Alper Korkmaz

Ozlem Ersoy Hepson, Alper Korkmaz, Idris Dag

The extended definition of the polynomial B-splines may give a chance to improve the results obtained by the classical cubic polynomial B-splines. Determination of the optimum value of the extension parameter can be achieved by scanning some intervals containing zero. This study aims to solve some initial boundary value problems con- structed for the Gardner equation by the extended cubic B-spline collocation method.The test problems are derived from some analytical studies to validate the efficiency and accuracy of the suggested method. The conservation laws are also determined to observe them remain constant as expected in theoretical aspect. The stability of the proposed method is investigated by the Von Neumann analysis.

Alper Korkmaz, Ozlem Ersoy, Idiris Dag

Occupation of an interval by self-replicating initial pulses is studied numerically. Two different approximates in different categories are proposed for the numerical solutions of some initial-boundary value problems. The sinc differential quadrature combined with third-fourth order implicit Rosenbrock and exponential B-spline collocation methods are setup to obtain the numerical solutions of the mentioned problems. The numerical simulations containing occupation of single initial pulse, non or slow occupation model and covering the domain with two initial pulses are demonstrated by using both proposed methods.

Ozlem Ersoy Hepsona, Alper Korkmaz, Idiris Dag

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.

Ozlem Ersoy, Idiris Dag, Alper Korkmaz

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.

Ozlem Ersoy Hepson, Alper Korkmaz, Idris Dag

The extended form of the classical polynomial cubic B-spline function is used to set up a collocation method for some initial boundary value problems derived for the Korteweg-de Vries-Burgers equation. Having nonexistence of third order derivatives of the cubic B-splines forces us to reduce the order of the term uxxx to give a coupled system of equations. The space discretization of this system is accomplished by the collocation method following the time discretization with Crank-Nicolson method. Two initial boundary value problems, one having analytical solution and the other is set up with a non analytical initial condition, have been simulated by the proposed method.

Alper Korkmaz

The time dependent complex Schrödinger equation with cubic nonlinearity is solved by constructing differential quadrature algorithm based on sinc functions. Reduction to a coupled system of real equations enables to approach the space derivative terms by the proposed method. The resulted ordinary differential equation system is integrated with respect to the time variable by using a bunch explicit methods of lower and higher orders. Some initial boundary value problems containing some analytical and non-analytical initial data are solved for experimental illustrations. The computational errors between the analytical and numerical solutions are measured by the discrete maximum error norm in case the analytical solution exists. The two conserved quantities are calculated by using the numerical results in all cases. The matrix stability analysis is implemented to control the time step size.

Ozlem Ersoy Hepson, Alper Korkmaz, Idiris Dag

Two cubic B-spline functions from different families are placed to the collocation method for the numerical solutions to the Gardner equation.Four models describing propagation of bell shaped single solitary, travel of a kink type wave, wave generation and interaction of two positive bell shaped solitaries propagating in the opposite directions are studied by both methods. The error between the numerical and the analytical solutions ismeasured by using the discrete maximum norm when the analytical solutions exist. The absolute changes of the lowest three conservation laws are also good indicators of valid results even when the analytical solutions do not exist. The stability of the proposed method is investigated by the Von Neumann analysis.

Alper Korkmaz

In this study, two initial boundary value problems for one dimensional advection-dispersion equation are solved by differential quadrature method based on sine cardinal functions. Pure advection problem modeling transport of conservative pollutants and fade out problem are simulated successfully by the proposed method. The time integration of the space discretized system is accomplished by using various single step and multi step methods covering forward, modified and improved Euler methods, Runge-Kutta, explicit Adams-Bashforth and implicit Adams-Moulton predictor-corrector methods of different orders. The errors between analytical and numerical solutions for both cases are measured by the use of discrete maximum norm. The numerical results are compared with some earlier results obtained by various methods.