

Cited by 19 publications

References 44 publications

Efficient approximate analytical methods for nonlinear fuzzy boundary value problem

Efficient approximate analytical methods for nonlinear fuzzy boundary value problem

Approximation treatment for Linear Fuzzy HIV Infection Model by Variational Iteration Method

Approximate Solution of Linear Fuzzy Initial Value Problems Using Modified Variaional Iteration Method

Contact Info

Product

Resources

About