A Galerkin-reproducing kernel method: Application to the 2D nonlinear coupled Burgers' equations

Variable Shape Parameter Strategy in Local Radial Basis Functions Collocation Method for Solving the 2D Nonlinear Coupled Burgers’ Equations

“…(14)- (16) We will compare the obtained solution in our proposed method with the results described in [11,35] because there is not the exact solution. Fake oscillations have been observed by using finite element method (FEM), finite-difference method (FDM), element free Galerkin method [32] and Galerkin-reproducing kernel method [11]. In [35], an adaptive upwind technique has been innovated to avoid wiggles in the recommended local RBF collocation method.…”

“…In [35], an adaptive upwind technique has been innovated to avoid wiggles in the recommended local RBF collocation method. As mentioned in [11], even a very fine grid cannot get rid of the oscillatory behavior caused by a sharp gradient. To omit the wiggles and defeat instabilities for Re = 1000, we have used RBFs with CSP as shown in Figure 18.…”

“…By following this utilization of kernels, we can solve the PDEs. An overview of kernel methods prior to the year 2006 is presented in [6], while their recent variations are in [7][8][9][10][11][12] and the related references.…”

Numerical solution of time-fractional Kawahara equation using reproducing kernel method with error estimate

“…In recent years, RKM and other methods are applied for some type of partial and ordinary equations. For instance, eighth-order boundary value problems Akram and Rehman (2013), Fredholm time-fractional partial integro-differential equation (Al-Smadi and Abu Arqub 2019), integro-differential equations of Robin functions types in Hilbert space , fractional advection-dispersion equation (Jiang and Lin 2010), nonlinear hyperbolic telegraph equation (Yao 2011), a special class of fractional partial differential equation (Wang et al 2013), Fredholm integro-differential equations (Abu Arqub et al 2013), fractional Riccati differential equations (Sakar et al 2017), time-fractional partial differential equations (Arqub 2017), nonlinear system of PDEs (Mohammadi and Mokhtari 2014), timefractional telegraph equation (Jiang and Lin 2011), reaction-diffusion equations (Lin and Zhou 2004), for nonlinear coupled Burgers equations (Mohammadi et al 2013), integrodifferential equations with Atangana-Baleanu fractional operator (Abu Arqub and Maayah 2018), time-fractional partial integro-differential equations with initial and Dirichlet boundary conditions (Abu Arqub and Al-Smadi 2018), and so on (Sakar et al 2018(Sakar et al , 2019Mohammadi et al 2018;Cui and Lin 2009;Arqub 2018;Arqub and Al-Smadi 2018;Beyrami et al 2017;Rezazadeh et al 2018Rezazadeh et al , 2019Osman 2017;Osman et al 2018).…”

A fast numerical method for solving coupled Burgers' equations

“…This system was first introduced by Bateman in , and was systematically studied by Burgers to capture some features of turbulent fluid in a channel caused by the interaction of the opposite effects of convection and diffusion . The coupled Burgers' equations have assorted applications in the field of science and engineering , such as turbulence and supersonic flows, wave propagation in a nonlinear thermoelastic media, shallow water waves, acoustic transmission, flow of a shock wave traveling in a viscous fluid, as well as sedimentation of two kinds of particles in fluid suspensions under the effect of gravity, see, for example, and the references therein.…”