A Meshless Method for Burgers’ Equation Using Multiquadric Radial Basis Functions With a Lie-Group Integrator

Abstract

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.

A composite method based on delta‐shaped basis functions and Lie group high‐order geometric integrator for solving Kawahara‐type equations

“…Hitherto, the geometric numerical integration techniques, which are proposed by Liu [43], have been used to solve various problems. To give some examples, by utilizing Lie group iterative scheme, Chang and Liu [44] solved nonlinear Klein-Gordon and sine-Gordon equations, a Lie group scheme combined with RBFs is applied to Burgers equation by Seydaoglu [45], a composition of reproducing kernel method and GPS is used for Lane-Emden equation [46], some heat conduction equations are investigated by Liu [47], Liu [48] solved Burgers equation by using finite difference and group preserving scheme (GPS), numerical solutions of some PDEs are obtained by Hajiketabi et al [49,50] utilizing meshless methods combined with geometric numerical integrators, and Klein-Gordon equation is considered by Gao et al [51]. Recently, KdV equation is solved by Polat and Oruç [52] via combination of Lie group scheme and DBFs, and accurate results are obtained.…”

Decatic B‐spline collocation scheme for approximate solution of Burgers' equation

“…Equation (1) is nonlinear partial differential equation which is applicable in many fields of sciences such as in approximation theory involving wave propagation in viscous fluid [16], heat conduction and continuous stochastic process [8], gas dynamics [34], longitudinal elastic waves in an isotropic solid [42]. Various efforts are taken to obtain the approximate solution of Equation (1) using different numerical schemes such as meshfree quasi‐interpolation scheme [33], variational scheme [1], optimized weighted essentially non‐oscillatory third order scheme [5], quasi‐linear technique [11], weighted average differential quadrature scheme [29], sinc differential quadrature scheme [31], septic B‐spline scheme [45], cubic B‐spline approaches [14, 51], non‐polynomial spline approach [44], hybrid numerical approach [27], quadratic and cubic B‐splines finite element scheme [13, 46], finite difference method [20, 32], finite element scheme [24, 41], automatic differentiation [6], differential quadrature scheme [36, 37], efficient numerical scheme [39], linearized implicit scheme [40], B‐spline Galerkin methods [15], Haar wavelet quasilinearization scheme [26], multiquadratic quasi‐interpolation technique [10], exponential twice continuously differentiable B‐spline scheme [18], quartic B‐spline collocation method [30], implicit and fully implicit exponential finite difference schemes [23], quintic B‐spline collocation scheme [47], combinations of the wavelet and finite volume scheme [38], semi‐implicit finite difference approach [43], modified cubic B‐spline scheme [35], improvised collocation scheme with cubic B‐spline as basis functions [19], radial basis functions with meshfree algorithms [28], two meshfree approaches [48], cubic trigonometric B‐spline approach [50], meshless technique by applying Lie group integrator with multiquadric radial basis functions [49], MIEELDLD technique [4], and so on.…”

A combined meshfree exponential Rosenbrock integrator for the third‐order dispersive partial differential equations

“…Recently, the applications of radial basis functions (RBFs) and exponential integrators to nonlinear differential equations and simulations are intriguing in the literature [5,6,10,13,15,16,23,28,29,35]. It is well understood with the robust and stable solutions that the RBF interpolation scheme, which has meshfree property, is suitable for traveling waves [5,10,11,16,28,30,33].…”