二阶记忆依赖型微分方程解的存在唯—性及其解法

Existence and Uniqueness of Solutions for Second Order Memory Dependent Differential Equations and Their Solutions

摘要: 记忆依赖型导数与分数阶导数相比，其核函数可以根据实际情况进行选择，具有更强的灵活性。为了进—步应用到实际定义了记忆依赖型微分方程，本文主要讨论二阶记忆依赖型微分方程解的存在性与唯—性。先对积分号进行处理，然后将变量的区间进行分割，利用构造皮卡迭代序列论证向量级数—致收敛，则当其时滞足够小，且核函数二阶可导时方程组的解存在且唯—。接下来证明了当核函数取特殊情况时，初值问题存在显示解。最后，根据图象观察当时滞取不同的值时， 初值问题解的变化情况。

Abstract: Compared with the fractional derivative, the kernel function of memory dependent derivative can be selected according to the actual situation, and it has more flexibility. The existence and uniqueness of solutions for second order memory dependent differ- ential equations are discussed in this paper. The integral number is processed first, then the interval of the variable is segmented, and the vector series is uniformly con- verged by constructing pickup iterative sequence, which proves that when the delay is small enough, the solution of the equations of the second order differentiable kernel function exists and is unique. Then it is proved that the initial value problem has a display solution when the kernel function takes a special case. Finally, the change of the solution of the initial value problem is observed according to the image.