Modeling And Numerical Study Of Memory-dependent Partial Differential Equations

Since the middle of the twentieth century,fractional derivatives have developed rapidly in mechanical,image processing,mechanics and so on.However,their kernel functions are fixed and can't be chosen freely according to the actual situation and the magnitude will increase with time,which will lead to the characterization failure.This is its inherent defect,thus developing memory-dependent derivatives and has been applied to viscoelastic and other fields.Compared with fractional derivative,its definition can describe the memory effect better.Its kernel function can be selected according to the actual situation and the differential equation composed of it is more expressive.In this paper,we use memory dependent derivative to model and expand numerical research.In this paper,a memory-dependent partial differential equation is established by replacing the rate of change of time with the second-order memory-dependent derivative in the classical string oscillation equation and the heat conduction equation.Then we study the behavior of the solution of the initial-boundary value problem and discuss the influence of kernel function,diffusion coefficient and time delay on the behavior of the solution.Finally,the similarities and differences between the new problem and the solutions of classical string vibration equation,heat conduction equation and fractional partial differential equation are discussed.The results show that the solution of the new problem is both decaying and fluctuating.The behavior of the solution is affected by kernel function,diffusion coefficient and time delay.The amplitude decreases with the increase of diffusion coefficient and time delay.The behavior of the solution under different kernel functions is also different.The solution of the new problem is between the classical string vibration equation and the heat conduction equation.The attenuation rate of the amplitude is slower than that of the heat conduction equation and the amplitude is much larger than that of the latter two.Compared with the numerical solution of fractional partial differential equations,the decay rate of the solutions is slower and the volatility is more obvious.Then,based on the phenomenon that the temperature rises slowly in the process of heat transfer,the physical process of heat transfer is re-modeled,a memorydependent heat transfer model is established,and the behavior of the solution is studied.The 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.The memory-dependent derivative studied in this paper is a new kind of derivative,which can get rid of the defect of the original fractional derivative and is more convenient in calculation and has great potential in application.The modeling method adopted in this paper can also be applied to other physical processes and can be used as a way to introduce memory-dependent derivatives into PDEs.