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A Hybrid Computational Scheme with Convergence Analysis for the Dependent Rosenau-Hyman Equation of Arbitrary Order Via Caputo-Fabrizio Operator

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Abstract

In this paper, a solution of the dependent Rosenau-Hyman (RH) equation is found by using a competent hybrid computational scheme q-homotopy analysis transform technique (q-HATT). The Liouville-Caputo fractional operator offers singularity in the kernel so fractional Caputo-Fabrizio operator with exponential non-singular kernel is chosen to describe memory effects well in analyzing solution. The q-HATT presents the solution in a convergent series at a large permissible domain. The convergence and uniqueness analysis of the time-fractional RH equation via q-HATT scheme is also presented. This scheme has an auxiliary parameter ħ which provides a fit way of handling convergence region. The given example confirms competency of q-HATT scheme. The behavior of solution by q-HATT for diverse orders of derivative is discussed by figures. The decreasing error between the successive approximations in the solution by q-HATT validates the convergence of acquired solution. The outcomes disclose that q-HATT scheme is reliable, attractive and also effective.

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Abbreviations

\(t\) :

Time variable

\(x\) :

Space coordinate

\(\alpha \) :

Order of the fractional derivative

\(^{{CF}}\!{_{0} D }_{t}^{\alpha } f\left( t \right) \) :

Caputo Fabrizio

\(\alpha \) :

Order derivative of f(t)

\(M(\alpha )\) :

Normalization function

\(h\left(x,t\right)\) :

Source term

\({\mathbb{N}}\) :

Set of natural numbers

\(N\) :

Nonlinear operator

\(q\) :

Embedding parameter

\(n (\ge 1)\) :

Parameter

\(\psi \left( {x,t;q} \right)\) :

A real-valued function

\(\hbar \) :

Auxiliary parameter

\(H(x,t)\) :

Auxiliary function

\({\mathrm{u}}_{0}(x,t)\) :

Initial value

\(L\) :

Laplace transform operator

\(B\) :

Banach Space

\(T\) :

Nonlinear mapping from \(B\) to \(B\)

\({\mathbb{R}}\) :

Set of real numbers

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Acknowledgements

Authors are appreciative to referees for valuable comments.

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Authors and Affiliations

  1. Department of Mathematics, Institute of Applied Sciences and Humanities, GLA University, Mathura, India

    Shivangi Gupta & Manish Goyal

  2. Department of Mathematics, National Institute of Technology, Kurukshetra, India

    Amit Prakash

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  1. Shivangi Gupta
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  2. Manish Goyal
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  3. Amit Prakash
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Contributions

SG conceptualized and investigated the results. MG visualized the solution of Rosenau-Hyman model, supervised the methodology used and was major contributor in writing manuscript. AP validated the solution. All the authors read the final manuscript and approved.

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Correspondence to Manish Goyal.

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Gupta, S., Goyal, M. & Prakash, A. A Hybrid Computational Scheme with Convergence Analysis for the Dependent Rosenau-Hyman Equation of Arbitrary Order Via Caputo-Fabrizio Operator. Int. J. Appl. Comput. Math 7, 259 (2021). https://doi.org/10.1007/s40819-021-01182-4

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