A General Iteration Formula of VIM for Fractional Heat- and Wave-Like Equations

Yin, Fukang; Song, Junqiang; Cao, Xiaoqun

doi:10.1155/2013/428079

Cited by 16 publications

(17 citation statements)

References 35 publications

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“…Example Let us assume that the one‐dimensional time fractional HWPM is written as follows:

^{Y A C} D_{t}^{μ} normalΥ false(η, t false) = \frac{1}{2} η^{2} D_{η η} normalΥ false(η, t false), 0 < μ \leq 1,

where the fractional operator

^{Y A C} D_{t}^{μ}

is taken in new YAC sense with boundary conditions

normalΥ false(0, t false) = 0, normalΥ false(1, t false) = e^{t}

and initial condition (IC)

normalΥ false(η, 0 false) = η^{2}

.…”

Section: Solutions Of Time‐fractional Multidimensional Heat Equationsmentioning

confidence: 99%

“…Example In this example, we consider the two‐dimensional time fractional HWPM, which is written as follows:

^{Y A C} D_{t}^{μ} normalΥ false(η, ξ, t false) = \frac{1}{2} ((), ξ^{2} D_{η η} normalΥ + η^{2} D_{ξ ξ} normalΥ), 0 < η, ξ < 1, 0 < μ \leq 1,

subject to Neumann boundary conditions

{normalΥ}_{η} false(0, ξ, t false) = 0, {normalΥ}_{η} false(1, ξ, t false) = 2 s i n h t

{normalΥ}_{ξ} false(η, 0, t false) = 0, {normalΥ}_{ξ} false(η, 1, t false) = 2 c o s h t

and with initial condition

normalΥ false(η, ξ, 0 false) = ξ^{2}

.…”

Section: Solutions Of Time‐fractional Multidimensional Heat Equationsmentioning

confidence: 99%

“…Zhang et al have been implemented variation iteration method (VIM) with He's polynomials for solution of HWPMs of arbitrary order in 2012. Yin et al suggested a correct function and proposed a modified technique to find the Lagrange multipliers by utilizing Laplace transform of arbitrary order. Recently, Safavi has been successfully employed the VIM to obtain the approximate analytical solutions of HWPMs of arbitrary order.…”

Section: Introductionmentioning

confidence: 99%

“…The major desire of present effort is to represent some new and interesting properties of new YAC fractional operator, and another purpose is to extend the HPM to solve the time fractional multidimensional heat equations where fractional operator is considered in new YAC sense . Let us consider the multidimensional heat equations with variable coefficient form, which is written as

^{Y A C} D_{t}^{μ} normalΥ false(η, t false) = f_{1} false(η, ξ, ω false) {normalΥ}_{η η} + f_{2} false(η, ξ, ω false) {normalΥ}_{ξ ξ} + f_{3} false(η, ξ, ω false) {normalΥ}_{ω ω},

where

0 < η < a, 0 < ξ < b, 0 < ω < c

. Subject to the Neumann boundary conditions

\begin{matrix} {normalΥ}_{η} false(0, ξ, ω, t false) & = g_{1} false(ξ, ω, t false), {normalΥ}_{η} false(a, ξ, ω, t false) = g_{2} false(ξ, ω, t false), \\ {normalΥ}_{ξ} false(η, 0, ω, t false) & = h_{1} false(η, ω, t false), {normalΥ}_{ξ} false(η, b, ω, t false) = h_{2} false(η, ω, t false), \\ {normalΥ}_{ω} false(η, ξ, 0, t false) & = <...> \end{matrix}

…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

179

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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

show abstract

“…Example Let us assume that the one‐dimensional time fractional HWPM is written as follows:

^{Y A C} D_{t}^{μ} normalΥ false(η, t false) = \frac{1}{2} η^{2} D_{η η} normalΥ false(η, t false), 0 < μ \leq 1,

where the fractional operator

^{Y A C} D_{t}^{μ}

is taken in new YAC sense with boundary conditions

normalΥ false(0, t false) = 0, normalΥ false(1, t false) = e^{t}

and initial condition (IC)

normalΥ false(η, 0 false) = η^{2}

.…”

Section: Solutions Of Time‐fractional Multidimensional Heat Equationsmentioning

confidence: 99%

“…Example In this example, we consider the two‐dimensional time fractional HWPM, which is written as follows:

^{Y A C} D_{t}^{μ} normalΥ false(η, ξ, t false) = \frac{1}{2} ((), ξ^{2} D_{η η} normalΥ + η^{2} D_{ξ ξ} normalΥ), 0 < η, ξ < 1, 0 < μ \leq 1,

subject to Neumann boundary conditions

{normalΥ}_{η} false(0, ξ, t false) = 0, {normalΥ}_{η} false(1, ξ, t false) = 2 s i n h t

{normalΥ}_{ξ} false(η, 0, t false) = 0, {normalΥ}_{ξ} false(η, 1, t false) = 2 c o s h t

and with initial condition

normalΥ false(η, ξ, 0 false) = ξ^{2}

.…”

Section: Solutions Of Time‐fractional Multidimensional Heat Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

^{Y A C} D_{t}^{μ} normalΥ false(η, t false) = f_{1} false(η, ξ, ω false) {normalΥ}_{η η} + f_{2} false(η, ξ, ω false) {normalΥ}_{ξ ξ} + f_{3} false(η, ξ, ω false) {normalΥ}_{ω ω},

where

0 < η < a, 0 < ξ < b, 0 < ω < c

. Subject to the Neumann boundary conditions

\begin{matrix} {normalΥ}_{η} false(0, ξ, ω, t false) & = g_{1} false(ξ, ω, t false), {normalΥ}_{η} false(a, ξ, ω, t false) = g_{2} false(ξ, ω, t false), \\ {normalΥ}_{ξ} false(η, 0, ω, t false) & = h_{1} false(η, ω, t false), {normalΥ}_{ξ} false(η, b, ω, t false) = h_{2} false(η, ω, t false), \\ {normalΥ}_{ω} false(η, ξ, 0, t false) & = <...> \end{matrix}

…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

179

View full text Add to dashboard Cite

show abstract

“…Gupta and S. Gupta worked out by using homotopy perturbation transform method ( HPTM) these types of equation tool [7], furthermore, A. Aslanov [3], F. Yin and et al [12] and A. Atangana and et al [2] researched for solving nonlinear heat and wave-like equation by using homotopy perturbation, variational iteration and homotopy decomposition methods respectively. Moreover, various techniques, such as homotopy analysis, perturbations, decompositions, iterations, di¤erential and Laplace transformation techniques have been used to handle similar types of these wave-like and also heat-like problems numerically and analytically as in references [1], [7], [10], [11].…”

Section: Introductionmentioning

confidence: 99%

Mixed of Elzaki Transform and Projected Differential Transform Method for a Nonlinear Wave-Like Equations with Variable Coefficients

Khalouta¹,

Kadem²

2018

Preprint

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Abstract. In this work, a mixture of Elzaki transform and projected di¤erential transform method is applied to solve a nonlinear wave-like equations with variable coe¢ cients. Nonlinear terms can be easily manipulated by using the projected di¤erential transformation method. The method gives the results show that the proposed method is very e¢ cient, simple and can be applied to other applications.

show abstract

A computational approach for fractional convection-diffusion equation via integral transforms

Singh

Swroop²,

Kumar

2018

Ain Shams Engineering Journal

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A General Iteration Formula of VIM for Fractional Heat- and Wave-Like Equations

Cited by 16 publications

References 35 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

Mixed of Elzaki Transform and Projected Differential Transform Method for a Nonlinear Wave-Like Equations with Variable Coefficients

A computational approach for fractional convection-diffusion equation via integral transforms

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