Residual Power Series Method for Fractional Diffusion Equations

Kumar, Amit; Kumar, Sunil; Yan, Song Y.

doi:10.3233/fi-2017-1488

Cited by 67 publications

(45 citation statements)

References 12 publications

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“…In this part, we now proceed to discuss the applications of new YAC fractional derivative through HPTM and residual power series method (RPSM). The essential theory of HPTM and RPSM and its utilization to the physical problems are given in published literature …”

Section: Solutions Of Time‐fractional Multidimensional Heat Equationsmentioning

confidence: 99%

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An analysis for heat equations arises in diffusion process using new Yang‐Abdel‐Aty‐Cattani fractional operator

Kumar

Ghosh

Samet

et al. 2020

Math Methods in App Sciences

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181

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The heat equation is parabolic partial differential equation and occurs in the characterization of diffusion progress. In the present work, a new fractional operator based on the Rabotnov fractional-exponential kernel is considered. Next, we conferred some fascinating and original properties of nominated new fractional derivative with some integral transform operators where all results are significant. The fundamental target of the proposed work is to solve the multi dimensional heat equations of arbitrary order by using analytical approach homotopy perturbation transform method and residual power series method, where new fractional operator has been taken in new Yang-Abdel-Aty-Cattani (YAC) sense. The obtained results indicate that solution converges to the original solution in language of generalized Mittag-Leffler function. Three numerical examples are discussed to draw an effective attention to reveal the proficiency and adaptability of the recommended methods on new YAC operator. KEYWORDS HPTM and RPSM, multidimensional heat equations, new fractional derivative, new YAC operator, Prabhakar or generalized Mittag-Leffler function MSC CLASSIFICATION 26A33; 34A08; 34A34; 60G22 Math Meth Appl Sci. 2020;43:6062-6080. wileyonlinelibrary.com/journal/mma

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Section: Solutions Of Time‐fractional Multidimensional Heat Equationsmentioning

confidence: 99%

“…The RPSM consists in expression the solution of Equation as a multiple FPS expansions about the initial point

t = t_{0}

. Let solution takes the following form.…”

Section: Solutions Of Time‐fractional Multidimensional Heat Equationsmentioning

confidence: 99%

An analysis for heat equations arises in diffusion process using new Yang‐Abdel‐Aty‐Cattani fractional operator

Kumar

Ghosh

Samet

et al. 2020

Math Methods in App Sciences

Self Cite

181

View full text Add to dashboard Cite

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“…There are different examples of fractional partial differential equations. Some of them are: the timefractional Boussinesq-type equation, the time-fractional ) 1 , 1 , 2 ( B -type equation and the time-fractional Klein-Gordon-type equation stated in Abu Arqub et al (2015), and time fractional diffusion equation stated in Kumar et al (2017), Cetinkaya and Kiymaz (2013), Kumar et al (2012) and so on.…”

Section: T CD mentioning

confidence: 99%

Analytic solutions of time fractional diffusion equations by fractional reduced differential transform method (FRDTM)

Kenea¹

2018

Afr. J. Math. Comput. Sci. Res.

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This paper examines a general recurrence relation by the use of fractional reduced differential transform and then a scheme (methodology) on how to find closed solutions of one dimensional time fractional diffusion equations with initial conditions in the form of infinite fractional power series and in terms of Mittag-Leffler function in one parameter as well as their exact solutions by the use of fractional reduced differential transform method. The new general recurrence relation and the methodology of the fractional reduced differential transform method were successfully developed. The obtained new general recurrence relation helps us to solve time-fractional diffusion equations with initial conditions and various external forces by using fractional reduced differential transform method. To see its effectiveness and applicability, five test examples were presented. The results show that the general recurrence relation works successfully in solving time-fractional diffusion equations in a direct way without using linearization, transformations, perturbation, discretization or restrictive assumptions by using fractional reduced differential transform method.

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“…4 Taking the fractal dimension into account, the extraneous fluids in the porous media 5 and the unsteady flow for power-law fluids 6 were also analyzed (also see Refs. [7][8][9].…”

Section: Introductionmentioning

confidence: 99%

Exact Traveling-Wave Solutions for One-Dimensional Modified Korteweg–de Vries Equation Defined on Cantor Sets

Gao

Yang

2019

Fractals

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The one-dimensional modified Korteweg–de Vries equation defined on a Cantor set involving the local fractional derivative is investigated in this paper. With the aid of the fractal traveling-wave transformation technology, the nondifferentiable traveling-wave solutions for the problem are discussed in detail. The obtained results are accurate and efficient for describing the fractal water wave in mathematical physics.

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Residual Power Series Method for Fractional Diffusion Equations

Cited by 67 publications

References 12 publications

An analysis for heat equations arises in diffusion process using new Yang‐Abdel‐Aty‐Cattani fractional operator

An analysis for heat equations arises in diffusion process using new Yang‐Abdel‐Aty‐Cattani fractional operator

Analytic solutions of time fractional diffusion equations by fractional reduced differential transform method (FRDTM)

Exact Traveling-Wave Solutions for One-Dimensional Modified Korteweg–de Vries Equation Defined on Cantor Sets

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