Numerical treatment for traveling wave solutions of fractional Whitham-Broer-Kaup equations

Ali, Amjad; Shah, Kamal; Khan, Rahmat Ali

doi:10.1016/j.aej.2017.04.012

Cited by 55 publications

(51 citation statements)

References 16 publications

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“…The exact solution for classical oreder ALW equations is Table 3 Comparative study between ADM [17], VIM [18], LADM [19], CRFDTM [15] and -HATM for the approximate solution at and for Ex. 6.2 ./ Table 4 Comparative study between ADM [17], VIM [18], LADM [19], CRFDTM [15] and -HATM for the approximate solution at and for Ex. 6.2…”

Section: Example 62mentioning

confidence: 99%

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A reliable technique for fractional modified Boussinesq and approximate long wave equations

et al. 2019

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In this paper, an efficient technique is employed to study the modified Boussinesq and approximate long wave equations of the Caputo fractional time derivative, namely -homotopy analysis transform method. These equations are playing a vital rule in describing the properties of shallow water waves through distinct dispersion relation. The convergence analysis and error analysis has been presented in the present investigation for the future scheme. We illustrate two examples to demonstrate the leverage and effectiveness of the proposed scheme, and the error analysis has been discussed to verify the accuracy. The numerical simulation has been conducted to ensure the exactness of the future technique. The obtained numerical and graphical results are divulge, the proposed scheme is computationally very accurate and straightforward to study and find the solution for fractional coupled nonlinear complex phenomena arised in science and technology.

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Section: Example 62mentioning

confidence: 99%

“…These equations are studied by several authors via distinct techniques like, ADM [17], VIM [18], CFRDTM [15], LADM [19] and other techniques [20][21][22]. These cited techniques require huge computation and has highly complicated procedure to solve coupled systems.…”

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confidence: 99%

A reliable technique for fractional modified Boussinesq and approximate long wave equations

et al. 2019

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“…Recently, numerical solutions of equations have been hot topic. There are a lot of researchers to study the topic by using different methods [1][2][3][4][5][6][7][8][9][10]. Dressing method [11,12] is presented by Zaharov and Shabat.…”

Section: Introductionmentioning

confidence: 99%

Explicit Solutions of Integrable Variable-coeffcient Cylindrical Toda Equations

Wang

Huang

2020

ARJOM

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“…Many physical phenomena that are modeled by partial differential equations and fractional‐order partial differential equations are solved by using the LADM. The numerical solution of the fractional‐order Whitham‐Broer‐Kaup equations are presented in Ali et al; the solution of linear and nonlinear fractional‐order partial differential equations are successfully found in Ahmed et al; the approximate solution of a nonlinear fractional‐order Volterra and Fredholm integro‐differential equations was discussed in Irfan et al; a system of delay differential equations was successfully described and investigated in Yousef and Ismail; the solution of some well‐known diffusion equations was given in Jafari et al; the application of the proposed method for other nonlinear partial differential equations can be found in Mohamed and Elzaki, and so on.…”

Section: Introductionmentioning

confidence: 99%

Some analytical and numerical investigation of a family of fractional‐order Helmholtz equations in two space dimensions

Srivástava

Shah

Khan

et al. 2019

Math Methods in App Sciences

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In this article, we aim at solving a family of two-dimensional fractional-order Helmholtz equations by using the Laplace-Adomian Decomposition Method (LADM). The fractional-order derivatives, which we use in this investigation, follows the Liouville-Caputo definition. Our results based upon the LADM are obtained in series form that helps us in analyzing the analytical solutions of the fractional-order Helmholtz equations considered here. For illustration and verification of the analytical procedure using the LADM, several numerical examples and graphical representations are presented for the analytical solution of the fractional-order Helmholtz equations. The mathematical analytic procedure, which we have used here, has shown that the LADM is a fairly accurate and computable method for the solution of problems involving fractional-order Helmholtz equations in two dimensions. In an analogous manner, one can apply the LADM for finding the analytical solution of other classes of fractional-order partial differential equations. KEYWORDS Adomian decomposition method (ADM), fractional calculus, fractional-order Helmholtz equations, finite-difference method (FDM), Laplace-Adomian decomposition method (LADM), Liouville-Caputo derivative operator, Mittag-Leffler functions, Riemann-Liouville derivative operator MSC CLASSIFICATION 35J15; 35J70; 58J10; 58J20 | INTRODUCTION AND MOTIVATIONFractional calculus plays an important rôle in applied mathematics because of its various applications in modeling of many different physical phenomena occurring in science and engineering. The concept of a derivative, D α { f (x)} with nonintegerorder α, has improved the early development of the ordinary derivative with nonnegative integer values of the order α. In fact, in the development of fractional calculus, many noted mathematical scientists, such as Euler, Riemann, Liouville, Weyl, Leibniz, Fourier, Bernoulli, l'Hôpital, Wallis, and others, made remarkable contributions to this subject and other related research areas. In recent years, researchers in the mathematical, physical, and engineering sciences have made significant contributions toward the theory and applications of fractional calculus. Fractional calculus has many applications in the fields of (for example) viscoelasticity,

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Numerical treatment for traveling wave solutions of fractional Whitham-Broer-Kaup equations

Cited by 55 publications

References 16 publications

A reliable technique for fractional modified Boussinesq and approximate long wave equations

A reliable technique for fractional modified Boussinesq and approximate long wave equations

Explicit Solutions of Integrable Variable-coeffcient Cylindrical Toda Equations

Some analytical and numerical investigation of a family of fractional‐order Helmholtz equations in two space dimensions

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