Ram Jiwari

A Haar wavelet quasilinearization approach for numerical simulation of Burgers’ equation

a b s t r a c t In this paper, an efficient numerical scheme based on uniform Haar wavelets and the quasilinearization process is proposed for the numerical simulation of time dependent nonlinear Burgers’ equation. The equation has great importance in many physical problems such as fluid dynamics, turbulence, sound waves in a viscous medium etc. The Haar wavelet basis permits to enlarge the class of functions used so far in the collocation framework. More accurate solutions are obtained by wavelet decomposition in the form of a multi-resolution analysis of the function which represents a solution of boundary value problems. The accuracy of the proposed method is demonstrated by three test problems. The numerical results are compared with existing numerical solutions found in the literature. The use of the uniform Haar wavelet is found to be accurate, simple, fast, flexible, convenient and has small computation costs.

Numerical simulation of two-dimensional sine-Gordon solitons by differential quadrature method

Abstract During the past few decades, the idea of using differential quadrature methods for numerical solutions of partial differential equations (PDEs) has received much attention throughout the scientific community. In this article, we proposed a numerical technique based on polynomial differential quadrature method (PDQM) to find the numerical solutions of two-dimensional sine-Gordon equation with Neumann boundary conditions. The PDQM reduced the problem into a system of second-order linear differential equations. Then, the obtained system is changed into a system of ordinary differential equations and lastly, RK4 method is used to solve the obtained system. Numerical results are obtained for various cases involving line and ring solitons. The numerical results are found to be in good agreement with the exact solutions and the numerical solutions that exist in literature. It is shown that the technique is easy to apply for multidimensional problems.

A differential quadrature algorithm to solve the two dimensional linear hyperbolic telegraph equation with Dirichlet and Neumann boundary conditions

In this article, we proposed a numerical technique based on polynomial differential quadrature method (PDQM) to find the numerical solutions of two dimensional hyperbolic telegraph equation with Dirichlet and Neumann boundary condition. The PDQM reduced the problem into a system of second order linear differential equation. Then, the obtained system is changed into a system of ordinary differential equations and lastly, RK4 method is used to solve the obtained system. 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 technique is easy to apply for multidimensional problems.

A hybrid numerical scheme for the numerical solution of the Burgers’ equation

Abstract In this article, a hybrid numerical scheme based on Euler implicit method, quasilinearization and uniform Haar wavelets has been developed for the numerical solutions of Burgers’ equation. Most of the numerical methods available in the literature fail to capture the physical behavior of the equations when viscosity ν → 0 . In Jiwari (2012), the author presented the numerical results up to ν = 0.003 and the scheme failed for values smaller than ν = 0.003 . The main aim in the development of the present scheme is to overcome the drawback of the scheme developed in Jiwari (2012). Lastly, three test problems are chosen to check the accuracy of the proposed scheme. The approximated results are compared with existing numerical and exact solutions found in literature. The use of uniform Haar wavelet is found to be accurate, simple, fast, flexible, convenient and at small computation costs.

Numerical solution of two-dimensional reaction-diffusion Brusselator system

Abstract In this paper, polynomial based differential quadrature method (DQM) is applied for the numerical solution of a class of two-dimensional initial-boundary value problems governed by a non-linear system of partial differential equations. The system is known as the reaction–diffusion Brusselator system. The system arises in the modeling of certain chemical reaction–diffusion processes. In Brusselator system the reaction terms arise from the mathematical modeling of chemical systems such as in enzymatic reactions, and in plasma and laser physics in multiple coupling between modes. The numerical results reported for three specific problems. Convergence and stability of the method is also examined numerically.

Lagrange interpolation and modified cubic B-spline differential quadrature methods for solving hyperbolic partial differential equations with Dirichlet and Neumann boundary conditions

In this article, the author proposed two differential quadrature methods to find the approximate solution of one and two dimensional hyperbolic partial differential equations with Dirichlet and Neumann’s boundary conditions. The methods are based on Lagrange interpolation and modified cubic B-splines respectively. The proposed methods reduced the hyperbolic problem into a system of second order ordinary differential equations in time variable. Then, the obtained system is changed into a system of first order ordinary differential equations and finally, SSP-RK3 scheme is used to solve the obtained system. The well known hyperbolic equations such as telegraph, Klein–Gordon, sine-Gordon, Dissipative non-linear wave, and Vander Pol type non-linear wave equations are solved to check the accuracy and efficiency of the proposed methods. The numerical results are shown in L∞,RMSandL2 errors form.

Differential Quadrature Method for Numerical Solution of Coupled Viscous Burgers’ Equations

In this paper, the coupled viscous Burgers’ equations have been solved by using the differential quadrature method. Two test problems considered by different researchers have been studied to demonstrate the accuracy and utility of the present method. The numerical results are found to be in good agreement with the exact solutions. The maximum absolute errors L ∞ between the exact solutions and the numerical solutions have been studied. A comparison of the computed solutions is made with those which are already available in the literature. It is shown that the present numerical scheme gives better solutions. Moreover, it is shown that the method can be easily applied to a wide class of higher-dimension, nonlinear partial differential equations with a little modification.

A numerical scheme based on differential quadrature method for numerical simulation of nonlinear Klein-Gordon equation

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 scheme gives very accu...

A computational modeling of two dimensional reaction-diffusion Brusselator system arising in chemical processes

In this article, the authors proposed a modified cubic B-spline differential quadrature method (MCB-DQM) to show computational modeling of two-dimensional reaction–diffusion Brusselator system with Neumann boundary conditions arising in chemical processes. The system arises in the mathematical modeling of chemical systems such as in enzymatic reactions, and in plasma and laser physics in multiple coupling between modes. The MCB-DQM reduced the Brusselator system into a system of nonlinear ordinary differential equations. The obtained system of nonlinear ordinary differential equations is then solved by a four-stage RK4 scheme. Accuracy and efficiency of the proposed method successfully tested on four numerical examples and obtained results satisfy the well known result that for small values of diffusion coefficient, the steady state solution converges to equilibrium point