Solution of damped generalized regularized long-wave equation using a modified homotopy analysis method

Akram, Ghazala; Sadaf, Maasoomah

doi:10.1007/s12648-017-1096-x

Cited by 6 publications

(7 citation statements)

References 25 publications

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“…Consider the fractional DGRLW equation with ABC derivative 46,47 given in Equation () by taking

μ = λ = 1

^{ABC} D_{ζ}^{ρ} ϕ - {false(normalΩ ϕ_{ξ ζ} false)}_{ξ} - ϕ_{ξ ξ} + ϕ_{ξ} + ϕ ϕ_{ξ} = 0, 0 < ρ \leq 1,

where

normalΩ false(ξ, ζ false) = 1 - 3 \frac{c o s h ((), \frac{1}{4} ξ - \frac{1}{3} ζ) s i n h ((), \frac{1}{4} ξ - \frac{1}{3} ζ)}{2 {c o s h}^{2} ((), \frac{1}{4} ξ - \frac{1}{3} ζ) - 3}

,…”

Section: Mathematical Analysis Of the Dgrlw Fractional Modelmentioning

confidence: 99%

“…The fractional DGRLW equation is given by 46,47

^{*} D_{ζ}^{ρ} ϕ - {false(normalΩ ϕ_{ξ ζ} false)}_{ξ} - μ ϕ_{ξ ξ} + ϕ_{ξ} + ϕ^{λ} ϕ_{ξ} = 0, 0 < ρ \leq 1,

where

ϕ

and

normalΩ

are all functions of

ξ

and

ζ

and

μ > 0

and

λ \geq 1

are integers. The derivative notation

^{*} D_{ζ}^{ρ}

in Equation () represents the

ζ

derivative of Caputo, ABC, and CFR operators of

ρ

order.…”

Section: Introductionmentioning

confidence: 99%

“…The DGRLW equation plays an important role in the study of nonlinear dispersive waves on account of its description to wide range of important physical phenomena such as shallow water waves and ion‐acoustic plasma 46,49 . For more details on physical importance of DGRLW equation, see Stoker and Waves 50…”

Section: Introductionmentioning

confidence: 99%

“…Among them, Achouri and Omrani 47 used the decomposition approximate technique to obtain the soliton‐like solutions of the DGRLW equations with a variable coefficient. In another technique, Akram and Sadaf 46 applied a modified algorithm for homotopy analysis method for the solution of nonlinear DGRLW equation. Nawaz et al 51 used a semianalytical technique optimal homotopy asymptotic method (OHAM) for deriving approximate solution of the homogeneous and nonhomogeneous nonlinear DGRLW equation.…”

Section: Introductionmentioning

confidence: 99%

“…Many other fractional models have been treated with those fractional-order derivatives. In this study, we investigate the fractional damped generalized regularized long-wave (DGRLW) equation 46,47 with the Caputo, 48 Atangana-Baleanu in the Caputo sense (ABC), 5 and Caputo-Fabrizio (CFR) 4 fractional-order derivatives.…”

Section: Introductionmentioning

confidence: 99%

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Analysis and numerical computations of the fractional regularized long‐wave equation with damping term

Yavuz

Sulaıman²,

Usta

2020

Math Methods in App Sciences

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This study explores the fractional damped generalized regularized long‐wave equation in the sense of Caputo, Atangana‐Baleanu, and Caputo‐Fabrizio fractional derivatives. With the aid of fixed‐point theorem in the Atangana‐Baleanu fractional derivative with Mittag‐Leffler–type kernel, we show the existence and uniqueness of the solution to the damped generalized regularized long‐wave equation. The modified Laplace decomposition method (MLDM) defined in the sense of Caputo, Atangana‐Baleanu, and Caputo‐Fabrizio (in the Riemann sense) operators is used in securing the approximate‐analytical solutions of the nonlinear model. The numerical simulations of the obtained solutions are performed with different suitable values of ρ, which is the order of fractional parameter. We have seen the effect of the various parameters and variables on the displacement in figures.

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“…Consider the fractional DGRLW equation with ABC derivative 46,47 given in Equation () by taking

μ = λ = 1

^{ABC} D_{ζ}^{ρ} ϕ - {false(normalΩ ϕ_{ξ ζ} false)}_{ξ} - ϕ_{ξ ξ} + ϕ_{ξ} + ϕ ϕ_{ξ} = 0, 0 < ρ \leq 1,

where

normalΩ false(ξ, ζ false) = 1 - 3 \frac{c o s h ((), \frac{1}{4} ξ - \frac{1}{3} ζ) s i n h ((), \frac{1}{4} ξ - \frac{1}{3} ζ)}{2 {c o s h}^{2} ((), \frac{1}{4} ξ - \frac{1}{3} ζ) - 3}

,…”

Section: Mathematical Analysis Of the Dgrlw Fractional Modelmentioning

confidence: 99%

“…The fractional DGRLW equation is given by 46,47

^{*} D_{ζ}^{ρ} ϕ - {false(normalΩ ϕ_{ξ ζ} false)}_{ξ} - μ ϕ_{ξ ξ} + ϕ_{ξ} + ϕ^{λ} ϕ_{ξ} = 0, 0 < ρ \leq 1,

where

ϕ

and

normalΩ

are all functions of

ξ

and

ζ

and

μ > 0

and

λ \geq 1

are integers. The derivative notation

^{*} D_{ζ}^{ρ}

in Equation () represents the

ζ

derivative of Caputo, ABC, and CFR operators of

ρ

order.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Analysis and numerical computations of the fractional regularized long‐wave equation with damping term

Yavuz

Sulaıman²,

Usta

2020

Math Methods in App Sciences

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show abstract

Effects of fractional order derivative on the solution of time-fractional Cahn–Hilliard equation arising in digital image inpainting

Sadaf

Akram

2020

Indian J Phys

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Application of Levenberg–Marquardt technique for electrical conducting fluid subjected to variable viscosity

et al. 2022

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In the present study, design of intelligent numerical computing through backpropagated neural networks (BNNs) is presented for numerical treatment of the fluid mechanics problems governing the dynamics of magnetohydrodynamic fluidic model (MHD-NFM) past a stretching surface embedded in porous medium along with imposed heat source/sink and variable viscosity. The original system model MHD-NFM in terms of PDEs is converted to nonlinear ODEs by introducing the similarity transformations. A reference dataset for BNNs approach is generated with Adams numerical solver for different scenarios of MHD-NFM by variation of parameter of viscosity, parameter of heat source and sink, parameter of permeability, magnetic field parameter, and Prandtl number. To calculate the approximate solution for MHD-NFM for different scenarios, the training, testing, and validation processes are conducted in parallel to adapt neural networks by reducing the mean square error (MSE) function through Levenberg–Marquardt backpropagation. The comparative studies and performance analyses through outcomes of MSE, error histograms, correlation and regression demonstrate the effectiveness of proposed BNNs methodology.

show abstract

Solution of damped generalized regularized long-wave equation using a modified homotopy analysis method

Cited by 6 publications

References 25 publications

Analysis and numerical computations of the fractional regularized long‐wave equation with damping term

Analysis and numerical computations of the fractional regularized long‐wave equation with damping term

Effects of fractional order derivative on the solution of time-fractional Cahn–Hilliard equation arising in digital image inpainting

Application of Levenberg–Marquardt technique for electrical conducting fluid subjected to variable viscosity

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