S. Kutluay

Application of a lumped Galerkin method to the regularized long wave equation

In this paper, a lumped Galerkin method based on quadratic B-spline finite elements is used to find numerical solutions of the one-dimensional regularized long wave (RLW) equation with a variant of initial and boundary conditions. The obtained numerical results show that the present method is a remarkably successful numerical technique for solving the equation. Results are compared with published numerical solutions. A linear stability analysis of the scheme is also investigated.

A numerical solution of the Stefan problem with a Neumann-type boundary condition by enthalpy method

In this paper, the enthalpy method based on suitable finite difference approximations has been applied to the one-dimensional moving boundary problem with a Neumann-type boundary condition known as the Stefan problem. The numerical results obtained by the hopscotch technique are compared with the exact solution of the problem. It is shown that all results are found to be in very good agreement with each other, and the numerical solution displays the expected convergence to the exact one as the mesh size is refined.

An analytical-numerical method for solving the Korteweg-de Vries equation

In this paper, an analytical-numerical method is applied to the one-dimensional Korteweg-de Vries equation with a variant of boundary and initial conditions to obtain its numerical solutions at small times. Two test problem with known exact solutions are studied to demonstrate the accuracy of the present method. The obtained results are compared with the exact solution of each problem and are found to be in good agreement with each other. The numerical scheme is also compared with earlier work and shown to be accurate and efficient.

A linearized numerical scheme for Burgers-like equations

A linearized implicit finite-difference method is presented to find numerical solutions of the one-dimensional Burgers-like equations. The method has been used successfully to obtain accurate numerical solutions even for small values of viscosity term @n. Results obtained by the present method using a direct technique for some values of @n have been compared with the exact values and are found to be in good agreement with each other.

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.

The G′G-expansion method for some nonlinear evolution equations

Abstract In this paper, the G ′ G -expansion method is applied to the Liouville, sine–Gordon and new coupled MKdV equations to obtain their some generalized exact travelling wave solutions.

An isotherm migration formulation for one-phase Stefan problem with a time dependent Neumann condition

In this paper, we present a numerical scheme based on an isotherm migration formulation for one-dimensional, one-phase Stefan problem with a time dependent Neumann condition on the fixed boundary and a constant Dirichlet condition on the moving boundary. The numerical results obtained by the present method have been compared with exact one and also those obtained by earlier authors, and are found to be in very good agreement with each other. It is also shown that the numerical solution displays the expected convergence to the exact one as the mesh size is refined.

Numerical schemes for one-dimensional Stefan-like problems with a forcing term

Variable space grid and boundary immobilisation schemes based on the explicit finite difference method are applied to the one-phase Stefan-like problems with a forcing term in order to evaluate the temperature distribution and the interface movement (location and speed). The numerical results obtained by the two schemes have been compared with the exact values and are found to be in good agreement with each other. It is shown that the numerical solutions exhibit the expected convergence to the exact one as the mesh size is reduced. A von-Neumann stability analysis of each scheme is also investigated.

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.

Numerical solutions of the thermistor problem with a ramp electrical conductivity

This paper presents approximate steady-state solutions of a positive temperature coefficient thermistor problem, having a ramp electrical conductivity that is a highly non-linear function of the temperature, using a standard explicit finite difference method. It is shown that numerical solutions exhibit the correct physical characteristics of the problem and, they are in good agreement with the exact solution.