In this paper, analytic solutions of TWO-DIMENSIONAL COUPLED BURGERS’ EQUATIONS are obtained by the Homotopy analysis and the Homotopy Pade methods. The obtained approximation by using Homotopy method contains an auxiliary parameter which is a simple way to control and adjust the convergence region and rate of solution series. The approximation solutions by [m,m] Homotopy Pade technique is often independent of auxiliary parameter and this technique accelerate the convergence of the related series.

The purpose of this paper is to design a fully discrete hybridized discon-tinuous Galerkin (HDG) method for solving a system of TWO-DIMENSIONAL (2D) COUPLED Burgers EQUATIONS over a specified spatial domain. The semi-discrete HDG method is designed for a nonlinear variational formulation on the spatial domain. By exploiting broken Sobolev approximation spaces in the HDG scheme, numerical fluxes are defined properly. It is shown that the proposed method is stable under specific mild conditions on the stabi-lization parameters to solve a well-posed (in the sense of energy method) 2D COUPLED Burgers EQUATIONS, which is imposed by Dirichlet boundary conditions. The fully discrete HDG scheme is designed by exploiting the Crank–Nicolson method for time discretization. Also, the Newton–Raphson method that has the order of at least TWO is nominated for solving the obtained nonlinear system of COUPLED Burgers EQUATIONS over the rect-angular domain. To reduce the complexity of the proposed method and the size of the linear system, we exploit the Schur complement idea. Numerical results declare that the best possible rates of convergence are achieved for approximate solutions of the 2D COUPLED Burgers EQUATIONS and their first-order derivatives. Moreover, the proposed HDG method is examined for TWO other types of systems, that is, a system with high Reynolds numbers and a system with an unavailable exact solution. The acceptable results of examples show the flexibility of the proposed method in solving various problems.

‎This paper is devoted to proposing hybridized discontinuous Galerkin (HDG) approximations for solving a system of COUPLED Burgers EQUATIONS (CBE) in a closed interval‎. ‎The noncomplete discretized HDG method is designed for a nonlinear weak form of one-DIMENSIONAL $x-$variable such that numerical fluxes are defined properly‎, ‎stabilization parameters are applied‎, ‎and broken Sobolev approximation spaces are exploited in this scheme‎. ‎Having necessary conditions on the stabilization parameters‎, ‎it is proven in a theorem and corollary that the proposed method is stable with imposed homogeneous Dirichlet and/or periodic boundary conditions to CBE‎. ‎The desired HDG method is stated by using the Crank-Nicolson method for time-variable discretization and the Newton-Raphson method for solving nonlinear systems‎. ‎Numerical experiences show that the optimal rate of convergence is gained for approximate solutions and their first derivatives‎.

In this paper an COUPLED Burgers' equation is considered and then a method entitled interval finite-difference method is introduced to find the approximate interval solution of interval model in level wise cases. Finally for more illustration, the convergence theorem is confirmed and a numerical example is solved.

As TWO-DIMENSIONAL COUPLED system of nonlinear partial differential EQUATIONS does not give enough smooth solutions, when approximated by linear, quadratic and cubic polynomials and gives poor convergence or no convergence. In such cases, approximation by zero degree polynomials like Haar wavelets (continuous functions with  nite jumps) are most suitable and reliable. Therefore, modi ed numerical method based on Taylor series expansion and Haar wavelets is presented for solving COUPLED system of nonlinear partial di erential EQUATIONS. E ciency and accuracy of the proposed method is depicted by comparing with classical methods.

In this paper, a TWO-DIMENSIONAL multi-wavelet is constructed in terms of Chebyshev polynomials. The constructed multi-wavelet is an orthonormal basis for L2 [0,1]2 space. By discretizing TWO-DIMENSIONAL Fredholm integral equation reduce to a algebraic system. The obtained system is solved by the Galerkin method in the subspace of L2 [0, 1]2 by using TWODIMENSIONAL multi-wavelet bases. Because the bases of subspaces are orthonormal, so the above mentioned system has a small dimension and also high accuracy in approximating solution of integral EQUATIONS. For one-DIMENSIONAL case, a similar works are done in [4,5], which they have small dimension and high accuracy. In this article, we extend one-DIMENSIONAL case to TWODIMENSIONAL by extending and by choosing good functions on TWO axes. Numerical results show that the above mentioned method has a good accuracy.

An infinitely long hollow cylinder containing isotropic linear elastic materials is considered to be under the effect of arbitrary boundary stress and thermal condition. The TWO-DIMENSIONAL COUPLED thermoelastodynamic PDEs are specified based on motion and energy EQUATIONS, which are unCOUPLED using Deresiewicz-Zorski potential functions. The Laplace integral transform and Bessel-Fourier series are used to derive solutions for the potential functions, then the displacements-, stresses- and temperature-potential relationships are used to determine displacements, stresses and temperature fields. It is shown that the formulation presented here is collapsed on the solution existed in the literature for a simpler case of axis-symmetric configuration. To solve the equation used and evaluate the displacements, stresses and temperature at any point and time, a numerical procedure is needed. In this case, the numerical inversion method proposed by Durbin is applied to evaluate the inverse Laplace transforms of different functions involved in this paper. For numerical inversion, there exist many difficulties such as singular points in the integrand functions, infinite limit of the integral and the time step of integration. Finally, the desired functions are numerically evaluated and the results show that the boundary conditions are accurately satisfied. The numerical evaluations are presented graphically to make engineering sense for the problem involved in this paper for different cases of boundary conditions. The results also indicate that although the thermal induced wave propagates with an infinite velocity, the time lag of receiving stress waves with significant amplitude is not zero. The effect of thermal boundary conditions are shown to be somehow oscillatory, which is due to reflective boundary conditions and may be used in designing of such an element.

‎In this paper‎, ‎a numerical method for finding the numerical solution of the Burgers-Fisher and Burgers' nonlinear EQUATIONS is proposed‎. ‎These EQUATIONS are very important in many physical problems such as fluid dynamics‎, ‎turbulence‎, ‎sound waves and etc‎. ‎We describe a meshless method to solve the nonlinear BURGERS’ equation as a stiff equation‎. ‎In the proposed method‎, ‎we also use the exponential time differencing (ETD) method‎. ‎In this method‎, ‎the moving least squares (MLS) method is used for the spatial part and the exponential time differencing(ETD) is used for the time part‎. ‎To solve these EQUATIONS‎, ‎we use the meshless method MLS to approximate the spatial derivatives‎, ‎and then use method ETDRK4 to obtain approximate solutions‎. ‎In order to improve the possible instabilities of method ETDRK4‎, ‎Approaches have been stated‎. ‎Method MLS provided good results for these EQUATIONS due to its high flexibility and high accuracy and having a moving window‎, ‎and obtains the solution at the shock point without any false oscillations‎. ‎The method is described in detail‎, ‎and a number of computational examples are presented‎. ‎The accuracy of the proposed method is demonstrated by several test simulations‎.