R. C. Mittal

Numerical solutions of nonlinear Burgers’ equation with modified cubic B-splines collocation method

Abstract In this paper a numerical method is proposed to approximate the solution of the nonlinear Burgers’ equation. The method is based on collocation of modified cubic B-splines over finite elements so that we have continuity of the dependent variable and its first two derivatives throughout the solution range. We apply modified cubic B-splines for spatial variable and derivatives which produce a system of first order ordinary differential equations. We solve this system by using SSP-RK43 or SSP-RK54 scheme. This method needs less storage space that causes less accumulation of numerical errors. The numerical approximate solutions to the Burgers’ equation have been computed without transforming the equation and without using the linearization. Illustrative eleven examples are included to demonstrate the validity and applicability of the technique. Easy and economical implementation is the strength of this method.

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.

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.

Numerical solution of second order one dimensional hyperbolic telegraph equation by cubic B-spline collocation method

The present paper uses new approach and methodology to solve second order one dimensional hyperbolic telegraph equation numerically by B-spline collocation method. It is based on collocation of modified cubic B-spline basis functions over the finite elements. The given equation is decomposed into 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 differential equations. The resulting system of equation subsequently has been solved by SSP-RK54 scheme. The efficacy of proposed approach has been confirmed with numerical experiments, which shows the results obtained are acceptable and in good agreement with earlier studies. The advantage of this scheme is that it can be conveniently used to solve the complex problems and it is also capable of reducing the size of computational work.

A numerical study of two dimensional hyperbolic telegraph equation by modified B-spline differential quadrature method

Abstract The present paper uses a relatively new approach and methodology to solve second order two dimensional hyperbolic telegraph equation numerically. We use modified cubic B-spline basis functions based differential quadrature method for space discretization that reduces the problem into an amenable system of ordinary differential equations. The resulting system of ODEs in time subsequently have been solved by SSP-RK43 scheme. Stability of the scheme is studied using matrix stability analysis and found to be stable. The efficacy of proposed approach has been confirmed with seven numerical experiments, where comparison is made with some earlier work. It is clear that the results obtained are acceptable and are in good agreement with earlier studies. However, we obtain these results in much less CPU time. The method is very simple, efficient and produces very accurate numerical results in considerably smaller number of nodes and hence saves computational effort.

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.

Numerical solutions of generalized Burgers–Fisher and generalized Burgers–Huxley equations using collocation of cubic B-splines

In this work, we propose a numerical scheme to obtain approximate solutions of generalized Burgers–Fisher and Burgers–Huxley equations. The scheme is based on collocation of modified cubic B-spline functions and is applicable for a class of similar diffusion–convection–reaction equations. We use modified cubic B-spline functions for space variable and for its derivatives to obtain a system of first-order ordinary differential equations in time. We solve this system by using SSP-RK54 scheme. The stability of the method has been discussed and it is shown that the method is unconditionally stable. The approximate solutions have been computed without using any transformation or linearization. The proposed scheme needs less storage space and execution time. The test problems considered by the different researchers have been discussed to demonstrate the strength and utility of the proposed scheme. The computed numerical solutions are in good agreement with the exact solutions and competent with those available in the literature. The scheme is simple as well as computationally efficient. The scheme provides approximate solution 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

In this paper, we propose a collocation–based numerical scheme to obtain approximate solutions of coupled Burgers’ equations. The scheme employs collocation of modified cubic B-spline functions. We have used modified cubic B-spline functions for unknown dependent variables u, v, and their derivatives w.r.t. space variable x. Collocation forms of the partial differential equations result in systems of first–order ordinary differential equations (ODEs). In this scheme, we did not use any transformation or linearization method to handle nonlinearity. The obtained system of ODEs has been solved by strong stability preserving the Runge-Kutta method. The proposed scheme needs less storage space and execution time. The test problems considered in the literature have been discussed to demonstrate the strength and utility of the proposed scheme. The computed numerical solutions are in good agreement with the exact solutions and competent with those available in earlier studies. The scheme is simple as well as easy to implement. The scheme provides approximate solutions not only at the grid points, but also at any point in the solution range.

A Higher Order Numerical Scheme for Some Nonlinear Differential Equations: Models in Biology

In this paper, we develop a numerical scheme based on differential quadrature method to solve nonlinear generalizations of the Fisher and Burgers’ equations with the zero flux on the boundary. In construction of the numerical scheme, quasilinearization is used to tackle the nonlinearity of the problem, which is followed by semi-discretization for spatial direction using differential quadrature method (DQM). Semi-discretization of the problem leads to a system of first order initial value problem. For total discretization, we discretize the system of first order initial value problem resulting from the space semi-discretization using the RK4 scheme with constant step length. The method is analyzed for stability and convergence. Finally, the efficiency of the method is derived via numerical comparison between their numerical solution and the exact solution. L 2 and L ∞ error norms demonstrate the accuracy of the proposed method.

Numerical solutions of two-dimensional Burgers’ equations using modified Bi-cubic B-spline finite elements

Purpose – The purpose of this paper is to develop an efficient numerical scheme for non-linear two-dimensional (2D) parabolic partial differential equations using modified bi-cubic B-spline functions. As a test case, method has been applied successfully to 2D Burgers equations. Design/methodology/approach – The scheme is based on collocation of modified bi-cubic B-Spline functions. The authors used these functions for space variable and for its derivatives. Collocation form of the partial differential equation results into system of first-order ordinary differential equations (ODEs). The obtained system of ODEs has been solved by strong stability preserving Runge-Kutta method. The computational complexity of the method is O(p log(p)), where p denotes total number of mesh points. Findings – Obtained numerical solutions are better than those available in literature. Ease of implementation and very small size of computational work are two major advantages of the present method. Moreover, this method provides...