高阶线性Gronwall不等式的推广及应用

Generalization and Application of High-order Linear Gronwall Inequality

针对一阶线性Gronwall不等式在求解高阶微分问题时容易出现的降阶错误,综合运用格林公式进行分部积分、构造辅助函数与指数函数相结合、Young不等式以及弗里德里克斯不等式等方法来推广一阶线性Gronwall不等式,将一阶Gronwall不等式推广到二阶和三阶以及含有2个函数的高阶微分形式的Gronwall不等式中,并且得到与一阶线性函数类似的结果。通过测试结果可知,此方法的计算量与准确率优于传统方法,适用于初边值问题的能量估计以及波动方程的能量估计。

In view of the order reduction error of first-order linear Gronwall inequality in solving higher-order differential problems,the authors comprehensively uses Green’s formula for partial integration,the combination of auxiliary function and exponential function,Young’s inequality and Friedrich’s inequality to generalize the first-order linear Gronwall inequality to the second-order and third-order Gronwall inequality and the higher-order differential form with two functions.The result is similar to that of the first-order linear function.The test results show that this method has better computational complexity and accuracy than the traditional method.The results are suitable for energy estimation of initial boundary value problems and wave equations.