Numerical solutions of the Kawahara type equations using radial basis functions

Dereli, Yılmaz; Dağ, İdris

doi:10.1002/num.20633

Cited by 15 publications

(4 citation statements)

References 28 publications

(20 reference statements)

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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 Dağ [22], Bagherzadeh [23] applied B‐spline collocation method, Crank–Nicolson differential quadrature method is applied by Korkmaz and Dağ [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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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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“…Bibi et al (2011) introduced a meshless method for Kawahara equation. Another radial basis method is used by Dereli and Dag (2012). Safavi and Khajehnasiri (2016) studied on time-and space-fractional Kawahara equation.…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of time-fractional Kawahara equation using reproducing kernel method with error estimate

2019

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We present a new approach depending on reproducing kernel method (RKM) for timefractional Kawahara equation with variable coefficient. This approach consists of obtaining an orthonormal basis function on specific Hilbert spaces. In this regard, some special Hilbert spaces are defined. Kernel functions of these special spaces are given and basis functions are obtained. The approximate solution is attained as serial form. Convergence analysis, error estimation and stability analysis are presented after obtaining the approximate solution. To show the power and effect of the method, two examples are solved and the results are given as table and graphics. The results demonstrate that the presented method is very efficient and convenient for Kawahara equation with fractional order.Fractional concept as a branch of mathematics investigated integration and derivatives for non-integer order. As of late, this concept has become an attractive field because of its significant researches and applications such as heat conduction, diffusion wave, dynamical system, oil industries, signal processing, cellular system and other subjects. These subjects Communicated by José Tenreiro Machado.

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“…We will adopt the SW solution to be our exact solution to compare with our numerical approximations. Finite differences, finite elements, and radial basis function have been used to get numerical integration of the Kawahara equation [7][8][9]. The present method has high accuracy; it is fast and easy to implement as seen in other works that deal with nonlinearity-see, for instance, [10].…”

Section: Introductionmentioning

confidence: 99%

Fourier Splitting Method for Kawahara Type Equations

Suarez

Morales²

2014

Journal of Computational Methods in Physics

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In this work, we integrate numerically the Kawahara and generalized Kawahara equation by using an algorithm based on Strang’s splitting method. The linear part is solved using the Fourier transform and the nonlinear part is solved with the aid of the exponential operator method. To assess the accuracy of the solution, we compare known analytical solutions with the numerical solution. Further, we show that astincreases the conserved quantities remain constant.

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Numerical solutions of the Kawahara type equations using radial basis functions

Cited by 15 publications

References 28 publications

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

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

Numerical solution of time-fractional Kawahara equation using reproducing kernel method with error estimate

Fourier Splitting Method for Kawahara Type Equations

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