In this paper, modi ed simple equation method has been applied to ob-tain generalized solutions of Burgers, Huxley equations and combined formsof these equations. The new exact solutions of these equations have beenobtained. It has been shown that the proposed method provides a very e ec-tive, and powerful mathematical tool for solving nonlinear partial di erentialequations.

‎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.

In this paper, a new three-step method based on cubic spline will be construct to the numerical solution of a class of partial differential equations well-known as Burgers’-Huxley and Burgers’-Fisher. As we know, the maximum order achieved using cubic spline for interpolating is O(h4), but this order is reduced when it is used for the solution of differential equations. Here we will find an O(h6) super convergent approximation for the solution of Burgers’-Huxley and Burgers’-Fisher equations by defining some proper end conditions and constructing a three step deferred-correction algorithm. We will discuss the convergence and error bounds of the method using Green’s function definition in details. In addition, to verify the obtained error bounds, some numerical examples will be presented. Finally, we will try to show the applicability and efficiency of the method by comparing the results with other existing methods.

In this paper, a reduced order model is reconstructed for boundary control and excitation of the unsteady viscous Burgers equation. First, the standard reduced order proper orthogonal decomposition model, which has been extracted from the governing equations without control inputs, was evaluated and illustrated the satisfactory results in short time period. Two approaches are used to imply the effects of boundaries excitations and the related control routines. In the first, a source term was added to the governing equation of the reduced order dynamical system and was contributed as   an expansion of the proper orthogonal decomposition modes without control input. For removing the inhomogeneities on the boundaries, the boundaries values are subtracted from all of the snapshots with an appropriate control input. The other approach is based on the rewriting of the diffusion term as      an expanded form which contains the effect of boundaries values explicitly. In both approaches, the obtained reduced order models will contain two parts, the effect of system states and the influence of boundaries control functions. The results obtained from the reduced order model without the control inputs demonstrate a good agreement to the benchmark direct numerical simulations data and prove the high accuracy of the model.

By applying finite difference formula to time discretization and the cubic B-splines for spatial variable, a numerical method for solving the inverse system of Burgers equations is presented. Also, the convergence analysis and stability for this problem are investigated and the order of convergence is obtained. By using two test problems, the accuracy of presented method is verified. Additionally, obtained numerical results of the cubic B-spline method are compared to trigonometric cubic B-spline method, exponential cubic Bspline method and radial basis function method. Implementation simplicity and less computational cost are the main advantages of proposed scheme compared to previous proposals.

Analysis of the pavements and their ingredients has always been important due to a better understanding of their behavior under different conditions and leads to better understanding and providing more accurate relations. Due to the extent of asphalt mixture application in the world, assessment of different behaviors of these mixes is very important from the various aspects of performance and safety. Given that the asphalt mixes are inherently very sensitive to temperature changes due to bitumen content, identification and analysis of the viscoelastic and visco-elasto-plastic behavior of the mixes is of particular importance. This research aims at performance evaluation of Burgers model and anew proposed model based on genetic programming techniqe in estimating visco-elastic behavior of asphalt concrete. For this purpose, a number of dynamic creep tests under various temperatures and different stress levels were done. Results showed that performance of the new proposed model based on genetic programming techniqe is quite satisfactory. Also, the new proposed model will help more researchers, willing to perform similar studies, without carrying out destructive tests.

In this paper, a Burgers-Huxley equation is solved by using variety of methods: the Adomian’s decomposition method, modified Adomian’s decomposition method, variational iteration method, modified variation al iteration method, homotopy perturbation method, modified homotopy perturbation method and homotopy analysis method. The approximate solution of this equation is calculated in the form of series whose components are computed by applying a recursive relation. Consequently, the existence and uniqueness of the solution and the convergence of the proposed methods are proved. Furthermore, a numerical example is studied to demonstrate the accuracy of the presented methods.

This study presents the characteristics of Cattaneo-Christov heat ux model for the boundary layer ow of Burgers' uid model. Instead of simple Fourier's law of heat conduction, we presented the Cattaneo-Christov model to analyze the thermal relaxation properties when the heat source/sink was present in the system. Mathematical modeling of the laws of momentum and energy were presented under the order analysis approach. It was revealed that the term \ B2 0u= " was for the hydro-magnetic rheology of the Newtonian model whereas the generalized magnetic  eld term was for the Burgers' model, which was incorporated in the current analysis. Suitable transformations were utilized for the conversion of partial di erential system into coupled nonlinear set of ordinary di erential equations, which was tackled analytically through homotopy analysis technique. The plots of various physical quantities are presented, showing the dynamics of the considered analysis. Streamlines for Burgers' and Newtonian models are presented, which show their di erence in rheology. Numerical values for skin friction and surface heat transfer rate are presented in tables.