记忆依赖型偏微分方程数值解研究

On Numerical Solution of the Memory Dependent Partial Differential Equations

摘要: 若将经典的弦振动方程与热传导方程中关于时间的变化率替换成记忆依赖型导数形式，其解的性态会发生怎样的变化呢?相对于普通导数而言，记忆依赖型导数可反映物理过程对过去状态的依赖性。与分数阶导数相比，其核函数的选取更自由，依赖区间也不会随时间的增长而增大，故而记忆依赖型偏微分方程也应具有更强表现力。本文针对核函数为线性函数的情况进行探讨。数值结果显示：1) 其解的性态介于弦振动方程和热传导方程之间，既有波动性又有衰减性，振幅随时滞和扩散系数的增大而减小。2) 与Caputo型分数阶偏微分方程相比，其波动性更强，振幅衰减更慢。

Abstract: If the rate of change respect to time in the classical string-vibration equation and heat-conduction equation is replaced by the type of memory-dependent derivative, what is the difference for the behavior of the solution? To compare with the ordinary derivative, the memory-dependent type can reflect clearly the dependence of physical process on their past states. To compare with the fractional derivative, the kernel function can be chosen freely and the interval for dependence doesn’t increase with time; so the memory-dependent partial differential equation should have strong expressive force. In this study, the case of kernel function with linear form is considered. Numerical results show that: 1) The characteristics of the solution lie between the string-vibration equation and the heat-conduction equation. It has both fluctuating and decaying properties. The amplitude of it decreases along with the increasing of time-delay and diffusion-coefficient. 2) To compare with the Caputo type of fractional partial differential equation, the fluctuation of the so-lution is stronger and the decay of amplitude is slower.