Solution of the Local Fractional Generalized KDV Equation Using Homotopy Analysis Method

Jafari, Hossein; Prasad, Jyoti Geetesh; Goswami, Pranay; Dubey, Ravi Shanker

doi:10.1142/s0218348x21400144

Cited by 22 publications

(8 citation statements)

References 19 publications

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“…Many methods have been proposed in the literature for getting exact and approximate solutions to these equations. Examples of these methods are, (G'/G)-expansion method [10,13], generalized F-expansion Method [23], Lie symmetry analysis method [7,8,11,14,19], generalized new auxiliary equation method [10][11], the invariant subspace method [3,4,15], homotopy perturbation methods [9,18], and reproducing Kernel Hilbert Space Method [17].…”

Section: Introductionmentioning

confidence: 99%

Some new exact solutions for a generalized variable coefficients KdV equation

Rajagopalan¹,

Kader²,

Latif³

et al. 2022

J. Math. Computer Sci.

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In this paper, the variable coefficients KdV equation with general power nonlinearities is proposed. Firstly, it is transformed into a generalized KdV equation with constant coefficients using a point transformation. Then, the traveling wave transformation is utilized to transform the obtained constant coefficients generalized KdV equation into a generalized ordinary differential equation. We give a classification for the obtained generalized ordinary differential equation using a suitable integrating factor. Some new solutions are obtained for the generalized KdV equation with constant coefficients. All the obtained solutions in this paper for the variable coefficients KdV equation with general power nonlinearities are new.

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Section: Introductionmentioning

confidence: 99%

Some new exact solutions for a generalized variable coefficients KdV equation

Rajagopalan¹,

Kader²,

Latif³

et al. 2022

J. Math. Computer Sci.

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“…Using linearization or perturbation approaches, only approximate answers can be obtained. All of these push us to develop a numerical approach for fractional differential equations that is both efficient and accurate [18][19][20][21]. Chaos theory, heat transfer, variational issues, and other fields have used the Atangana-Baleanu fractional differential extensively.…”

Section: Introductionmentioning

confidence: 99%

Analytical Investigation of Some Dynamical Systems by ZZ Transform with Mittag–Leffler Kernel

2022

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In this work, ZZ transformation is combined with the Adomian decomposition method to solve the dynamical system of fractional order. The derivative of fractional order is represented in the Atangana–Baleanu derivative. The numerical examples are combined for their approximate-analytical solution. It is explored using graphs that indicate that the actual and approximation results are close to each other, demonstrating the method’s usefulness. Fractional-order solutions are the most in line with the dynamics of the targeted problems, and they provide an endless number of options for an optimal mathematical model solution for a particular physical phenomenon. This analytical approach produces a series form solution that is quickly convergent to exact solutions. The acquired results suggest that the novel analytical solution technique is simple to use and very successful at assessing complicated problems that arise in related fields of research and technology.

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“…Homogeneous beam's transverse vibrations are controlled by fractional single fourth-order parabolic partial differential equations (PDEs). Such problem types occur in viscoelastic and inelastic flow mathematical modeling, layer deflection theories, and beam deformation [1][2][3][4][5][6][7][8][9][10][11][12]. Analyses of these problems have taken several physicist's and mathematician's attention [13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Numerical Analysis of Fractional-Order Parabolic Equations via Elzaki Transform

Naeem

Azhar

Zidan

et al. 2021

Journal of Function Spaces

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This research article is dedicated to solving fractional-order parabolic equations, using an innovative analytical technique. The Adomian decomposition method is well supported by Elzaki transformation to establish closed-form solutions for targeted problems. The procedure is simple, attractive, and preferred over other methods because it provides a closed-form solution for the given problems. The solution graphs are plotted for both integer and fractional-order, which shows that the obtained results are in good contact with problems’ exact solution. It is also observed that the solution of fractional-order problems is convergent to the integer-order problem. Moreover, the validity of the proposed method is analyzed by considering some numerical examples. The theory of the suggested approach is fully supported by the obtained results for the given problems. In conclusion, the present method is a straightforward and accurate analytical technique that can solve other fractional-order partial differential equations.

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Solution of the Local Fractional Generalized KDV Equation Using Homotopy Analysis Method

Cited by 22 publications

References 19 publications

Some new exact solutions for a generalized variable coefficients KdV equation

Some new exact solutions for a generalized variable coefficients KdV equation

Analytical Investigation of Some Dynamical Systems by ZZ Transform with Mittag–Leffler Kernel

Numerical Analysis of Fractional-Order Parabolic Equations via Elzaki Transform

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