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.