Bülent Saka

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.

Galerkin method for the numerical solution of the RLW equation using quadratic B-splines

A numerical solution of the Regularised Long Wave (RLW) Equation is obtained using space-splitting technique and quadratic B-spline Galerkin finite element method. Solitary wave motion, interaction of two solitary waves and wave generation are studied using the proposed method. Comparisons are made with analytical solutions and with some spline finite element method calculations at selected times. Accuracy and efficiency are discussed by computing the numerical conserved laws and L 2, L ∞ norms.

B-Spline collocation methods for numerical solutions of the RLW equation

The numerical solution of the RLW equation is obtained by using a splitting up technique and both quadratic and cubic B-splines. Both quadratic and cubic B-spline collocation methods are applied to the resulting equation. Solutions without splitting the RLW equation are also obtained with the method of the cubic collocation method. Results are substantiated by studying propagation of a solitary wave and undular bore development. Comparison is made with results of the proposed schemes.

A numerical study of the Burgers’ equation

Abstract Time and space splitting techniques are applied to the Burgers’ equation and the modified Burgers’ equation, and then the quintic B-spline collocation procedure is employed to approximate the resulting systems. Some numerical examples are studied to demonstrate the accuracy and efficiency of the proposed method. Comparisons with both analytical solutions and some published numerical results are done in computational section.

Algorithms for numerical solution of the modified equal width wave equation using collocation method

Quintic B-spline collocation algorithms for numerical solution of the modified equal width wave (MEW) equation have been proposed. The algorithms are based on Crank-Nicolson formulation for time integration and quintic B-spline functions for space integration. Quintic B-spline collocation method over the finite intervals is also applied to the time split MEW equation and space split MEW equation. Results for the three algorithms are compared by studying the propagation of the solitary wave, interaction of the solitary waves, wave generation and birth of solitons.

A B‐spline algorithm for the numerical solution of Fisher's equation

Purpose – This paper seeks to develop an efficient B‐spline Galerkin scheme for solving the Fishers equation, which is a nonlinear reaction diffusion equation describing the relation between the diffusion and nonlinear multiplication of a species.Design/methodology/approach – The solution domain is partitioned into uniform mesh and, using the quartic B‐spline functions, the Galerkin method is applied to the Fishers equation.Findings – The method yields stable accurate solutions. Obtained results are acceptable and in unison with some earlier studies.Originality/value – Using the uniform mesh, quartic B‐spline Galerkin method is employed for finding the numerical solutions of Fishers equation.

A finite element method for equal width equation

Abstract A numerical solution of the equal width (EW) equation is obtained using space-splitting technique and quadratic B-spline Galerkin finite element method. Solitary wave motion, interaction of two solitary waves, wave undulation and wave generation are studied using the proposed method. Comparisons are made with analytical solutions and with some spline finite element method calculations at selected times. Accuracy and efficiency are discussed by computing the numerical conserved laws and L 2 , L ∞ error norms.

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.