Approximate damped oscillatory solutions for compound KdV-Burgers equation and their error estimates

In this paper,we focus on studying approximate solutions of damped oscillatory solutions of the compound KdV-Burgers equation and their error estimates.We employ the theory of planar dynamical systems to study traveling wave solutions of the compound KdV-Burgers equation.We obtain some global phase portraits under different parameter conditions as well as the existence of bounded traveling wave solutions.Furthermore,we investigate the relations between the behavior of bounded traveling wave solutions and the dissipation coefficient r of the equation.We obtain two critical values of r,and find that a bounded traveling wave appears as a kink profile solitary wave if |r| is greater than or equal to some critical value,while it appears as a damped oscillatory wave if |r| is less than some critical value.By means of analysis and the undetermined coefficients method,we find that the compound KdV-Burgers equation only has three kinds of bell profile solitary wave solutions without dissipation.Based on the above discussions and according to the evolution relations of orbits in the global phase portraits,we obtain all approximate damped oscillatory solutions by using the undetermined coefficients method.Finally,using the homogenization principle,we establish the integral equations reflecting the relations between exact solutions and approximate solutions of damped oscillatory solutions.Moreover,we also give the error estimates for these approximate solutions.     

朴实自然也出巧

<正>题目1已知a、b、c>0,求证:a3b+b3c+c3 a≥abc(a+b+c).以上题目1是文[1]的例1,且文[1]用了大量篇幅,运用多元均值不等式及巧妙神奇的待定系数法给出该题三种解法,可谓"优雅华贵堪称妙".其实本题仅用初中数学知识,不用多元均值不等式,利用作差比较法即可获证,方法更自然简单,有道是"朴实自然也出巧",现展示如下,与初中、高中同学们共享.证明由于待证不等式是轮换对称不等式,故不失一般性,可设a最大.     

梯形电阻网络等效电阻通项公式求法

探讨如何从物理和数学的角度,运用3种不同的方法求解一个二端无限梯形电阻网络等效电阻的通项公式,开阔视野活化思维,提高学生分析处理问题能力。