Dr. Md. Shafiqul Islam - Academia.edu (original) (raw)

Papers by Dr. Md. Shafiqul Islam

Runge-Kutta discontinuous Galerkin (RKDG) method is a high order finite element method for solvin... more Runge-Kutta discontinuous Galerkin (RKDG) method is a high order finite element method for solving hyperbolic conservation laws employing useful features from high resolution finite volume schemes, such as the exact or approximate Riemann solvers serving as numerical fluxes, TVD Runge-Kutta time discretizations and limiters. In most of the RKDG papers in the literature, the Lax-Friedrichs numerical flux is used due to its simplicity, although there are many other numerical fluxes which could also be used. This paper presents a Runge-Kutta discontinuous Galerkin (RKDG) method for the Euler equations of gas dynamics from the viewpoint of kinetic theory. The aim is also to define a way of taking into account high-order space discretization in limiting process, to make use of all the expansion terms of the approximate solution. A limiting procedure is carefully designed to suppress numerical oscillations. It is demonstrated by the numerical experiments that the proposed RKDG methods giv...

GANIT: Journal of Bangladesh Mathematical Society, 2014

This paper is devoted to find the numerical solutions of the fourth order linear and nonlinear di... more This paper is devoted to find the numerical solutions of the fourth order linear and nonlinear differential equations using piecewise continuous and differentiable polynomials, such as Bernstein, Bernoulli and Legendre polynomials with specified boundary conditions. We derive rigorous matrix formulations for solving linear and non-linear fourth order BVP and special care is taken about how the polynomials satisfy the given boundary conditions. The linear combinations of each polynomial are exploited in the Galerkin weighted residual approximation. The derived formulation is illustrated through various numerical examples. Our approximate solutions are compared with the exact solutions, and also with the solutions of the existing methods. The approximate solutions converge to the exact solutions monotonically even with desired large significant digits. GANIT J. Bangladesh Math. Soc. Vol. 33 (2013) 53-64 DOI: http://dx.doi.org/10.3329/ganit.v33i0.17659

GANIT: Journal of Bangladesh Mathematical Society, 2011

In this paper, we use the integration technique together with the finite element method to approx... more In this paper, we use the integration technique together with the finite element method to approximate the numerical solution of an initial value problem of differential equations. The function of two variables is expanded into Taylor’s series up to order two. We exploit Gauss- Legendre quadrature rules evaluating the integrals arising in the formulation of the present method to get the better accuracy. GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 30 (2010) 51-58 DOI: http://dx.doi.org/10.3329/ganit.v30i0.8503

Mathematical Theory and Modeling, 2014

In this paper, we investigate numerical solutions of odd higher order differential equations, par... more In this paper, we investigate numerical solutions of odd higher order differential equations, particularly the fifth, seventh and ninth order linear and nonlinear boundary value problems (BVPs) with two point boundary conditions. We exploit Galerkin weighted residual method with Legendre polynomials as basis functions. Special care has been taken to satisfy the corresponding homogeneous form of boundary conditions where the essential types of boundary conditions are given. The method is formulated as a rigorous matrix form. Several numerical examples, of both linear and nonlinear BVPs available in the literature, are presented to illustrate the reliability and efficiency of the proposed method. The present method is quite efficient and yields better results when compared with the existing methods.

Runge-Kutta discontinuous Galerkin (RKDG) method is a high order finite element method for solvin... more Runge-Kutta discontinuous Galerkin (RKDG) method is a high order finite element method for solving hyperbolic conservation laws employing useful features from high resolution finite volume schemes, such as the exact or approximate Riemann solvers serving as numerical fluxes, TVD Runge-Kutta time discretizations and limiters. In most of the RKDG papers in the literature, the Lax-Friedrichs numerical flux is used due to its simplicity, although there are many other numerical fluxes which could also be used. This paper presents a Runge-Kutta discontinuous Galerkin (RKDG) method for the Euler equations of gas dynamics from the viewpoint of kinetic theory. The aim is also to define a way of taking into account high-order space discretization in limiting process, to make use of all the expansion terms of the approximate solution. A limiting procedure is carefully designed to suppress numerical oscillations. It is demonstrated by the numerical experiments that the proposed RKDG methods giv...

GANIT: Journal of Bangladesh Mathematical Society, 2014

This paper is devoted to find the numerical solutions of the fourth order linear and nonlinear di... more This paper is devoted to find the numerical solutions of the fourth order linear and nonlinear differential equations using piecewise continuous and differentiable polynomials, such as Bernstein, Bernoulli and Legendre polynomials with specified boundary conditions. We derive rigorous matrix formulations for solving linear and non-linear fourth order BVP and special care is taken about how the polynomials satisfy the given boundary conditions. The linear combinations of each polynomial are exploited in the Galerkin weighted residual approximation. The derived formulation is illustrated through various numerical examples. Our approximate solutions are compared with the exact solutions, and also with the solutions of the existing methods. The approximate solutions converge to the exact solutions monotonically even with desired large significant digits. GANIT J. Bangladesh Math. Soc. Vol. 33 (2013) 53-64 DOI: http://dx.doi.org/10.3329/ganit.v33i0.17659

GANIT: Journal of Bangladesh Mathematical Society, 2011

In this paper, we use the integration technique together with the finite element method to approx... more In this paper, we use the integration technique together with the finite element method to approximate the numerical solution of an initial value problem of differential equations. The function of two variables is expanded into Taylor’s series up to order two. We exploit Gauss- Legendre quadrature rules evaluating the integrals arising in the formulation of the present method to get the better accuracy. GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 30 (2010) 51-58 DOI: http://dx.doi.org/10.3329/ganit.v30i0.8503

Mathematical Theory and Modeling, 2014

In this paper, we investigate numerical solutions of odd higher order differential equations, par... more In this paper, we investigate numerical solutions of odd higher order differential equations, particularly the fifth, seventh and ninth order linear and nonlinear boundary value problems (BVPs) with two point boundary conditions. We exploit Galerkin weighted residual method with Legendre polynomials as basis functions. Special care has been taken to satisfy the corresponding homogeneous form of boundary conditions where the essential types of boundary conditions are given. The method is formulated as a rigorous matrix form. Several numerical examples, of both linear and nonlinear BVPs available in the literature, are presented to illustrate the reliability and efficiency of the proposed method. The present method is quite efficient and yields better results when compared with the existing methods.