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

Abstract

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,

Numerical Solutions of the Gardner Equation via Trigonometric Quintic B-spline Collocation Method

“…Use Kink type wave solution of the Gardner equation is [17] We choose homogeneous Neumann conditions. We compute the numerical solutions using the selected values α=1, β=-5 and µ=1 with different values of time step size Δt.…”

“…Recently trigonometric B-splines have been adapted to construct numerical techniques for getting solutions of differential equations such as diffusion problems, Fisher equation and Burger equations in [12,13,14]. Quintic trigonometric Bspline functions are newly defined basis function that are different from polynomial [15,16], exponential [17,18] ones but in the family of trigonometric B-splines. Also trigonometric quintic B-spline collocation method was applied to solve coupled Burgers' equation system in [19].…”

Interpolation and Sampling with Exponential Splines of Real Order

“…Furthermore, in [2] it is shown that exponential splines play an important role in setting up a one-to-one correspondence between dual pairs of Gabor frames and dual pairs of wavelet frames. For an application to some numerical methods, we refer the interested reader to [5] and [12]. In [15], a new class of more general cardinal exponential B-splines, so-called cardinal exponential B-splines of complex order, was introduced, some properties derived and connections to fractional differential operators and sampling theory exhibited.…”

An efficient cubic B‐spline and bicubic B‐spline collocation method for numerical solutions of multidimensional nonlinear stochastic quadratic integral equations

“…Also, they applied collocation of cubic B‐splines finite element method to solve symmetric regularized long wave equations . Furthermore, there are lots of paper about collocation method to solve partial differential equations using B‐splines . In this paper, we use the collocation method and interpolation method via cubic B‐spline functions and bicubic B‐spline functions to solve one dimensional and two dimensional nonlinear stochastic quadratic integral equations, respectively.…”