Local Fractional Variational Iteration and Decomposition Methods for Wave Equation on Cantor Sets within Local Fractional Operators

Băleanu, Dumitru; Machado, J. A. Tenreiro; Cattani, Carlo; Baleanu, Mihaela Cristina; Yang, Xiao‐Jun

doi:10.1155/2014/535048

Cited by 64 publications

(86 citation statements)

References 28 publications

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“…To illustrate the idea of this method, let us consider the fractional order problem [30][31][32][33][34]:…”

Section: Adomian Decomposition Methods (Adm)mentioning

confidence: 99%

“…Therefore, approximation and numerical techniques can be used. Picard method [26][27][28][29] and the Adomian decomposition method [30][31][32][33][34] are two powerful approaches of these techniques which can be used in simple manner and short time to obtain analytical approximations to nonlinear problems and they are particularly valuable as tools for researchers, because they provide immediate and visible symbolic terms of analytic solutions, as well as numerical approximate solutions to nonlinear differential equations without linearization or discretization.…”

Section: Introductionmentioning

confidence: 99%

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Solution of the Fractional Form of Unsteady Squeezing Flow through Porous Medium

Hemeda¹,

Eladdad²,

Lairje³

2017

Mathematical Problems in Engineering

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We propose two friendly analytical techniques called Adomian decomposition and Picard methods to analyze an unsteady axisymmetric flow of nonconducting, Newtonian fluid. This fluid is assumed to be squeezed between two circular plates passing through porous medium channel with slip and no-slip boundary conditions. A single fractional order nonlinear ordinary differential equation is obtained by means of similarity transformation with the help of the fractional calculus definitions. The resulting fractional boundary value problems are solved by the proposed methods. Convergence of the two methods' solutions is confirmed by obtaining various approximate solutions and various absolute residuals for different values of the fractional order.Comparison of the results of the two methods for different values of the fractional order confirms that the proposed methods are in a well agreement and therefore they can be used in a simple manner for solving this kind of problems. Finally, graphical study for the longitudinal and normal velocity profiles is obtained for various values of some dimensionless parameters and fractional orders.

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“…To illustrate the idea of this method, let us consider the fractional order problem [30][31][32][33][34]:…”

Section: Adomian Decomposition Methods (Adm)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Solution of the Fractional Form of Unsteady Squeezing Flow through Porous Medium

Hemeda¹,

Eladdad²,

Lairje³

2017

Mathematical Problems in Engineering

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show abstract

“…Recently, there appeared a large part of scientific research concerning local fractional differential equations or local fractional partial differential, adopted in its entirety on the above mentioned methods to solve this new types of equations. For example, among these research we find, local fractional Adomian decomposition method ( [9], [10], [15]), local fractional homotopy perturbation method ( [11], [12]), local fractional homotopy perturbation Sumudu transform method [13], local fractional variational iteration method ( [14], [15]), local fractional variational iteration transform method ( [16]- [18]), local fractional Fourier series method ( [19]- [21]), Laplace transform series expansion method [22], local fractional Sumudu transform method ( [23], [24]), local fractional Sumudu transform series expansion method ( [25], [26]), local fractional Sumudu decomposition method for linear partial differential equations with local fractional derivative [27].…”

Section: Introductionmentioning

confidence: 99%

Exact solutions for linear systems of local fractional partial differential equations

Ziane¹,

Cherif²,

Belghaba³

2018

Malaya J. Mat.

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The basic motivation of the present study is to apply the local fractional Sumudu decomposition method to solve linear system of local fractional partial differential equations. The local fractional Sumudu decomposition methodl (LFSDM) can easily be applied to many problems and it's capable of reducing the size of computational work to find non-differentiable solutions for similar problems. Some illustrative examples are given, revealing the effectiveness and convenience of this method. KeywordsLocal fractional derivative operator, local fractional Sumudu decomposition method, linear systems of local fractional partial differential equations.

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“…The heat conduction equation was discussed by the help of local fractional derivative (LFD) [26]. Some numerical methods are applied to many non-differentiable problems in Cantor sets by using LFD [21][22][23][24][25][26][27][28][29].…”

Section: Introductionmentioning

confidence: 99%

Analytical Approximate Solutions of (n + 1)-Dimensional Fractal Heat-Like and Wave-Like Equations

Acan

Băleanu

Qurashi

et al. 2017

Entropy

Self Cite

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Abstract:In this paper, we propose a new type (n + 1)-dimensional reduced differential transform method (RDTM) based on a local fractional derivative (LFD) to solve (n + 1)-dimensional local fractional partial differential equations (PDEs) in Cantor sets. The presented method is named the (n + 1)-dimensional local fractional reduced differential transform method (LFRDTM). First the theories, their proofs and also some basic properties of this procedure are given. To understand the introduced method clearly, we apply it on the (n + 1)-dimensional fractal heat-like equations (HLEs) and wave-like equations (WLEs). The applications show that this new technique is efficient, simply applicable and has powerful effects in (n + 1)-dimensional local fractional problems.

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Local Fractional Variational Iteration and Decomposition Methods for Wave Equation on Cantor Sets within Local Fractional Operators

Cited by 64 publications

References 28 publications

Solution of the Fractional Form of Unsteady Squeezing Flow through Porous Medium

Solution of the Fractional Form of Unsteady Squeezing Flow through Porous Medium

Exact solutions for linear systems of local fractional partial differential equations

Analytical Approximate Solutions of (n + 1)-Dimensional Fractal Heat-Like and Wave-Like Equations

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