Alper Korkmaz

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.

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.

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.

Hyperbolic tangent solution to the conformable time fractional Zakharov-Kuznetsov equation in 3D space

In the study, hyperbolic tangent (tanh) ansatz solution is investigated for the conformable time fractional Zakharov-Kuznetsov Equation (fZKE) in 3D space. Transformation of the fZKE to an ODE by the compatible wave transformation is the first step of the methodology. It is assumed that there exists a solution of positive integer power of hyperbolic tangent form. Determining the power of the predicted solution follows some algebra to find the relations among the other parameters given in the solution. The final step is transforming the solution into original variables.