Analytical Solutions of Fractional-Order Heat and Wave Equations by the Natural Transform Decomposition Method

Khan, Hassan; Shah, Rasool; Kumam, Poom; Arif, Muhammad

doi:10.3390/e21060597

Cited by 58 publications

(53 citation statements)

References 31 publications

(30 reference statements)

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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%

“…In 2018, Singh has been coupled a new integral transform and HPTM for solutions of HWPMs of arbitrary order. Khan et al have been coupled a natural transform with decomposition method for solutions of HWPMs of arbitrary order, and this new method is known as natural transform decomposition method. Further, Atangana and Alabaraoye have been used a relatively advanced analytical method, namely, the homotopy decomposition method, to determine 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

178

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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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“…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

178

View full text Add to dashboard Cite

show abstract

“…To find the answer of this question, the mathematicians have managed to open a new window of opportunities to improve the mathematical modeling of real world problems, which has given birth to many new questions and intriguing results. These newly established results have numerous implementation in many areas of engineering [1,2], such as fractional-order Buck master and diffusion problems [3], fractional-order telegraph model [4,5], fractional KdV-Burger-Kuramoto equation [6], fractal vehicular traffic flow [7], fractional Drinfeld-Sokolov-Wilson equation [8], fractional-order anomalous sub-diffusion model [9], fractional design of hepatitis B virus [10], fractional modeling chickenpox disease [11], fractional blood ethanol concentration model [12], fractional model for tuberculosis [13], fractional vibration equation [14], fractional Black-Scholes option pricing equations [15], fractionally damped beams [16], fractionally damped coupled system [17], fractional-order heat, wave and diffusion equations [18,19], fractional order pine wilt disease model [20], fractional diabetes model [21] etc.…”

Section: Introductionmentioning

confidence: 99%

Analytical Solutions of (2+Time Fractional Order) Dimensional Physical Models, Using Modified Decomposition Method

et al. 2019

Self Cite

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In this article, a new analytical technique based on an innovative transformation is used to solve (2+time fractional-order) dimensional physical models. The proposed method is the hybrid methodology of Shehu transformation along with Adomian decomposition method. The series form solution is obtained by using the suggested method which provides the desired rate of convergence. Some numerical examples are solved by using the proposed method. The solutions of the targeted problems are represented by graphs which have confirmed closed contact between the exact and obtained solutions of the problems. Based on the novelty and straightforward implementation of the method, it is considered to be one of the best analytical techniques to solve linear and non-linear fractional partial differential equations.

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Fractional view of heat‐like equations via the Elzaki transform in the settings of the Mittag–Leffler function

Rashid

Kubra

Abualnaja

2021

Math Methods in App Sciences

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In this article, the Elzaki homotopy perturbation transform method (EHPTM) is profusely employed to discover the approximate solutions of fractional-order (FO) heat-like equations. To show this, we first establish the Elzaki transform in the context of the Atangana-Baleanu fractional derivative in the Caputo sense (ABC) and then extend it to heat-like equations. Our suggested approach has been reinforced by convergence and error analysis. The validity of the novel technique is tested with the aid of some illustrative examples. Comparative analysis has been established for both fractional and integer-order solutions.EHPTM is considered to be an appropriate and convenient approach for solving FO time-dependent linear and nonlinear partial differential problems. Plots and tables are being used to reveal the findings. The relatively high validity and reliability of the present approach are also reflected by the comparative solution analysis by means of statistical analysis.

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Analytical Solutions of Fractional-Order Heat and Wave Equations by the Natural Transform Decomposition Method

Cited by 58 publications

References 31 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

Analytical Solutions of (2+Time Fractional Order) Dimensional Physical Models, Using Modified Decomposition Method

Fractional view of heat‐like equations via the Elzaki transform in the settings of the Mittag–Leffler function

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