A differential quadrature based numerical method for highly accurate solutions of Burgers' equation

Abstract

In this article, we introduce a new, simple, and accurate computational technique for one‐dimensional Burgers' equation. The idea behind this method is the use of polynomial based differential quadrature (PDQ) for the discretization of both time and space derivatives. The quasilinearization process is used for the elimination of nonlinearity. The resultant scheme has simulated for five classic examples of Burgers' equation. The simulation outcomes are validated through comparison with exact and secondary data in the literature for small and large values of kinematic viscosity. The article has deduced that the proposed scheme gives very accurate results even with less number of grid points. The scheme is found to be very simple to implement. Hence, it applies to any domain requires quick implementation and computation.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2023–2042, 2017

On adaptive mesh for the initial boundary value singularly perturbed delay Sobolev problems

“…Singularly perturbed partial differential equations relate an unknown function to its derivatives evaluated at the same instance. We can see these types of problems in [32][33][34][35][36][37][38]. Singularly perturbed partial differential equations have been studied extensively by many authors and developed thoroughly over the recent decades [29,30].…”

“…But, when the delay is of big order of the singular perturbation parameter, the use of Taylor's series expansion for the term containing delay may lead to a bad approximation. We can see these types of problems in [32][33][34][35][36][37][38]. In this article, we suppose that r > 0 the delay parameter is big and proposes a special method for discretization of continuous time-delay.…”

High‐order numerical solution of viscous Burgers' equation using a Cole‐Hopf barycentric Gegenbauer integral pseudospectral method

“…Now, suppose that Assumption 1 holds, ie, 1]. Through Equations (6d), (9), and (10), we can define the function 2 explicitly in terms of as follows…”

“…The breadth and the range of the applications of Burgers' equation are certainly part of its establishment as an important nonlinear mathematical model in applied mathematics. [3][4][5][6][7][8][9] The development of both analytical and numerical methods for solving Burgers' equation provided with various types of constraints continues to be an area of interest to scholars and researchers who endeavor to enrich profound understanding of such important nonlinear problems and to scrutinize the quality of diverse numerical methods as a natural first step towards developing methods for computations of complex flows. In fact, Burgers' equation is one of few nonlinear partial differential equations that can be solved exactly for a restricted set of initial and boundary functions.…”

Numerical simulation and solutions of the two‐component second order KdV evolutionarysystem

“…Nonlinear evolution equations are often used to describe some physical aspects that arise in the various fields of nonlinear sciences, such as plasma physics, quantum mechanics, biological sciences, chemistry, chemical physics, and so forth. Various powerful techniques have been formulated and used by different scholars to find the solutions of some NLEEs, such as the sine-Gordon expansion method [1][2][3], the generalized Kudryashov method [4,5], the extended tanh method [6,7], the new generalized and improved (G /G)-expansion method [8], the Jacobi elliptic function method [9,10], the improved Bernoulli subequation function method [11], the tanh method [12,13], the sine-cosine method [14], the Lie group analysis method [15][16][17], the homogeneous balance method [18], the modified simple equation method [19,20], the meshless method of radial basis functions [21], He's variational iteration method [22], the explicit multistep Galerkin finite element method [23], the differential quadrature based numerical method [24], the partitioned second-order method [25], the adaptive pseudo-transient-continuation-Galerkin methods [26]. In general, various efficient techniques have been implemented to explore the search for the solutions of the different kind of NLEEs [27][28][29][30][31][32][33][34][35][36][37][38].…”