Study On Properties Of The Solution Of Memory Dependent Differential Equations

Fractional derivative(FD)is an extension of the ordinary derivative,Provides methods for people to study more complex systems and phenomena,and the second half of the twentieth century,In the mechanics,FD has been widely used in fields such as image processing,but it cannot get rid of its dependence on fixed points,The memory-dependent interval length increases over time,Thus failing its memory effect,and its nuclear function is fixed,No selection cannot be made.So on this basis,Memory-dependent type derivative(MDD)is proposed,It is widely used in generalized thermal viscoelasticity.Compared to FD,MDD's nuclear function is according to practice,More flexibility.Otherwise,The dependence of the fractional derivative on fixed points is free.Its constructed memory-dependent differential equations are more expressive.MDD are introduced into the first and second order differential equations in this paper to form the first and second order memory dependent differential equations.Then we discuss the unique dominant solution of the equation when the conditions are satisfied.According to the extension theorem of the solution,the solution on the interval of the equation group can be extended,and the continuous dependence on the initial value can be discussed by using the method of step estimation and scaling.Then the method of solving the exact solution of the equation is considered when the kernel function is fixed.Finally,the relationship between the solution and the solution of the corresponding ordinary differential equations is observed by using the image.The results show that there is a very small gap between the numerical solutions of the memory dependent differential equations and the ordinary differential equations,but when the time increases,the gap gradually becomes obvious.And the larger the time lag,the smaller the gap between the solutions of the two equations.For the components,the two equations show a slow downward trend in the early stage,but the attenuation speed of the ordinary equations increases with the increase of time.For components,the two equations show a slow upward trend in the early stage,but with the increase of time,the common equations appear a downward trend.For the second order memory dependent differential equation,when discussing whether there is a unique solution,the transformed integral equation contains both the first order integral and the second order integral,so it is transformed by limiting its conditions.Then the method of solving the exact solution of the equation is considered when the kernel function is fixed.Finally,the influence of time delay on its solution is considered.The results show that when different values are taken,the larger the solution of the second order memory dependent differential equation is in the early and later stages,the larger the solution is,and when the middle period is the opposite,the change of the delay has no effect on the ordinary differential equation.A memory dependent differential equation composed of ordinary differential equations is studied in this paper.MDD is the extension of FD,which solves the problem of FD,and its calculation process is simple and has certain application potential.