Reconstruct the Heat Conduction Model with Memory Dependent Derivative

The classical heat conduction equation is derived from the assumption that the temperature increases immediately after heat transfer, but the increase of temperature is a slow process, so the memory-dependent heat conduction model has been reconstructed. Numerical results show that the solution of the initial boundary value problem of the new model is similar to that of the classical heat conduction equation, but its propagation speed is slower than that of the latter. In addition, the propagation speed of the former is also affected by time delay and kernel function.


Introduction
Fractional calculus is an important branch of mathematics. The original fractional differential operator has three forms: Grumwald-Letnikov definition, Riemann-Liouville definition and Caputo definition. Caputo type fractional derivative is more convenient to solve practical problems. In 2011, Wang & Li [1] referring to Caputo type fractional derivative proposed a new derivative called memory-dependent derivative. Compared with fractional derivative, its kernel function can be selected according to actual conditions and the memory-dependent interval does not increase with time t. It always concentrates the dependency interval on a limited time period related to the past state [ ] , t t τ − , here τ is a time delay. The concept of derivative has attracted many scholars' attention. Ezzat et al. used memory-dependent derivatives in generalized ther-ry-dependent differential equations and studied them deeply in [5]. In [6], we studied the classical string vibration equation and heat conduction equation. The time derivative is replaced by the memory-dependent derivative. The numerical results simulated by MATLAB are between the wave equation and the heat conduction equation. With time, both the wave propagation and the energy attenuation occur. This change conforms to the characteristic of the "weighted period" function [7]. The diffusion velocity is lower than the heat conduction equation, and the wave amplitude is much higher than the wave equation under the same initial condition and boundary condition. Compared with the numerical solutions of fractional partial differential equations, the attenuation speed of the numerical solutions is slower and the wave phenomenon is more obvious.
The above research is only a simple substitution for the time derivative, and does not construct a model from the actual physical background. The classical heat conduction equation has obvious physical background. Its setting is derived without considering the motion of the medium, when the temperature rises immediately in a certain region of heat transfer, the problem of infinite velocity will arise.
From the definition of memory-dependent derivative, we can see that it is reflected in the average of the overall rate of change over a period of time, compared with the ordinary derivative, it can reflect the dependence of physical process on past state. It is more in line with the physical fact that the temperature rises slowly when heat propagates in the medium of less dense gas in theory.
Is the heat conduction model made of it more realistic? In this paper, we use modeling method to explore.

Memory-Dependent Derivatives (MDD)
Based on Caputo fractional differential operator this kind of derivatives, so memory-dependent derivatives are more reasonable.
We find that the memory-dependent derivative in Equation (2.2) is approximately equal to half of ( ) ( ) m f t . In order to keep the values of memory-dependent derivative and ordinary derivative basically unchanged, an improved definition is given.
Definition 2 [6]: Let m be a natural number ( Ν m ∈ ), then for an m-times it is called m-order "memory-dependent derivative" of f at τ relative to the is a m-times differentiable about t and s.

Model Construction
Considering that the temperature rises slowly with time after heat is transferred into an object, that is, the rise of temperature is delayed relative to the heat transfer process. From the physical process, a one-dimensional heat conduction model with memory-dependent derivatives on a cylinder is constructed. The specific derivation process is as follows.
Consider a cylinder (assuming that the cross section area is 1, the object is homogeneous and isotropic, without considering thermal expansion). A function ( ) , u x t is used to represent the temperature at the position x and t at the cylinder. As shown in Figure 1, an infinitesimal segment [ ] The influx of heat changes the temperature inside the body, but the increase in temperature is a slow process. Suppose it will delay τ periods. When temperature is changed from ( ) here c is the specific heat, ρ is density,

Numerical Simulation
In the previous section, a memory-dependent heat conduction model is deduced.
In this section, we mainly study the solution of the initial-boundary value problem under the first boundary condition, discuss the effect of time delay and kernel function on the solution and compare it with the classical heat conduction equation.

The Initial Boundary Value Problem of the New Model
The initial boundary value problem of the memory-dependent heat conduction model is discussed The temperature function of the first 1 n − time period is used in numerical calculation, which is discrete form of classical heat conduction equation,

Comparison with Classical Heat Conduction Equation
The heat conduction equation we know well.
the numerical solution has obvious characteristics. Comparison questions (3.1) and problems (3.2).
The above problems are discretized by the finite difference method, and the following Figure 3 is achieved by MATLAB software.
As can be seen from Figure 3, the classical heat conduction equation is a Applied Mathematics   a)), but the diffusion rate is slightly slower than that of the latter (see (c)). It is found that the new model is more effective in describing the real heat transfer phenomena.

Conclusion
The classical heat conduction equation is proposed on the basis of the instantaneous rise of temperature after heat transfer. In fact, the rise of temperature is not instantaneous, but a slow process. Numerical results show that: 1) The solution of the new model is diffusion type. When the kernel function is unchanged, the diffusion rate slows down with the increase of time delay. 2) Compared with the classical heat conduction equation, the two properties are similar, but the diffusion rate of the new model is even slower.
3) It is found that the kernel function is also one of the factors affecting the diffusion rate, but how to influence the diffusion rate needs further study.