Convergence of Variational Iteration Method for Solving Singular Partial Differential Equations of Fractional Order

Elbeleze, Asma Ali; Kılıçman, Adem; Taib, Bachok M.

doi:10.1155/2014/518343

Cited by 2 publications

(5 citation statements)

References 17 publications

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“…There are many references discussing the convergence of approximate solutions by variational iteration method, one can see [33] for example. In ref.…”

Section: Variational Iteration Laplace Transform Methodsmentioning

confidence: 99%

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Series Solutions of High-Dimensional Fractional Differential Equations

2021

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In the present paper, the series solutions and the approximate solutions of the time–space fractional differential equations are obtained using two different analytical methods. One is the homotopy perturbation Sumudu transform method (HPSTM), and another is the variational iteration Laplace transform method (VILTM). It is observed that the approximate solutions are very close to the exact solutions. The solutions obtained are very useful and significant to analyze many phenomena, and the solutions have not been reported in previous literature. The salient feature of this work is the graphical presentations of the third approximate solutions for different values of order α.

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“…There are many references discussing the convergence of approximate solutions by variational iteration method, one can see [33] for example. In ref.…”

Section: Variational Iteration Laplace Transform Methodsmentioning

confidence: 99%

“…In ref. [33], based on the sufficient condition that guarantees the existence of a unique solution, the authors proved that the series solution is convergence.…”

Section: Variational Iteration Laplace Transform Methodsmentioning

confidence: 99%

Series Solutions of High-Dimensional Fractional Differential Equations

2021

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“…Some properties of the Gamma Function 𝛤(𝛼 + 1) = 𝛼𝛤(𝛼) = 𝛼!. Definition 2.2 [30], [31] The Riemann-Liouville fractional integral operator of order 𝛼 ≥ 0 is defined as follows:…”

Section: Definition 21[29]mentioning

confidence: 99%

“…Definition 2.4 [30], [31] The Caputo time-fractional derivative operator of order 𝛼 > 0 is defined as follows: (…”

Section: -mentioning

confidence: 99%

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Application of the Variational Iteration Method for the time-fractional Kaup-Kupershmidt Equation and the Boussinesq-Burger equation

Shihab,

Taha,

Hameed

et al. 2023

Iraqi Journal of Science

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The variational iteration method is used to deal with linear and nonlinear differential equations. The main characteristics of the method lie in its flexibility and ability to accurately and easily solve nonlinear equations. In this work, a general framework is presented for a variational iteration method for the analytical treatment of partial differential equations in fluid mechanics. The Caputo sense is used to describe fractional derivatives. The time-fractional Kaup-Kupershmidt (KK) equation is investigated, as it is the solution of the system of partial differential equations via the Boussinesq-Burger equation. By comparing the results that are obtained by the variational iteration method with those obtained by the two-dimensional Legendre multiwavelet, the optimal homotopy asymptotic method (OHAM), the q-homotopy analysis transform method, the Laplace Adomian Decomposition Method, and the homotopy perturbation method, the first method proved to be very effective and convenient. The main methodology in this work is anticipated to be applied to various fractional calculus, linear, and nonlinear problems.

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Convergence of Variational Iteration Method for Solving Singular Partial Differential Equations of Fractional Order

Cited by 2 publications

References 17 publications

Series Solutions of High-Dimensional Fractional Differential Equations

Series Solutions of High-Dimensional Fractional Differential Equations

Application of the Variational Iteration Method for the time-fractional Kaup-Kupershmidt Equation and the Boussinesq-Burger equation

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