The Exponential Cubic B-Spline Algorithm for Korteweg-de Vries Equation

Ersoy, Özlem; Dağ, İdris

doi:10.1155/2015/367056

Cited by 28 publications

(21 citation statements)

References 22 publications

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“…The set {B −1 (x), B 0 (x), ..., B N +1 (x)} constitutes a basis for the functions defined over the interval [a, b]. Since introduced by McCartin, exponential cubic B-spline functions have been used to solve some engineering and physics problems numerically [15][16][17].…”

Section: Exponential Cubic B-spline Collocation Method(ecc)mentioning

confidence: 99%

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Exponential B-Splines for Numerical Solutions to Some Boussinesq Systems for Water Waves

2016

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In the study, the collocation method based on exponential cubic B-spline functions is proposed to solve one dimensional Boussinesq systems numerically. Two initial boundary value problems for Regularized and Classical Boussinesq systems modeling motion of traveling waves are considered. The accuracy of the method is validated by measuring the error between the numerical and analytical solutions. The numerical solutions obtained by various values of free parameter p are compared with some solutions in literature.Keywords: Boussinesq sytems; solitary waves; exponential cubic B-spline; collocation. MSC2010: 35Q99;35C07;76B25;65D07;65M70. Boussinesq SystemsIn the light of the d' Alembert solution, describing two distinct waves moving in the opposite directions for the Cauchy problem for one dimensional wave equation, many physical problems modeled by linear and nonlinear partial differential equations have been solved in wave forms covering traveling waves, solitary waves, harmonic waves, etc. Waves are an important solution class for the model problems in physics, chemistry, and many fields of engineering. This solution type is widely constructed for different models in various media covering solids and fluids. Class of surface bell-shaped solitary waves, having long amplitude when compared with its width, is one of the most prominent classes of waves. In addition to linear equations, solitary wave solutions have also been studied for well known nonlinear equations and systems such as Korteweg-deVries(KdV), Schrödinger, Boussinesq, etc. Having analytical solutions only for some particular cases including additional restrictions and conditions, many numerical methods have been developed to obtain solutions for nonlinear problems. Dougalis et al.[1] studied the numerical behavior of solitary waves of a member of Boussinesq systems family (Bona-Smith system). In that study, long time stability and possible blow-up solutions were examined under small and large perturbations covering perturbation of amplitudes, system coefficients, etc. Numerical models were constructed with the Galerkin-finite element method on the space S h of smooth, periodic, cubic splines. Quintic B-spline collocation tecnique based on Crank-Nicolson formulation was set up for numerical solutions of Boussines type coupled-BBM system [2]. Two initial boundary value problems describing motion of single solitary wave and interaction of solitary waves were simulated by the proposed method. The numerical results were compared with the analytical ones. * Ozlem Ersoy,

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Section: Exponential Cubic B-spline Collocation Method(ecc)mentioning

confidence: 99%

“…Rearranging the initial and boundary conditions (17) gives N + 1 equations with N + 3 unknowns. Using …”

Section: The Initial Statementioning

confidence: 99%

Exponential B-Splines for Numerical Solutions to Some Boussinesq Systems for Water Waves

2016

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“…Each exponential cubic B-spline B m (x) has two continuous first and second order derivatives defined in the interval [x m−2 , x m+2 ]. Exponential B-splines are used as basis functions in some methods to solve problems appearing in various fields [23,24,25,26,27,28]. The functional and derivative values of the exponential B-splines are summarized in Table 1 .…”

Section: Exponential B-spline Approachmentioning

confidence: 99%

Exponential B-spline collocation solutions to the Gardner equation

Hepson

Korkmaz

Dağ

2019

International Journal of Computer Mathematics

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The exponential B-spline basis function set is used to develop a collocation method for some initial boundary value problems (IBVPs) to the Gardner equation. The Gardner equation has two nonlinear terms, namely quadratic and cubic ones. The order reduction of the equation is resulted in a coupled system of PDEs that enables the exponential B-splines to be implemented. The system is integrated in time by Crank-Nicolson implicit method. The validity of the method is investigated by calculating the discrete maximum error norm and observing the absolute relative changes of the conservation laws at the end of the simulations.

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“…For example, it describes surface waves of long wavelength and small amplitude on shallow water and internal waves in a shallow density-strati ed uid [28]. This nonlinear wave equation has been studied extensively by numerical methods [29][30][31][32][33][34][35]. The generalized KdV equation with power law nonlinearity is given by: U t + "U p U x + U xxx = 0;…”

Section: Introductionmentioning

confidence: 99%