Solutions of the  modified  Kawahara equation with time-and space-fractional derivatives

Safavi, Mostafa; Khajehnasiri, A. A.

doi:10.20454/jmmnm.2016.1044

Cited by 11 publications

(5 citation statements)

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“…where a, b, and λ are arbitrary constants that occurs in many branches of physics, such as shallow water waves, plasma waves, capillary-gravity water waves, and water waves with surface tension. Numerous types of methods have been used for resolving these equations for the different values of the fractional derivatives α and β [27][28][29][30][31][32][33]. Furthermore, by combining the modified Kawahara and the Kawahara equations, we obtain the Extended Kawahara equation (sometimes the Gardner Kawahara equation), which could be employed for examining various nonlinear structures in optical fibers, the physics of plasma, etc., close to the decisive values of the appropriate physical arguments that make the coefficients of dispersions and nonlinearity closed to zero [2].…”

Section: Introductionmentioning

confidence: 99%

Solitary and Periodic Wave Solutions of the Space-Time Fractional Extended Kawahara Equation

Bayram

2023

Fractal Fract

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The extended Kawahara (Gardner Kawahara) equation is the improved form of the Korteweg–de Vries (KdV) equation, which is one of the most significant nonlinear evolution equations in mathematical physics. In that research, the analytical solutions of the conformable fractional extended Kawahara equation were acquired by utilizing the Jacobi elliptic function expansion method. The given expansion method was applied to different fractional forms of the extended Kawahara equation, such as the fraction that occurs in time, space, or both time and space by suitably changing the variables. In addition, various types of fractional problems are exhibited to expose the realistic application of the given method, and some of the obtained solutions were illustrated in two- or three-dimensional graphics as proof of the visualization.

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Section: Introductionmentioning

confidence: 99%

Solitary and Periodic Wave Solutions of the Space-Time Fractional Extended Kawahara Equation

Bayram

2023

Fractal Fract

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“…In [35], the author developed wave solutions for some Kawahara equations. The authors of [36] applied the Adomian decomposition method for treating the modified Kawahara equation.…”

Section: Introductionmentioning

confidence: 99%

A computational strategy for nonlinear time-fractional generalized Kawahara equation using new eighth-kind Chebyshev operational matrices

Ahmed,

Hafez,

Abd-Elhameed

2024

Phys. Scr.

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This paper uses a new method to numerically solve the nonlinear time-fractional generalized Kawahara equations (NTFGKE) with uniform initial boundary conditions (IBCs). A class of modified shifted eighth-kind Chebyshev polynomials (MSEKCPs) is defined to satisfy the given IBCs. The proposed method is based on using operational matrices (OMs) for the ordinary derivatives (ODs) and fractional derivatives (FDs) of MSEKCPs. These OMs are employed together with the spectral collocation method (SCM). Our presented algorithm enables the extraction of efficient and accurate numerical solutions. The convergence of the suggested method and the error analysis have been developed. Three numerical examples are presented to demonstrate the applicability and accuracy of our algorithm. Some comparisons of the presented numerical results with other existing ones are offered to validate the efficiency and superiority of our approach. Tables and graphs demonstrate that the proposed approach produces approximate solutions with high accuracy.

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“…Also, a lot of researchers have studied numerical solutions of the modified Kawahara equation. For instance, Fourier splitting method is used by Suarez and Morales [1], the meshless method of lines is proposed in Bibi et al [19], Gong et al [20] utilized multisymplectic Fourier pseudo-spectral method, collocation methods based on RBFs are developed by Zarebnia and Aghili [21] and Dereli and Dag [22], Bagherzadeh [23] applied B-spline collocation method, Crank-Nicolson differential quadrature method is applied by Korkmaz and Dag [24], Yuan et al [25] employed dual Petrov-Galerkin method, new multisymplectic integrator is introduced by Wen-Jun and Yu-Shun [26], and Marinov and Marinova [27] devised finite difference method [28]. Recently, RBF-finite difference method is used to solve Kawahara equation by Rasoulizadeh and Rashidinia [29], and differential quadrature method is applied to the Kawahara-type equations by Başhan [30].…”

Section: Introductionmentioning

confidence: 99%

A composite method based on delta‐shaped basis functions and Lie group high‐order geometric integrator for solving Kawahara‐type equations

Oruç,

Polat

2023

Math Methods in App Sciences

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In this paper, we devise a novel method to solve Kawahara‐type equations numerically. In this novel method, for spatial discretization, we use delta‐shaped basis functions and generate differentiation matrices for spatial derivatives of the Kawahara‐type equations. For discretization of temporal variable, we utilize a high‐order geometric numerical integrator based on Lie group methods. For illustration of efficiency of the suggested method, we consider some test problems. We calculate errors and make some comparisons with other results that exist in literature. We also report changes in conservation laws during numerical simulations, and we indicate that the suggested method can preserve the conservation laws pretty good. Outcomes of numerical simulations indicate that the suggested method in this paper is reliable and effective for nonlinear partial differential equations (PDEs).

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Solutions of the modified Kawahara equation with time-and space-fractional derivatives

Cited by 11 publications

References 0 publications

Solitary and Periodic Wave Solutions of the Space-Time Fractional Extended Kawahara Equation

Solitary and Periodic Wave Solutions of the Space-Time Fractional Extended Kawahara Equation

A computational strategy for nonlinear time-fractional generalized Kawahara equation using new eighth-kind Chebyshev operational matrices

A composite method based on delta‐shaped basis functions and Lie group high‐order geometric integrator for solving Kawahara‐type equations

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