Finding exact solutions of nonlinear PDEs using the natural decomposition method

Rawashdeh, Mahmoud S.; Maitama, Shehu

doi:10.1002/mma.3984

Cited by 54 publications

(41 citation statements)

References 9 publications

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“…To present the basic theory and solution procedure of the proposed method, we consider a system of fractional order nonlinear partial differential equation of the form

D_{t}^{α} scriptu ((),,, x, y, t) + R scriptv ((),,, x, y, t) + F scriptu ((),,, x, y, t) = h_{1} ((),,, x, y, t), D_{t}^{β} scriptv ((),,, x, y, t) + R scriptu ((),,, x, y, t) + F scriptv ((),,, x, y, t) = h_{1} ((),,, x, y, t)

with initial conditions

scriptu ((),,, x, y, 0) = g_{1} ((),, x, y), scriptv ((),,, x, y, 0) = g_{2} ((),, x, y),

where

D^{α} scriptu ((),,, x, y, t)

and

D^{β} scriptv ((),,, x, y, t)

respectively denote the Caputo fractional derivatives of the functions

scriptu ((),,, x, y, t)

and

scriptv ((),,, x, y, t)

, R is the linear differential operator, F represents the general nonlinear differential operator, and h 1 ( x , y , t ) and h 2 ( x , y , t ) are the source terms. On employing NT and using Theorem , Equation yields

scriptU ((),,,, x, y, s, w) = \frac{u^{α}}{s^{α}} \sum_{k = 0}^{n - 1} \frac{s^{α - (k + 1)}}{w^{α - k}} {[D^{k} u (x, y, t)]}_{t = 0} + \frac{w^{α}}{s^{α}} ℕ^{+}

…”

Section: Fundamental Idea Of Fndmmentioning

confidence: 99%

“…Last three decades have been witnessed for the discovery of a number of new methods to explain non‐linear differential system having fractional order and in parallel to the development of new computational algorithms and symbolic programming. In 2014, Rawashdeh and Maitama introduced and nurtured the new technique, known as natural decomposition method (NDM) . The NDM is an elegant amalgamation of the Adomian decomposition method (ADM) and the natural transform method (NTM) .…”

Section: Introductionmentioning

confidence: 99%

“…In 2014, Rawashdeh and Maitama introduced and nurtured the new technique, known as natural decomposition method (NDM). 26 The NDM is an elegant amalgamation of the Adomian decomposition method (ADM) 27 and the natural transform method (NTM). 28 The proposed algorithm provides solution in terms infinite series, and the obtained series rapidly converge to a closed form exact solution; for instance, Belgacem and Silambarasan 29 provide a proof for two fundamental results of fractional natural transform by employing duality of Natural and Laplace transform.…”

mentioning

confidence: 99%

“…are the Adomian polynomials and they characterized the nonlinear terms. Using these terms, we can rewrite Equation26 in the form…”

mentioning

confidence: 99%

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Numerical solution for (2 + 1)‐dimensional time‐fractional coupled Burger equations using fractional natural decomposition method

Prakasha

Veeresha

Rawashdeh

2019

Math Methods in App Sciences

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The aim of the present work is to find the numerical solutions for time‐fractional coupled Burgers equations using a new novel technique, called fractional natural decomposition method (FNDM). Two examples are considered in order to illustrate and validate the efficiency of the proposed algorithm. The numerical simulation has been conducted to ensure the exactness of the present method, and the obtained solutions are offered graphically to reveal the applicability and reliability of the FNDM. The outcomes of the study reveal that the FNDM is computationally very effective and accurate to study the (2 + 1)‐dimensional coupled Burger equations of arbitrary order.

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“…To present the basic theory and solution procedure of the proposed method, we consider a system of fractional order nonlinear partial differential equation of the form

D_{t}^{α} scriptu ((),,, x, y, t) + R scriptv ((),,, x, y, t) + F scriptu ((),,, x, y, t) = h_{1} ((),,, x, y, t), D_{t}^{β} scriptv ((),,, x, y, t) + R scriptu ((),,, x, y, t) + F scriptv ((),,, x, y, t) = h_{1} ((),,, x, y, t)

with initial conditions

scriptu ((),,, x, y, 0) = g_{1} ((),, x, y), scriptv ((),,, x, y, 0) = g_{2} ((),, x, y),

where

D^{α} scriptu ((),,, x, y, t)

and

D^{β} scriptv ((),,, x, y, t)

respectively denote the Caputo fractional derivatives of the functions

scriptu ((),,, x, y, t)

and

scriptv ((),,, x, y, t)

scriptU ((),,,, x, y, s, w) = \frac{u^{α}}{s^{α}} \sum_{k = 0}^{n - 1} \frac{s^{α - (k + 1)}}{w^{α - k}} {[D^{k} u (x, y, t)]}_{t = 0} + \frac{w^{α}}{s^{α}} ℕ^{+}

…”

Section: Fundamental Idea Of Fndmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

“…are the Adomian polynomials and they characterized the nonlinear terms. Using these terms, we can rewrite Equation26 in the form…”

mentioning

confidence: 99%

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Numerical solution for (2 + 1)‐dimensional time‐fractional coupled Burger equations using fractional natural decomposition method

Prakasha

Veeresha

Rawashdeh

2019

Math Methods in App Sciences

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“…Later, Belgacem and Silambarasan defined inverse of natural transform and studied some additional fundamental properties of its [19]. Recently, N-transform is used to find the solution of differential and integral equations [21,22,23,24,25,26,27,28].…”

Section: The Natural Transformmentioning

confidence: 99%

The approximate solutions of fractional gas dynamics equations by means of fractional natural decomposition method

Ibis¹

2017

ntmsci

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Abstract:In this work, a new practical method, which is named by the Fractional Natural Decomposition Method (FNDM), is proposed to obtain the approximate analytical solutions of fractional gas dynamics equations. The FNDM is a mixture of the Natural Transform Method and the Adomian Decomposition Method. In this method, the fractional derivatives are considered as Caputo sense and the nonlinear terms are determined by virtue of Adomian polynomials. Some test examples are given to demonstrate the efficiency and accuracy of the FNDM.

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The fractional natural decomposition method: theories and applications

Rawashdeh

2016

Math Methods in App Sciences

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In this paper, we propose a new method called the fractional natural decomposition method (FNDM). We give the proof of new theorems of the FNDM, and we extend the natural transform method to fractional derivatives. We apply the FNDM to construct analytical and approximate solutions of the nonlinear time‐fractional Harry Dym equation and the nonlinear time‐fractional Fisher's equation. The fractional derivatives are described in the Caputo sense. The effectiveness of the FNDM is numerically confirmed. Copyright © 2016 John Wiley & Sons, Ltd.

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Finding exact solutions of nonlinear PDEs using the natural decomposition method

Cited by 54 publications

References 9 publications

Numerical solution for (2 + 1)‐dimensional time‐fractional coupled Burger equations using fractional natural decomposition method

Numerical solution for (2 + 1)‐dimensional time‐fractional coupled Burger equations using fractional natural decomposition method

The approximate solutions of fractional gas dynamics equations by means of fractional natural decomposition method

The fractional natural decomposition method: theories and applications

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