Fractional natural decomposition method for solving a certain class of nonlinear time-fractional wave-like equations with variable coefficients

Khalouta, Ali; Kadem, Abdelouahab

doi:10.2478/ausm-2019-0009

Cited by 14 publications

(9 citation statements)

References 7 publications

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“…Taking α = 2 in equalities (5.13) and (5.15), the solution will be as follows v(x, t) = x 2 tE 2,2 (−t 2 ) = x 2 sin t, which is exactly the same solution obtained by FNDM ( [8]) and FRPSM ( [9]). These figures afirm that when the order of the fractional derivative α tends to 2, the approximate solutions obtained by FNVIM and FNHPM tends continuously to the exact solutions.…”

Section: Numerical Applicationsmentioning

confidence: 66%

“…The main objective of this paper is to introduce a numerical comparison of two powerful methods, the fractional natural variational iteration method (FNVIM) and the fractional natural homotopy perturbation method (FNHPM) for solving certain type of nonlinear Caputo time-fractional partial differential equations in particular, nonlinear Caputo time-fractional wave-like equation with variable coefficients of the form ( [8], [9])…”

Section: Introductionmentioning

confidence: 99%

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Numerical Comparison of FNVIM and FNHPM for Solving a Certain Type of Nonlinear Caputo Time-Fractional Partial Differential Equations

Khalouta

Kadem

2020

Annales Mathematicae Silesianae

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This work presents a numerical comparison between two efficient methods namely the fractional natural variational iteration method (FNVIM) and the fractional natural homotopy perturbation method (FNHPM) to solve a certain type of nonlinear Caputo time-fractional partial differential equations in particular, nonlinear Caputo time-fractional wave-like equations with variable coefficients. These two methods provided an accurate and efficient tool for solving this type of equations. To show the efficiency and capability of the proposed methods we have solved some numerical examples. The results show that there is an excellent agreement between the series solutions obtained by these two methods. However, the FNVIM has an advantage over FNHPM because it takes less time to solve this type of nonlinear problems without using He’s polynomials. In addition, the FNVIM enables us to overcome the difficulties arising in identifying the general Lagrange multiplier and it may be considered as an added advantage of this technique compared to the FNHPM.

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Section: Numerical Applicationsmentioning

confidence: 66%

Section: Introductionmentioning

confidence: 99%

Numerical Comparison of FNVIM and FNHPM for Solving a Certain Type of Nonlinear Caputo Time-Fractional Partial Differential Equations

Khalouta

Kadem

2020

Annales Mathematicae Silesianae

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“…To mention a few, we have the homotopy perturbation method (HPM) [6], the Adomian decomposition method (ADM) [7], the Laplace decomposition method (LDM) [8], the homotopy perturbation transform method (HPTM) [9], and so on. Besides using the Laplace-type integral transform [10,11], some new efficient iterative techniques with the Caputo fractional derivative [12] and Atangana-Baleanu fractional derivative [13] are developed, for example, see [14][15][16][17][18][19][20][21][22][23][24][25]. Those iterative algorithms are successfully applied to many applications in applied physical science.…”

Section: Introductionmentioning

confidence: 99%

Homotopy perturbation Shehu transform method for solving fractional models arising in applied sciences

Maitama¹,

Zhang²

2021

J. Appl. Math. Comput. Mech.

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Using the recently proposed homotopy perturbation Shehu transform method (HPSTM), we successfully construct reliable solutions of some important fractional models arising in applied physical sciences. The nonlinear terms are decomposed using He's polynomials, and the fractional derivative is calculated in the Caputo sense. Using the analytical method, we obtained the exact solution of the fractional diffusion equation, fractional wave equation and the nonlinear fractional gas dynamic equation.

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“…Since then, ADM has been implemented in nonlinear ODEs and PDEs without using perturbation or linearization procedure. e Shehu decomposition method (SDM) is a mixture of ADM and Shehu transform [51][52][53][54].…”

Section: Introductionmentioning

confidence: 99%

The Fractional View Analysis of Polytropic Gas, Unsteady Flow System

Khan

Islam

Arif

2021

Mathematical Problems in Engineering

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Generally, the differential equations of integer order do not properly model various phenomena in different areas of science and engineering as compared to differential equations of fractional order. The fractional-order differential equations provide the useful dynamics of the physical system and thus provide the innovative and effective information about the given physical system. Keeping in view the above properties of fractional calculus, the present article is related to the analytical solution of the time-fractional system of equations which describe the unsteady flow of polytropic gas dynamics. The present method provides the series form solution with easily computable components and a higher rate of convergence towards the targeted problem’s exact solution. The present techniques are straightforward and effective for dealing with the solutions of fractional-order problems. The fractional derivatives are expressed in terms of the Caputo operator. The targeted problems’ solutions are calculated using the Adomian decomposition method and variational iteration methods along with Shehu transformation. In the current procedures, we first applied the Shehu transform to reduce the problems into a more straightforward form and then implemented the decomposition and variational iteration methods to achieve the problems’ comprehensive results. The solution of the nonlinear equations of unsteady flow of a polytropic gas at various fractional orders of the derivative is the core point of the present study. The solution of the proposed fractional model is plotted via two- and three-dimensional graphs. It is investigated that each problem’s solution-graphs are best fitted with each other and with the exact solution. The convergence of fractional-order problems can be observed towards the solution of integer-order problems. Less computational time is the major attraction of the suggested methods. The present work will be considered a useful tool to handle the solution of fractional partial differential equations.

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Fractional natural decomposition method for solving a certain class of nonlinear time-fractional wave-like equations with variable coefficients

Cited by 14 publications

References 7 publications

Numerical Comparison of FNVIM and FNHPM for Solving a Certain Type of Nonlinear Caputo Time-Fractional Partial Differential Equations

Numerical Comparison of FNVIM and FNHPM for Solving a Certain Type of Nonlinear Caputo Time-Fractional Partial Differential Equations

Homotopy perturbation Shehu transform method for solving fractional models arising in applied sciences

The Fractional View Analysis of Polytropic Gas, Unsteady Flow System

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