numerical solutions the nonlinear burgers-fisher and burgers’ equations with adaptive numerical method

‎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.