Dursun Irk

Application of cubic B-splines for numerical solution of the RLW equation

Cubic B-spline functions have been used to develop a collocation method to solve the regularized long wave (RLW) equation, which is used to model solitary waves, undular bore development and wave generation. Performance of the scheme is tested by computing a solitary wave solution of the RLW equation and made comparison with analytical results. Undular bore development and wave generation is shown to be in good agreement with available results.

Least-squares finite element method for the advection-diffusion equation

The space-time least-squares finite element methods are constructed for the advection-diffusion equation by using both linear shape functions and quadratic B-spline shape functions. Two test problems are studied to demonstrate the accuracy of the present methods. Results of the two schemes have been compared.

Solitary wave solutions of the CMKdV equation by using the quintic B-spline collocation method

In this paper, the method based on the collocation method with quintic B-spline finite elements is set up to simulate the solitary wave solution of the complex modified Korteweg–de Vries (CMKdV) equation. The Crank–Nicolson central differencing scheme has been used for the time integration and quintic B-spline functions have been used for the space integration. Propagation of the solitary wave and the interaction of two solitary waves are studied.

A small time solutions for the Korteweg-de Vries equation using spline approximation

In this paper the one-dimensional nonlinear Korteweg-de Vries (KdV) equation is numerically solved using second order spline approximation. The test problems concerning the propagation of a solution and two solution interaction are used to validate the proposal scheme and it is found to be both accurate and efficient at small times. Also, it is shown that the second order spline approximation may be used effectively at small times when the exact solution of the KdV equation is not known.

Numerical integration of the RLW equation using cubic splines

A numerical solution of the RLW equation is presented using a cubic spline collocation method. Basic cubic spline relations are outlined and incorporated into the numerical solution procedure. Two test problems are studied to show the robustness of the proposed procedure.

Quintic B-spline collocation method for the generalized nonlinear Schrödinger equation

Abstract In this study, the generalized nonlinear Schrodinger (GNLS) equation is solved numerically by the quintic B-spline collocation finite element method. In the method, Crank–Nicolson scheme is used for the time integration, and the space variable is discretized by means of quintic B-spline functions. Finally, we investigate properties of the numerically computed solutions of the GNLS equation; in particular we study the generation of solitary waves, interaction of solitons and blow up.

A NUMERICAL SOLUTION OF THE ADVECTION-DIFFUSION EQUATION BY USING EXTENDED CUBIC B-SPLINE FUNCTIONS

In this paper, numerical solution of the advection-diffusion equation is obtained by using extended cubic B-spline functions. For space discretization, the extended cubic B-spline Galerkin method is used to integrate the advection-diffusion equation and for time discretization, the Crank-Nicolson method is employed to obtain the fully integrated advection-diffusion equation. The maximum error norm has been used to show the accuracy of the method. Robustness of the suggested method is shown by studying some classical test problems and comparing the results with some earlier ones.

Simulations of solitary waves of RLW equation by exponential B-spline Galerkin method

In this paper, the exponential B-spline functions are used for the numerical solution of the RLW equation. Three numerical examples related to propagation of single solitary wave, interaction of two solitary waves and wave generation are employed to illustrate the accuracy and the efficiency of the method. Obtained results are compared with some early studies.

Wave Propagation by Way of Exponential B-Spline Galerkin Method

In this paper, the exponential B-spline Galerkin method is set up for getting the numerical solution of the Burgers’ equation. Two numerical examples related to shock wave propogation and travelling wave are studied to illustrate the accuracy and the efficiency of the method. Obtained results are compared with some early studies.