Solution of fractional partial differential equations using iterative method

Dhaigude, Chandradeepa D.; Nikam, Vasant R.

doi:10.2478/s13540-012-0046-8

Cited by 37 publications

(20 citation statements)

References 29 publications

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“…But these fractional differential equations are difficult to get their exact solutions [9,12,13,21]. So, these types of equations are solved by various methods such as Adomian decomposition method [1,22], variational iteration method [7,20], differential transform method [2,6], homotopy perturbation method [11,15], an iterative method [4,23], finite element method [19,25], finite difference method [3,18], etc..…”

Section: Introductionmentioning

confidence: 99%

A new Mittag-Leffler function undetermined coefficient method and its applications to fractional homogeneous partial differential equations

Li¹,

Sun²,

Yin³

et al. 2017

J. Nonlinear Sci. Appl.

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In this paper, we develop a new application of the Mittag-Leffler function that will extend the application to fractional homogeneous differential equations, and propose a Mittag-Leffler function undetermined coefficient method. A new solution is constructed in power series. When a very simple ordinary differential equation is satisfied, no matter the original equation is linear or nonlinear, the method is valid, then combine the alike terms, compare the coefficient with identical powers, and the undetermined coefficient will be obtained. The fractional derivatives are described in the Caputo sense. To illustrate the reliability of the method, some examples are provided, and the solutions are in the form of generalized Mittag-Leffler function. The results reveal that the approach introduced here are very effective and convenient for solving homogeneous differential equations with fractional order.

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Section: Introductionmentioning

confidence: 99%

A new Mittag-Leffler function undetermined coefficient method and its applications to fractional homogeneous partial differential equations

Li¹,

Sun²,

Yin³

et al. 2017

J. Nonlinear Sci. Appl.

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“…For example, in [9] the solutions for both linear and nonlinear initial value problem for time-space fractional transport (reaction-advection) equations and fractional diffusionwave equations are obtained by using iterative method. In [10] the multiterm time-space Caputo-Riesz fractional advection diffusion equations with Dirichlet nonhomogeneous boundary conditions are considered.…”

Section: Introductionmentioning

confidence: 99%

Reaction-Advection-Diffusion Equations with Space Fractional Derivatives and Variable Coefficients on Infinite Domain

Japundžić

Rajter-Ćirić

2015

FCAA

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In this paper we consider a space fractional reaction-advection-diffusion equation which is actually a semi-linear Cauchy problem with a spatial fractional derivative operator of order α, 0 < α < 2. We establish and prove assertions concerning the existence and uniqueness of solution within certain Colombeau space. The proofs are given in the case when the left Liouville fractional derivative is involved, but the results are also valid in the case of the right Liouville fractional derivative as well as for the Riesz fractional derivative. The solutions are obtained by using the theory of generalized uniformly continuous semigroups of operators. We have also proved that the non-regularized and corresponding regularized operators, as well as solutions of non-regularized and corresponding regularized equations are L 2 −associated.MSC 2010 : 26A33, 35R11, 46F30

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“…In addition, using polynomials to approximate the fractional system is an effective method as well, such as Jacobi polynomials [22], Bernstein polynomials [23], and Chebyshev and Legendre polynomials [24]. In this paper, we introduce a type of iterative method, based on decomposing the nonlinearity term, for solving a class of functional equations [25][26][27][28][29][30].…”

Section: Introductionmentioning

confidence: 99%

An Iterative Method for Time-Fractional Swift-Hohenberg Equation

Pang

2018

Advances in Mathematical Physics

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We study a type of iterative method and apply it to time-fractional Swift-Hohenberg equation with initial value. Using this iterative method, we obtain the approximate analytic solutions with numerical figures to initial value problems, which indicates that such iterative method is effective and simple in constructing approximate solutions to Cauchy problems of time-fractional differential equations.

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Solution of fractional partial differential equations using iterative method

Abstract: The purpose of this paper is to obtain solutions for both linear and nonlinear initial value problems (IVPs) for fractional transport equations and fractional diffusion-wave equations using the iterative method.MSC 2010 : Primary 26A33; Secondary 31B10

Cited by 37 publications

References 29 publications

A new Mittag-Leffler function undetermined coefficient method and its applications to fractional homogeneous partial differential equations

A new Mittag-Leffler function undetermined coefficient method and its applications to fractional homogeneous partial differential equations

Reaction-Advection-Diffusion Equations with Space Fractional Derivatives and Variable Coefficients on Infinite Domain

An Iterative Method for Time-Fractional Swift-Hohenberg Equation

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