The Non-traveling Wave Solutions For A Nonlinear Reaction Diffusion Equation And Traveling Wave Solutions For K(m, N) Equations

Abstract: The additional generating condition method and the reduction oforder for solving di?erential equations are developed to study several types ofnonlinear partial di?erential equations. The analytical expressions of the exactsolutions for these equations are obtained under certain circumstances. Themain factors leading to the quantitative change in the physical structures ofsolutions are highlighted.Chapter 1 focuses on investigating the exact solution for a generalizedBurger - type equation. By employing the additional generating conditionmethod and two sorts of ansatzes, several non-traveling wave solutions are ob-tained.In chapters 2, a mathematical technique based on the reduction of orderfor solving dierential equations has been developed to find exact solutions forthe nonlinear equation K(m,n) in five cases:K(m,1), K(1,n), K((n+1)/2,n),K((n+1)/2 1,n) and K(n,n). The formulas for the traveling wave solutions ofthe equation for the five cases are derived. It is shown that, the solutions,depending on the values of the coe?cient a, the exponent n and the wavespeed, have dierent forms including solitary patterns, solitons, compactons, periodic solutions. Our results obtained in this section include those presentedin Wazwaz's paper [4].