A. Özdeş

Numerical solution of one-dimesional Burgers equation: explicit and exact-explicit finite difference methods

This paper presents finite-difference solution and analytical solution of the finite-difference approximations based on the standard explicit method to the one-dimensional Burgers equation which arises frequently in the mathematical modelling used to solve problems in fluid dynamics. Results obtained by these ways for some modest values of viscosity have been compared with the exact (Fourier) one. It is shown that they are in good agreement with each other.

The numerical solution of one-phase classical Stefan problem

In this paper, variable space grid and boundary Immobilisation Techniques based on the explicit finite difference are applied to the one-phase classical Stefan problem. It is shown that all the results obtained by the two methods are in good agreement with the exact solution, and exhibit the expected convergence as the mesh size is refined.

Numerical solution of Korteweg–de Vries equation by Galerkin B-spline finite element method

A finite element solution of the KdV equation is presented. To demonstrate the efficiency of the method two test problems are considered. The numerical solutions of the KdV equation are compared with both the exact solutions and other numerical solutions in the literature. The numerical solutions are found to be in good agreement with the exact solutions.

A numerical solution of Burgers’ equation based on least squares approximation

Abstract Burgers’ equation which is one-dimensional non-linear partial differential equation was converted to p non-linear ordinary differential equations by using the method of discretization in time. Each of them was solved by the least squares method. For various values of viscosity at different time steps, the numerical solutions obtained were compared with the exact solutions. It was seen that both of them were in excellent agreement. While the exact solution was not available for viscosity smaller than 0.01, it was shown that mathematical structure of the equation for the obtained numerical solutions did not decay.

A small time solutions for the Korteweg-de Vries equation

In this paper a heat balance integral (HBI) method is applied to the one-dimensional non-linear Korteweg-de Vries (KdV) equation prescribed by appropriate homogenous boundary conditions and a set of initial conditions to obtain its approximate analytical solutions at small times. It is shown that the HBI solutions obtained by the method may be used effectively at small times when the exact solution of the KdV equation is not known.

A variety of finite difference methods to the thermistor with a new modified electrical conductivity

We consider the numerical solution of a one-dimensional thermistor (thermo-electric) problem with a new modified step function electrical conductivity which is an inherently non-linear function of the temperature. A variety of finite difference methods are applied to solve the problem using a new modification of the step function electrical conductivity to be satisfied the physical phenomena of the problem.

A variational approximation to the problem of the deflection of a bar

In this paper, a mathematical modelling of the deflection of a bar lying on an elastic medium. It was shown that results obtained were reasonably in good agreement with Ritz solution.