Papers by Ramesh Chand Mittal
International Journal of Applied and Computational Mathematics, 2017
In this paper we have developed the quintic B-spline basis functions based differential quadratur... more In this paper we have developed the quintic B-spline basis functions based differential quadrature method for the numerical solution of nonlinear Schrödinger equation. The numerical simulations such as motion of single solitary wave, interaction of two solitary waves, generation of solitary waves using the Maxwellian initial condition, birth of mobile soliton, bound state of solitons have been carried out and results are compared with the analytical solution and some published work. Conserved quantities are also computed numerically for all cases. The stability is discussed using matrix stability method. The scheme is found stable and provides very accurate results as compared to other numerical techniques.
International Journal of Computational Mathematics, 2014
We present a technique based on collocation of cubic B-spline basis functions to solve second ord... more We present a technique based on collocation of cubic B-spline basis functions to solve second order one-dimensional hyperbolic telegraph equation with Neumann boundary conditions. The use of cubic B-spline basis functions for spatial variable and its derivatives reduces the problem into system of first order ordinary differential equations. The resulting system subsequently has been solved by SSP-RK54 scheme. The accuracy of the proposed approach has been confirmed with numerical experiments, which shows that the results obtained are acceptable and in good agreement with the exact solution.
International Journal of Partial Differential Equations, 2014
Modified cubic B-spline collocation method is discussed for the numerical solution of one-dimensi... more Modified cubic B-spline collocation method is discussed for the numerical solution of one-dimensional nonlinear sine-Gordon equation. The method is based on collocation of modified cubic B-splines over finite elements, so we have continuity of the dependent variable and its first two derivatives throughout the solution range. The given equation is decomposed into a system of equations and modified cubic B-spline basis functions have been used for spatial variable and its derivatives, which gives results in amenable system of ordinary differential equations. The resulting system of equation has subsequently been solved by SSP-RK54 scheme. The efficacy of the proposed approach has been confirmed with numerical experiments, which shows that the results obtained are acceptable and are in good agreement with earlier studies.
International Journal of Applied and Computational Mathematics, 2017
In this paper we have developed the quintic B-spline basis functions based differential quadratur... more In this paper we have developed the quintic B-spline basis functions based differential quadrature method for the numerical solution of nonlinear Schrödinger equation. The numerical simulations such as motion of single solitary wave, interaction of two solitary waves, generation of solitary waves using the Maxwellian initial condition, birth of mobile soliton, bound state of solitons have been carried out and results are compared with the analytical solution and some published work. Conserved quantities are also computed numerically for all cases. The stability is discussed using matrix stability method. The scheme is found stable and provides very accurate results as compared to other numerical techniques.
International Journal of Computational Mathematics, 2014
We present a technique based on collocation of cubic B-spline basis functions to solve second ord... more We present a technique based on collocation of cubic B-spline basis functions to solve second order one-dimensional hyperbolic telegraph equation with Neumann boundary conditions. The use of cubic B-spline basis functions for spatial variable and its derivatives reduces the problem into system of first order ordinary differential equations. The resulting system subsequently has been solved by SSP-RK54 scheme. The accuracy of the proposed approach has been confirmed with numerical experiments, which shows that the results obtained are acceptable and in good agreement with the exact solution.
International Journal of Partial Differential Equations, 2014
Modified cubic B-spline collocation method is discussed for the numerical solution of one-dimensi... more Modified cubic B-spline collocation method is discussed for the numerical solution of one-dimensional nonlinear sine-Gordon equation. The method is based on collocation of modified cubic B-splines over finite elements, so we have continuity of the dependent variable and its first two derivatives throughout the solution range. The given equation is decomposed into a system of equations and modified cubic B-spline basis functions have been used for spatial variable and its derivatives, which gives results in amenable system of ordinary differential equations. The resulting system of equation has subsequently been solved by SSP-RK54 scheme. The efficacy of the proposed approach has been confirmed with numerical experiments, which shows that the results obtained are acceptable and are in good agreement with earlier studies.
Journal of Computational and Applied Mathematics, 2004
The proposed method of linear time invariant discrete system order reduction is based on multipoi... more The proposed method of linear time invariant discrete system order reduction is based on multipoint step response matching for both pole and zero evaluation of the low order model. Depending on the number of zeros and poles of the low order model, the number of points are selected on the time axis of the unit step response such that the unknown poles and zeros can be determined by solving a set of nonlinear equations using Newton's method.
Applied Mathematical Modelling, 2005
Order reduction of linear discrete systems using classical methods of optimization is well unders... more Order reduction of linear discrete systems using classical methods of optimization is well understood and developed by various workers. The present effort is towards development of a method of linear discrete system order reduction using a genetic algorithm (GA) to get rid of usual difficulties of classical methods. The method developed is applied to a variety of systems and the reduced order models are obtained using the method. The results reported are encouraging and more work can be initiated in this area using a GA.
In this paper, we have proposed a Haar wavelet quasilinearization<br> method to solve the w... more In this paper, we have proposed a Haar wavelet quasilinearization<br> method to solve the well known Blasius equation. The<br> method is based on the uniform Haar wavelet operational matrix<br> defined over the interval [0, 1]. In this method, we have proposed the<br> transformation for converting the problem on a fixed computational<br> domain. The Blasius equation arises in the various boundary layer<br> problems of hydrodynamics and in fluid mechanics of laminar<br> viscous flows. Quasi-linearization is iterative process but our<br> proposed technique gives excellent numerical results with quasilinearization<br> for solving nonlinear differential equations without any<br> iteration on selecting collocation points by Haar wavelets. We have<br> solved Blasius equation for 1≤α ≤ 2 and the numerical results are<br> compared with the available results in literature. Finally, we<br> conclude that proposed met...
Applied Mathematical Modelling, 2014
In this paper, we present a numerical scheme using uniform Haar wavelet approximation and quasili... more In this paper, we present a numerical scheme using uniform Haar wavelet approximation and quasilinearization process for solving some nonlinear oscillator equations. In our proposed work, quasilinearization technique is first applied through Haar wavelets to convert a nonlinear differential equation into a set of linear algebraic equations. Finally, to demonstrate the validity of the proposed method, it has been applied on three type of nonlinear oscillators namely Duffing, Van der Pol, and Duffing-van der Pol. The obtained responses are presented graphically and compared with available numerical and analytical solutions found in the literature. The main advantage of uniform Haar wavelet series with quasilinearization process is that it captures the behavior of the nonlinear oscillators without any iteration. The numerical problems are considered with force and without force to check the efficiency and simple applicability of method on nonlinear oscillator problems.
A numerical scheme based on differential quadrature method for numerical simulation of nonlinear Klein-Gordon equation
International Journal of Numerical Methods for Heat & Fluid Flow, 2014
Purpose – The purpose of this paper is to propose a numerical scheme based on forward finite diff... more Purpose – The purpose of this paper is to propose a numerical scheme based on forward finite difference, quasi-linearisation process and polynomial differential quadrature method to find the numerical solutions of nonlinear Klein-Gordon equation with Dirichlet and Neumann boundary condition. Design/methodology/approach – In first step, time derivative is discretised by forward difference method. Then, quasi-linearisation process is used to tackle the non-linearity in the equation. Finally, fully discretisation by differential quadrature method (DQM) leads to a system of linear equations which is solved by Gauss-elimination method. Findings – The accuracy of the proposed method is demonstrated by several test examples. The numerical results are found to be in good agreement with the exact solutions and the numerical solutions exist in literature. The proposed scheme can be expended for multidimensional problems. Originality/value – The main advantage of the present scheme is that the...
Differential Quadrature Method for Two-Dimensional Burgers' Equations
International Journal for Computational Methods in Engineering Science and Mechanics, 2009
In this paper we propose a rapid convergent differential quadrature method (DQM) for calculating ... more In this paper we propose a rapid convergent differential quadrature method (DQM) for calculating the numerical solutions of nonlinear two-dimensional Burgers&amp;amp;amp;#x27; equations with appropriate initial and boundary conditions. The two dimensional Burgers&amp;amp;amp;#x27; equations arise in various kinds of phenomena such as a mathematical model of turbulence and the approximate theory of flow through a shock wave traveling in a viscous fluid. To the best of the authors&amp;amp;amp;#x27; knowledge, this is the first attempt that the system has been solved up to big ...
Journal of Computational and Applied Mathematics, 2001
In this work, the solution of a sparse system of linear algebraic equations is obtained by using ... more In this work, the solution of a sparse system of linear algebraic equations is obtained by using the Cramer rule. The determinants are computed with the help of the numerical structure approach deÿned in Suchkov (Graphs of Gearing Machines, Leningrad, Quebec, 1983) in which only the non-zero elements are used. Cramer rule produces the solution directly without creating ÿll-in problem encountered in other direct methods. Moreover, the solution can be expressed exactly if all the entries, including the right-hand side, are integers and if all products do not exceed the size of the largest integer that can be represented in the arithmetic of the computer used. The usefulness of Suchkov numerical structure approach is shown by applying on seven examples. Obtained results are also compared with digraph approach described in Mittal and Kurdi (J. Comput. Math., to appear). It is shown that the performance of the numerical structure approach is better than that of digraph approach.
Computers & Mathematics with Applications, 2002
In this work, the solution of a large sparse linear system of equations with an arbitrary sparsit... more In this work, the solution of a large sparse linear system of equations with an arbitrary sparsity pattern is obtained by using LU-decomposition method as well as numerical structure approach. The LU-decomposition method is based on Doolittle's method while the numerical structure approach is based on Cramer's rule. The numerical structure approach produces direct solution without facing fill-in problems as encountered in LU-decomposition. In order to reduce the 'fill-ins' in the decomposition, the powers of a Boolean matrix, obtained from the coefficient matrix A are taken so that the 'fill-ins' in the structure of A can be known in advance. The position of fill-ins in A are thus determined in the best choice manner, that is, it is very effective and memory-wise cheap. We also outline a method by using numerical structure with reduced computation efforts. Finally, experiments are performed on eight examples to compare the efficiency of the proposed methods. The results obtained are reported in a table. It is found that the LU-decomposition method is much better than numerical structure. The usefulness of numerical structure approach is also discussed. (~) 2001 Elsevier Science Ltd. All rights reserved.
Numerical Solutions of Symmetric Regularized Long Wave Equations Using Collocation of CubicB-splines Finite Element
International Journal for Computational Methods in Engineering Science and Mechanics, 2015
In this work, we present a new methodology to obtain numerical solutions of symmetric regularized... more In this work, we present a new methodology to obtain numerical solutions of symmetric regularized long wave equations. Our approach is based on collocation of modified cubic B-spline basis functions over the finite elements. First, we discretize the given equation in space and then we use collocation of modified cubic B-spline basis functions for unknown dependent variables and their derivatives w.r.t. space variable x, which produces a system of first-order ordinary differential equations. This system of differential equations has been solved by SSP-RK54 scheme. The stability of the method has been discussed and shown that it is unconditionally stable. The efficiency of the proposed method has been confirmed with numerical experiments, which shows that the obtained results are acceptable and are in good agreement with earlier studies. Ease of implementation and very small size of computational work are two major advantages of this approach. Moreover, this method provides approximate solutions not only at the grid points, but also at any point in the solution range.
A Collocation Method for Numerical Solutions of Coupled Burgers’ Equations
International Journal for Computational Methods in Engineering Science and Mechanics, 2014
ABSTRACT Disclaimer: This is a version of an unedited manuscript that has been accepted for publi... more ABSTRACT Disclaimer: This is a version of an unedited manuscript that has been accepted for publication. As a service to authors and researchers we are providing this version of the accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proof will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to this version also. PLEASE SCROLL DOWN FOR ARTICLE Taylor &amp; Francis makes every effort to ensure the accuracy of all the information (the &quot;Content&quot;) contained in the publications on our platform. However, Taylor &amp; Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor &amp; Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &amp; Conditions of access and use can be found at
Communications in Numerical Analysis, 2012
In this paper a numerical method is proposed to approximate the solution of the nonlinear general... more In this paper a numerical method is proposed to approximate the solution of the nonlinear general Rosenau-RLW Equation. The method is based on collocation of quintic B-splines over finite elements so that we have continuity of the dependent variable and its first four derivatives throughout the solution range. We apply quintic B-splines for spatial variable and derivatives which produce a system of first order ordinary differential equations. We solve this system by using SSP-RK54 scheme. This method needs less storage space that causes to less accumulation of numerical errors. The numerical approximate solutions to the nonlinear general Rosenau-RLW Equation have been computed without transforming the equation and without using the linearization. Illustrative example is included for different value of p = 2, 3 and 6, to demonstrate the validity and applicability of the technique. Easy and economical implementation is the strength of this method.
Computers & Mathematics with Applications, 2003
main idea of this paper is in determination of the pattern of nonzero elements of the LU factors ... more main idea of this paper is in determination of the pattern of nonzero elements of the LU factors of a given matrix A. The idea is based on taking the powers of the Boolean matrix derived from A. This powers of a Boolean matrix strategy (PBS) is an efficient, effective, and inexpensive approach. Construction of an ILU preconditioner using PBS is described and used in solving large nonsymmetric sparse linear systems. Effectiveness of the proposed ILU preconditioner in solving large nonsymmetric sparse linear systems by the GMRES method is also shown. Numerical experiments are performed which show that it is possible to considerably reduce the number of GMRES iterations when the ILU preconditioner constructed here is used. In numerical examples, the influence of lc, the dimension of the Krylov subspace, on the performance of the GMRES method using an ILU preconditioner is tested. For all the tests carried out, the best value for Ic is found to be 10.
A differential quadrature method for numerical solutions of Burgers'-type equations
International Journal of Numerical Methods for Heat & Fluid Flow, Sep 14, 2012
Purpose–The purpose of this paper is to use the polynomial differential quadrature method (PDQM) ... more Purpose–The purpose of this paper is to use the polynomial differential quadrature method (PDQM) to find the numerical solutions of some Burgers'-type nonlinear partial differential equations. Design/methodology/approach–The PDQM changed the nonlinear partial differential equations into a system of nonlinear ordinary differential equations (ODEs). The obtained system of ODEs is solved by Runge-Kutta fourth order method.
Cryptology ePrint Archive: Report 2010/036, 2010
In 2003, Shen et al [4] proposed a timestamp-based password authentication scheme in which remote... more In 2003, Shen et al [4] proposed a timestamp-based password authentication scheme in which remote server does not need to store the passwords or verification table for users authentication. Unfortunately Wang and Li[6], E.J.Yoon [8], Lieu et al.[3], analyzed independently the Shen Lin Scheme [4] and was found to be vulnerable to some deadly attacks. In continuation to it, this paper analyzes few attacks and finally proposes an improved Timestamp-based password remote user authentication scheme so that it can withstand the existing forged attacks.
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Papers by Ramesh Chand Mittal