Analytical solution of a fractional diffusion equation by variational iteration method

The demand of many scientific areas for the usage of fractional partial differential equations (FPDEs) to explain their real-world systems has been broadly identified. The solutions may portray dynamical behaviors of various particles such as chemicals and cells. The desire of obtaining approximate solutions to treat these equations aims to overcome the mathematical complexity of modeling the relevant phenomena in nature. This research proposes a promising approximate-analytical scheme that is an accurate technique for solving a variety of noninteger partial differential equations (PDEs). The proposed strategy is based on approximating the derivative of fractional-order and reducing the problem to the corresponding partial differential equation (PDE). Afterwards, the approximating PDE is solved by using a separation-variables technique. The method can be simply applied to nonhomogeneous problems and is proficient to diminish the span of computational cost as well as achieving an approximate-analytical solution that is in excellent concurrence with the exact solution of the original problem. In addition and to demonstrate the efficiency of the method, it compares with two finite difference methods including a nonstandard finite difference (NSFD) method and standard finite difference (SFD) technique, which are popular in the literature for solving engineering problems.

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“…The interested readers are refereed for more details to the following monographs [1,2,5,17]. Definition 1.…”

Section: Notes On Fractional Calculusmentioning

confidence: 99%

“…Consequently, the solution of FPDEs represents nowadays a vigorous research area for scientists and finding approximate and exact solutions to FPDEs is an important task. However, PDEs are commonly hard to tackle, and their fractional-order types are more complicated [1,2,17,18].…”

Section: Introductionmentioning

confidence: 99%

An Accurate Approximate-Analytical Technique for Solving Time-Fractional Partial Differential Equations

et al. 2017

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“…Numerical solutions for various types of FDEs by applying several techniques were proposed, for instance, Adomian's decomposition method [2,3], the Taylor collocation method [4], the variational iteration method [5], the finite difference method [6,7] and the ultraspherical wavelets method [8,9]. In addition, orthogonal polynomials have been widely used for obtaining numerical solutions for different types of FDEs.…”

Section: Introductionmentioning

confidence: 99%

A Novel Operational Matrix of Caputo Fractional Derivatives of Fibonacci Polynomials: Spectral Solutions of Fractional Differential Equations

Abd-Elhameed

Youssri

2016

Entropy

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Herein, two numerical algorithms for solving some linear and nonlinear fractional-order differential equations are presented and analyzed. For this purpose, a novel operational matrix of fractional-order derivatives of Fibonacci polynomials was constructed and employed along with the application of the tau and collocation spectral methods. The convergence and error analysis of the suggested Fibonacci expansion were carefully investigated. Some numerical examples with comparisons are presented to ensure the efficiency, applicability and high accuracy of the proposed algorithms. Two accurate semi-analytic polynomial solutions for linear and nonlinear fractional differential equations are the result.

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“…Several forms of fractional differential equations have been proposed in standard models, and there has been significant interest in developing numerical schemes for their solutions. There are numerous methods that deal with these types of equations; some of these methods are Laplace transform [20], Adomian decomposition method (ADM) [14], variational iteration method (VIM) [8], fractional difference method (FDM) [18], a quadrature tau method [4] and a shifted Jacobi spectral method [9].Recently, there have been a number of schemes devoted to the solution of fractional differential equations. These schemes can be broadly classified into two classes, numerical and analytical.The homotopy perturbation method [19], [16], [3]and [1] homotopy analysis method [5] and [21] Taylor matrix method [13] and Haar wavelet method [17] have been used to solve the fractional-order Riccati differential equation.…”

Section: Introductionmentioning

confidence: 99%

Application of Generalized Differential Transform Method to Fractional Order Riccati Differential Equation and Numerical Result

Bansal¹,

Jain²

2015

Int. J. of Pure and Appl. Math.

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Analytical solution of a fractional diffusion equation by variational iteration method

Cited by 176 publications

References 9 publications

An Accurate Approximate-Analytical Technique for Solving Time-Fractional Partial Differential Equations

An Accurate Approximate-Analytical Technique for Solving Time-Fractional Partial Differential Equations

A Novel Operational Matrix of Caputo Fractional Derivatives of Fibonacci Polynomials: Spectral Solutions of Fractional Differential Equations

Application of Generalized Differential Transform Method to Fractional Order Riccati Differential Equation and Numerical Result

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