Alaattin Esen

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.

A numerical solution of the equal width wave equation by a lumped Galerkin method

In this paper, the equal width wave (EW) equation is solved by a numerical technique based on a lumped Galerkin method using quadratic B-spline finite elements to investigate the motion of a single solitary wave, development of two solitary waves interaction and an undular bore. The obtained results are compared with published numerical solutions. A linear stability analysis of the method is also investigated.

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 Galerkin Finite Element Method to Solve Fractional Diffusion and Fractional Diffusion-Wave Equations

Abstract In the present study, numerical solutions of the fractional diffusion and fractional diffusion-wave equations where fractional derivatives are considered in the Caputo sense have been obtained by a Galerkin finite element method using quadratic B-spline base functions. For the fractional diffusion equation, the L1 discretizaton formula is applied, whereas the L2 discretizaton formula is applied for the fractional diffusion-wave equation. The error norms L 2 and L ∞ are computed to test the accuracy of the proposed method. It is shown that the present scheme is unconditionally stable by applying a stability analysis to the approximation obtained by the proposed scheme.

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.

A Haar wavelet-finite difference hybrid method for the numerical solution of the modified Burgers’ equation

In this paper, we investigate the numerical solutions of one dimensional modified Burgers’ equation with the help of Haar wavelet method. In the solution process, the time derivative is discretized by finite difference, the nonlinear term is linearized by a linearization technique and the spatial discretization is made by Haar wavelets. The proposed method has been tested by three test problems. The obtained numerical results are compared with the exact ones and those already exist in the literature. Also, the calculated numerical solutions are drawn graphically. Computer simulations show that the presented method is computationally cheap, fast, reliable and quite good even in the case of small number of grid points.

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.

Application of the Exp-function method to the two dimensional sine-Gordon equation

In this paper, the Exp-function method is used to obtain some new generalized solitary wave solutions of the two dimensional sine-Gordon equation. In solving some other nonlinear evolution equations arising in mathematical physics, Exp-function method provides a straightforward and powerful mathematical tool.

A B-spline finite element method for the thermistor problem with the modified electrical conductivity

In this paper, approximate steady-state solutions of a one-dimensional positive temperature coefficient thermistor problem with a modified step function electrical conductivity are obtained by using the Galerkin cubic B-spline finite element method. It is shown that the computational results obtained by the method display the correct physical characteristics of the problem, and they are found to be in very good agreement with the exact solution. It is also shown that the numerical solution exhibits the expected convergence to the exact one as the mesh size is refined. Further a Fourier stability analysis of the method is investigated.