İdris Dağ

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.

A differential quadrature algorithm for simulations of nonlinear Schrödinger equation

Numerical simulations of Nonlinear Schrodinger Equation are studied using differential quadrature method based on cosine expansion. Propogation of a soliton, interaction of two solitons, birth of standing and mobile solitons and bound state solutions are simulated. The accuracy of the method (DQ) is measured using maximum error norm. The results are compared with some earlier works. The lowest two conserved quantities are computed numerically for all cases.

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.

Cubic B‐spline differential quadrature methods for the advection‐diffusion equation

Purpose – Cubic B‐spline differential quadrature methods have been introduced. As test problems, two different solutions of advection‐diffusion equation are chosen. The first test problem, the transportion of an initial concentration, and the second one, the distribution of an initial pulse, are simulated. The purpose of this paper is to simulate the test problems.Design/methodology/approach – The cubic B‐spline functions are chosen as test functions in order to construct the differential quadrature method. The error between the numerical solutions and analytical solutions are measured using various error norms.Findings – The cubic B‐spline differential quadrature methods have produced acceptable solution for advection‐diffusion equation.Originality/value – The advection‐diffusion equation has never been solved by any differential quadrature method based on cubic B‐splines.

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.

SOLITARY WAVE SIMULATIONS OF COMPLEX MODIFIED KORTEWEG-DE VRIES EQUATION USING DIFFERENTIAL QUADRATURE METHOD

Abstract Complex Modified Korteweg–deVries Equation is solved numerically using differential quadrature method based on cosine expansion. Three test problems, motion of single solitary wave, interaction of solitary waves and wave generation, are simulated. The accuracy of the method is measured via the discrete root mean square error norm L 2 , maximum error norm L ∞ for the motion of single solitary wave since it has an analytical solution. A rate of convergency analysis for motion of single solitary wave containing both real and imaginary parts is also given. Lowest three conserved quantities are computed for all test problems. A comparison with some earlier works is given.

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.