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# The space-time graph of the estimated solution of Burgers’ equation by using DMQQI for x∈[0,1], t∈[0,1], υ=0.1 (a), and υ=0.01 (b) of Experiment 1.

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The nonlinear Burgers’ equation is a simple form of Navier-Stocks equation. The nonlinear nature of Burgers’ equation has been exploited as a useful prototype differential equation for modeling many phenomena. This paper proposes two meshfree methods for solving the one-dimensional nonlinear nonhomogeneous Burgers’ equation. These methods are based...

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... where t ≥ 1 and t 0 = exp(0.125/ν) are considered as in [16], [22] and [23]. The solution and the corresponding numerical results are shown in Fig. 1, using ν = 0.005, ∆x = 0.02 and ∆t = 0.02. ...
... are considered. Unlike in [16] and [22], periodic boundary condition is applied for 0 ≤ x ≤ 1, ν = 0.01, ∆x = 0.02, and ∆t = 0.005. The results are plotted in Fig. 2. Numerical results, exact solution and error norms at t = 1 are presented in Table 4 and 5. Similar features can be seen as in Example 1. ...
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An integral-like approach established on spline polynomial interpolations is applied to the one-dimensional Burgers' equation. The Hopf-Cole transformation that converts non-linear Burgers' equation to linear diffusion problem is emulated by using Taylor series expansion. The diffusion equation is then solved by using analytic integral formulas. Four experiments were performed to examine its accuracy, stability and parallel scalability. The correctness of the numerical solutions is evaluated by comparing with exact solution and assessed error norms. Due to its integral-like characteristic, large time step size can be employed without loss of accuracy and numerical stability. For practical applications, at least cubic interpolation is recommended. Parallel efficiency seen in the weak-scaling experiment depends on time step size but generally adequate.
... The solution to this system of ordinary differential equations is obtained by constructing first or second order finite difference schemes. The numerical solutions of Eq. (1) based on multiquadratic quasiinterpolation operator and radial basis function network schemes are obtained by [9]. In these methods the solution or its space derivative is quasi interpolated by using Hardy basis functions. ...
... The numerical solutions of this example obtained for k = 5, β = 2 in x ∈ [−1, 1] is used to compute L 2 and L ∞ errors. The comparison of L 2 and L ∞ errors obtained in [9] is given in Table 8. From Table 8 it is seen that, the numerical solutions obtained by the present method are compatible with solutions in [9]. ...
... The comparison of L 2 and L ∞ errors obtained in [9] is given in Table 8. From Table 8 it is seen that, the numerical solutions obtained by the present method are compatible with solutions in [9]. We have computed L 2 and L ∞ errors in the solution for N = 20, k = 100 and β = 1 at t = 1 for different values of ∆t. ...
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In this paper second order explicit Galerkin finite element method based on cubic B-splines is constructed to compute numerical solutions of one dimensional nonlinear forced Burgers' equation. Taylor series expansion is used to obtain time discretization. Galerkin finite element method is set up for the constructed time discretized form. Stability of the corresponding linearized scheme is studied by using von Neumann analysis. The accuracy, efficiency, applicability and reliability of the present method is demonstrated by comparing numerical solutions of some test examples obtained by the proposed method with the exact and numerical solutions available in literature.
... Xie et al. [14] presented the method of particular solutions based on RBFs and finite difference scheme to solve one-dimensional time-dependent inhomogeneous Burgers' equations. Two mesh-free methods, in which the MQ quasi-interpolation method is applied in direct and indirect forms for the numerical solution of the Burgers' equation, are proposed in [15]. Fan et al. [16] developed a mesh-free numerical method based on a combination of the multiquadric RBFs (LRBFCM) and the fictitious time integration method (FTIM) to solve the two-dimensional Burgers' equations. ...
... ζ n = (a n − 1) f n .u n + b n u n f n f n 2 (15) u n+1 = b n ( f n .u n ) + a n u n f n f n . ...
... and ζ n can be computed from Equation (15). One can write Equations (5), (17), (18) and boundary conditions (2) in the matrix form as follows ...
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An efficient technique is proposed to solve the one-dimensional Burgers’ equation based on multiquadric radial basis function (MQ-RBF) for space approximation and a Lie-Group scheme for time integration. The comparisons of the numerical results obtained for different values of kinematic viscosity are made with the exact solutions and the reported results to demonstrate the efficiency and accuracy of the algorithm. It is shown that the numerical solutions concur with existing results and the proposed algorithm is efficient and can be easily implemented.
... So they did not have a system where they had to invert a matrix but an iterative relationship easy to implement. Sarboland et al [181], who had developed mesh-free method in [169] came up with two more mesh-free methods based on the multiquadric (MQ) quasi-interpolation operator L W 2 and direct and indirect radial basis function network (RBFNs) schemes. Ganaie and Kukreja [182] in 2014 discussed cubic Hermite collocation method (CHCM) for Burgers' equation. ...
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Even if numerical simulation of the Burgers’ equation is well documented in the literature, a detailed literature survey indicates that gaps still exist for comparative discussion regarding the physical and mathematical significance of the Burgers’ equation. Recently, an increasing interest has been developed within the scientific community, for studying non-linear convective–diffusive partial differential equations partly due to the tremendous improvement in computational capacity. Burgers’ equation whose exact solution is well known, is one of the famous non-linear partial differential equations which is suitable for the analysis of various important areas. A brief historical review of not only the mathematical, but also the physical significance of the solution of Burgers’ equation is presented, emphasising current research strategies, and the challenges that remain regarding the accuracy, stability and convergence of various schemes are discussed. One of the objectives of this paper is to discuss the recent developments in mathematical modelling of Burgers’ equation and thus open doors for improvement. No claim is made that the content of the paper is new. However, it is a sincere effort to outline the physical and mathematical importance of Burgers’ equation in the most simplified ways. We throw some light on the plethora of challenges which need to be overcome in the research areas and give motivation for the next breakthrough to take place in a numerical simulation of ordinary / partial differential equations.
... Numerical techniques are therefore of much interest to meet the requirement of the wide range of solutions of the Burgers' equation. Among the common numerical techniques used are finite difference schemes, differential quadrature methods, finite volume methods, finite element methods, Haar wavelet methods, spectral methods, pseudospectral methods, method of lines, meshless methods, Adomian decomposition methods, B-spline methods, discontinuous Galerkin methods, reproducing kernel functions, etc. [10,[20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]. However, such methods either lack the exponential convergence enjoyed by spectral and pseudospectral methods, or they enjoy exponential convergence rate in the spatial direction, but suffer from low-order convergence rate in the temporal direction, or suffer from degradation of the observed precision due to the ill-conditioning of the employed numerical differential operators to the extent that the development of efficient preconditioners becomes extremely crucial, or subject to serious time step restrictions that could be more severe than those predicted by the standard stability theory; cf. ...
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We present a novel, high-order numerical method to solve viscous Burger's equation with smooth initial and boundary data. The proposed method combines Cole-Hopf transformation with well conditioned integral reformulations to reduce the problem into either a single easy-to-solve integral equation with no constraints, or an integral equation provided by a single integral boundary condition. Fully exponential convergence rates are established in both spatial and temporal directions by embracing a full Gegenbauer collocation scheme based on Gegenbauer-Gauss (GG) mesh grids using apt Gegenbauer parameter values and the latest technology of barycentric Gegenbauer differentiation and integration matrices. The global collocation matrices of the reduced algebraic linear systems were derived allowing for direct linear system solvers to be used. Rigorous error and convergence analyses are presented in addition to two easy-to-implement pseudocodes of the proposed computational algorithms. We further show three numerical tests to support the theoretical investigations and demonstrate the superior accuracy of the method even when the viscosity paramter $\nu \to 0$, in the absence of any adaptive strategies typically required for adaptive refinements.
... This approach is based on inverse multiquadric (IMQ) RBF interpolation, and Wu and Schaback's operator L D that have the advantages of high approximation order. Up to now, MQ quasi-interpolation is applied for solving different types of PDEs, see (Jiang & Wang, 2012;Sarboland & Aminataei, 2014, 2015b, 2015a. ...
... where the basis functions ψ i (x) are a linear combination of functions ψ i (x) and Sarboland and Aminataei (2014) for details of compactness approach. ...
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In this paper, a meshfree method is presented to solve time fractional partial differential equations. It is based on the multiquadric quasi-interpolation operator . In the present scheme, quadrature formula is used to discretise the temporal Caputo fractional derivative of order and the quasi-interpolation is used to approximate the solution function and its spatial derivatives. Our numerical results are compared with the exact solutions as well as the results obtained from the other numerical schemes. It can be easily seen that the proposed method is a reliable and effective method to solve fractional partial differential equation. Furthermore, the stability analysis of the method is surveyed.
... Adomian's decomposition method was tried by Abbasbandy and Darvishi [27], [28]. Several mesh free methods [29], [30], [31], [32] have also been developed for solving Burgers' equation. In 2015, Vijitha Mukundan and Ashish Awasthi [33] presented new eecient numerical techniques for solving Burgers' equation. ...
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In this paper, a numerical θ scheme is proposed for solving nonlinear Burgers’ equation. By employing Hopf-Cole transformation, the nonlinear Burgers’ equation is linearized to the linear Heat equation. The resulting Heat equation is further solved by cubic B-splines. The time discretization of linear Heat equation is carried out using Crank-Nicolson scheme θ=12 as well as backward Euler scheme (θ = 1). Accuracy in temporal direction is improved by using Richardson extrapolation. This method hence possesses fourth order accuracy both in space and time. The system of matrix which arises by using cubic splines is always diagonal. Therefore, working with splines has the advantage of reduced computational cost and easy implementation. Stability of the schemes have been discussed in detail and shown to be unconditionally stable. Three examples have been examined and the L2 and L∞ error norms have been calculated to establish the performance of the method. The numerical results obtained on applying this method have shown to give more accurate results than existing works of Kutluay et al. [1], Ozis et al. [2], Dag et al. [3], Salkuyeh et al. [4] and Korkmaz et al. [5].
... wherein the basis functions ψ i (x) can be obtained by using Equations (8)-(11), i.e., see [27]. By writing operator L W 2 in the compact form (12), we can use it in the indirect form for the numerical solution of partial differential equations. ...
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This paper's purpose is to provide a numerical scheme to approximate solutions of the nonlinear Klein-Gordon equation by applying the multiquadric quasi-interpolation scheme and the integrated radial basis function network scheme. Our scheme uses θ-weighted scheme for discretization of the temporal derivative and the integrated form of the multiquadric quasi-interpolation scheme for approximation of the unknown function and its spatial derivative. To confirm the accuracy and ability of the presented scheme, this scheme is applied on some test experiments and the numerical results have been compared with the exact solutions. Furthermore, it should be emphasized that with the presently available computing power, it has become possible to develop realistic mathematical models for the complicated problems in science and engineering. The mathematical description of various processes such as the nonlinear Klein-Gordon equation occurring in mathematical physics leads to a nonlinear partial differential equation. However, the mathematical model is only the first step towards the solution of the problem under consideration. The development of the well-documented, robust and reliable numerical tech- nique for handing the mathematical model under consideration is the next step in the solution of the problem. This second step is at least as important as the first one. Therefore, the robustness, the efficiency and the reliability of the numerical technique have to be checked carefully.
... where the basis functions () i x  are obtained by substituting Equations (8), (9) and (11) into (10), see [22]. ...
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A collocation scheme based on the use of the multiquadric quasi-interpolation operator 2 W L , integrated radial basis function networks (IRBFNs) method and three order finite difference method is applied to the nonlinear Klein-Gordon equation. In the present scheme, the three order finite difference method is used to discretize the temporal derivative and the integrated form of the multiquadric quasi-interpolation scheme is used to approximate the unknown function and its spatial derivatives. Several numerical experiments are provided to show the efficiency and the accuracy of the given method.
... The key advantage of EFG method is that only nodal data is required and no element connectivity is needed, when moving least-squares (MLS) interpolation is used to construct trial and test functions. It is currently widely used in computational mechanics and other areas, such as by Sarboland and Aminataei [6] for nonlinear nonhomogeneous Burgers Equation and Pirali et al. [7] for crack discontinuities problem. In the meantime, some new techniques are used to improve the performance of the MLS, complex variable moving least-squares [8][9][10] and improved complex variable moving least-squares [11], the moving least-squares with singular weight function [12], and so forth. ...
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