# 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

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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.

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